ï»¿ WPS6599
Policy Research Working Paper 6599
Economic Growth and Equality
of Opportunity
Vito Peragine
Flaviana Palmisano
Paolo Brunori
The World Bank
Development Economics Vice Presidency
Partnerships, Capacity Building Unit
September 2013
Policy Research Working Paper 6599
Abstract
The paper proposes an approach to understand the infer the role of growth in the evolution of inequality of
relationship between inequality and economic growth opportunity over time. The paper shows the relevance of
obtained by shifting the analysis from the space of final the introduced framework by providing two empirical
achievements to the space of opportunities. To this analyses, one for Italy and the other for Brazil. These
end, it introduces a formal framework based on the analyses show the distributional impact of the recent
concept of the Opportunity Growth Incidence Curve. growth experienced by Brazil and the recent crisis
This framework can be used to evaluate the income suffered by Italy from both the income inequality and
dynamics of specific groups of the population and to opportunity inequality perspectives.
This paper is a product of the Partnerships, Capacity Building Unit, Development Economics Vice Presidency. 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 v.peragine@dse.uniba.it, a-viana.palmisano@gmail.com, and paolo.brunori@uniba.it.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
Economic Growth and Equality of Opportunityâˆ—
Vito Peragineâ€ Flaviana Palmisanoâ€¡and Paolo BrunoriÂ§
September 9, 2013
JEL Classiï¬?cation codes: D63, E24, O15, O40.
Keywords: social development (SDV); income inequality; inequality of opportunity; economic
growth.
âˆ— The authors are grateful to Francisco Ferreira, Dirk Van de Gaer, and to 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 dâ€™Etudes Mondiale, Paris. The authors also thank Francisco Ferreira and Maria
Ana Lugo for kindly providing them with access to data.
â€ Vito Peragine (corresponding author) is an associate professor at University of Bari Aldo Moro, Italy; his email
address is v.peragine@dse.uniba.it.
â€¡ Flaviana Palmisano is a post-doc fellow at University of Bari Aldo Moro, Italy; her email address is ï¬‚a-
viana.palmisano@gmail.com.
Â§ Paolo Brunori is an assistant professor at University of Bari Aldo Moro, Italy; his email address is
paolo.brunori@uniba.it.
1
INTRODUCTION
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 eï¬€ect of growth on distributional phenomena such as income poverty
and income inequality. Indices for measuring the pro-poorness of growth have been proposed,1
and the Growth Incidence Curve (GIC), measuring the quantile-speciï¬?c 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 also been investigated (Bourguignon 2004). All of these tools are now used
extensively in the ï¬?eld of development economics to evaluate and compare diï¬€erent 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 â€œspaceâ€? of distributional assessments.
In contrast, recent literature in the ï¬?eld of normative economics has argued that equity judg-
ments 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 eï¬€ort, 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 countries can be found in World Bank (2006) and de Barros et al.
1 See Essama-Nssah and Lambert (2009) for a comprehensive survey.
2
(2009).
The growing interest in EOp, in addition to the intrinsic normative justiï¬?cations, 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 relationship between
inequality and growth can be obtained by shifting the analysis from the space of ï¬?nal achievements
to the space of opportunities. If two growth processes have, say, the same impact in terms of
poverty and inequality reduction, but in the ï¬?rst 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 diï¬€erences 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 proposed to explain changes in outcome inequality as the result of diï¬€erences in growth for
individuals with diï¬€erent initial outcomes, to the best of our knowledge, the relationship between
the change in IOp and growth has never been investigated.
Our aim is to address this measurement problem2 by proposing a framework and a set of
simple tools that can be used to investigate the distributional eï¬€ects of growth from an opportunity
egalitarian viewpoint. In particular, with reference to a given growth episode, we address the
following questions: is growth reducing 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
2 Hence, we investigate the relationship between growth and inequality of opportunity using a â€œmicro approachâ€?;
an alternative â€œmacro approachâ€? would also be possible by investigating the relationship between growth and IOp
from a cross-country or longitudinal perspective (see Marrero and Rodriguez 2010).
3
the space of opportunities. Hence, we introduce the concept of the Opportunity Growth Incidence
Curve (OGIC), which is intended to capture the eï¬€ect 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 deï¬?ned 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 eï¬€ect of growth in terms of increasing or reducing aggregate IOp; the type OGIC, in
contrast, allows us to track the evolution of speciï¬?c 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 speciï¬?c 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 opportunity. In fact, the two-period framework could easily be adapted for the comparison of
pre- and post-public intervention distributionsâ€“for instance, if one is interested in evaluating the
distributive impact of a certain ï¬?scal reform in the space of opportunities.
In this paper, we adopt this theoretical framework to analyze the distributional impact of growth
in two diï¬€erent countries, Italy and Brazil, in recent years. These two countries experienced very
diï¬€erent 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 2002â€“04 and 2004â€“06 periods, the
average household income increased by 2% and 2.6%, respectively, whereas the equivalent disposable
income of Italian households was characterized by a long spell of negative growth during the recent
economic crisis: it decreased by 2.6% in the 2006â€“10 period and by 0.6% between 2008 and 2010
(Banca dâ€™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% GDP yearly growth in
the last decade), and this growth, as shown in the literature, was markedly progressive. In fact, the
4
Gini index for the entire distribution decreased during the period considered from 60.01 in 2001 to
54.7 in 2009 (see contributions by Ferreira et al. 2008, World Bank 2012).
Therefore, it is interesting to examine how the perspective of opportunity inequality can add
elements of knowledge to the analysis of two markedly diï¬€erent distributional dynamics.
We use the Bank of Italyâ€™s â€œSurvey on Household Income and Wealthâ€? (SHIW) to assess the
distributional impact of growth in Italy. In particular, we consider four of the most recent available
waves to compare the 2002â€“06 growth episode with the 2006â€“10 episode. We use the â€œPesquisa
Ä±liosâ€? (PNAD), provided by the Istituto Brazilero de Geograpia e
Nacional por Amostra de DomicÂ´
Estatistica, to analyze growth in Brazil, and we focus on the 2002â€“05 growth episode against the
2005â€“08 episode.
As far as Italy is concerned, when we focus on each single growth episode, some relevant insights
arise. For instance, when the 2002â€“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 2006â€“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 ï¬?rst period dominates the second according 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 diï¬€erences 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 experienced by the
individual outcome in 2002â€“05 appears considerable for the whole distribution (with the exception
of the top 15%), the growth experienced in terms of opportunities is less prominent. Indeed, most of
the types suï¬€er a reduction in the value of the opportunity during the growth process.3 In contrast,
the 2005â€“08 growth episode appears to be beneï¬?cial for the whole population regardless of the focus
of the analysis (whether outcome or opportunity). Our analysis shows that the 2005â€“08 growth
process is not only generally progressive but that it also leads to a reduction in the IOp (progressive
3 To obtain this conï¬‚ict between type OGIC and GIC, it is necessary that rich individuals experiencing losses are
spread across the majority of socioeconomic groups.
5
individual OGIC). Furthermore, the initially disadvantaged groups of the population seem to beneï¬?t
more from growth than those that were initially advantaged (decreasing type OGIC). When the two
processes are compared, the dominance of the 2002â€“05 growth episode over the 2005â€“08 episode is
evident for every perspective 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 diï¬€er 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 eï¬€ect of growth and in the EOp literature. 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.
I. 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 growth4 in the space of ï¬?nal achievements. 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 Âµ (yt ),
and let yt (p) be the quantile function of F (yt ), representing the income corresponding to quantile
4 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
p in F (yt ). To evaluate the growth taking place from t to t + 1, Ravallion and Chen (2003) deï¬?ne
the Growth Incidence Curve (GIC) as follows5 :
yt+1 (p) L (p)
g (p) = âˆ’ 1 = t+1 (Î³ + 1) âˆ’ 1, for all p âˆˆ [0, 1] , (1)
yt (p) Lt (p)
where L (p) is the ï¬?rst derivative of the Lorenz curve at percentile p and Î³ = Âµ (yt+1 ) /Âµ (yt ) âˆ’ 1 is
the overall mean income growth rate. The GIC plots the percentile-speciï¬?c rate of income growth
in a given period of time. Clearly, g (p) â‰¥ 0 (g (p) < 0) indicates positive (negative) growth at p.
A downward-sloping GIC indicates that growth contributes to equalize the distribution of income
(i.e., g (p) decreases as p increases), whereas an upward-sloping GIC indicates 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 aï¬€ects the diï¬€erent parts of
the distribution. In addition, they are used as criteria to rank diï¬€erent growth episodes. Ravallion
and Chen (2003) apply ï¬?rst-order dominance criteria based on the GIC: ï¬?rst-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 conditions,
requiring that the mean growth rate up to the p poorest percentile in a growth episode - or the
â€œcumulative GICâ€? - be everywhere larger than in another. In this case, the cumulative GIC is given
p p
by G (p) = 0
g (q ) yt (q ) dq/ 0
yt (q ) dq for all p âˆˆ [0, 1].
Building on the concept of the GIC, the literature has provided a variety of aggregate mea-
sures of growth. We recall, among these, the rate of pro-poor growth proposed6 by Essama-Nssah
1
(2005): RP P GEN = 0
v (p) g (p) dp, where v (p) > 0, and v (p) â‰¤ 0 is a normalized social weight,
decreasing with the rank in the income distribution. Hence, RP P GEN represents a rank-dependent
aggregation of each point of the GIC and measures the overall extent of growth, giving more im-
5 For a longitudinal perspective on the evaluation of growth, see Bourguignon (2011) and Jenkins and Van Kerm
(2011).
6 In the original paper, RPPG
EN is applied to discrete distributions. Here, we use a continuous version of the
same index to be consistent with our notation.
7
portance to the growth experienced by the income of the poorest individuals.7 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 individual income
at a given time, t âˆˆ {1, ..., T } , yt , is assumed to be a function of two sets of characteristics: the
circumstances, c, belonging to a ï¬?nite set â„¦ and the level of eï¬€ort, et âˆˆ Î˜ âŠ† R+ . The individual
cannot be held responsible for c, which is ï¬?xed over time; he is, instead, responsible for the eï¬€ort et
that he autonomously decides to exert in every period of time. Income is generated by a production
function g : â„¦ Ã— Î˜ â†’ R+ :
yt = g (c, et ). (2)
This is a reduced form model in which circumstances and eï¬€ort are assumed to be orthogonal, and
the function g is assumed to be monotonic in both arguments. Although the monotonicity of g
is a fairly reasonable assumption, the orthogonality assumption rests on the theoretical argument
that it would be hardly sustainable to hold people accountable for factor et if it were dependent on
exogenous 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 = 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 Âµi (yt ).
Given this analytical framework, the focus is on the income prospects of individuals of the same
type, represented by the type-speciï¬?c income distribution Fi (yt ). This distribution is interpreted
as the set of opportunities open to each individual in type i. In other words, the observable actual
incomes of all individuals in a given type is used to proxy the unobservable ex ante opportunities
7 Ravallion Ht
and Chen (2003) also propose the RP P GRC = 0 g (p) dp/Ht where Ht is the initial poverty
headcount ratio. RP P GRC measures the proportionate income change of the poorest individuals.
8
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 multivariate 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-deï¬?ned 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 speciï¬?c version of the EOp model, which is called â€œutilitarianâ€?, further assumes that the
value of the opportunity set Fi (yt ) can be summarized by the mean Âµi (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 artiï¬?cial 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 formally, by ordering the
types on the basis of their mean such that Âµ1 (yt ) â‰¤ ... â‰¤ Âµj (yt ) â‰¤ ... â‰¤ Âµn (yt ) , the smoothed
distribution corresponding to F (yt ) is deï¬?ned as Yts = Âµt t t
1 , ..., Âµj , ..., ÂµN . N is the total size of
the population, and Âµt
j is the smoothed income, interpreted as the value of the opportunity set,
j
of the individual ranked N in Yts . Hence, in this model, measuring opportunity inequality simply
amounts to measuring inequality in the smoothed distribution Yts .
Some authors have questioned this â€œutilitarianâ€? approach (see Fleurbaey 2008 for a discussion
of the issue). For instance, some authors argue that in addition to circumstances and eï¬€ort, an
additional factor, luck, plays a role in determining 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 diï¬€erences in eï¬€ort. In particular, the unequal outcomes resulting
from â€œbruteâ€? luck should be compensated for.8 Furthermore, these authors argue, individuals may
8 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
diï¬€erent meanings of luck.
9
be risk averse; hence, the within-type inequality 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.9
The model adopted in this paper, based on the assumption of within-type inequality 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 heterogeneity is very demanding
in terms of data. It is often the case that the small size of the samples used makes it diï¬ƒcult to
obtain easily comparable within-type distributions. This approach makes our empirical analysis
fully consistent with most of the analyses performed in the existing literature.10 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 ï¬?nal methodological consideration is in order here and concerns the issue of omitted circum-
stance variables. We use a pure deterministic model where, given a set of selected circumstances,
any residual variation in individual income is attributed to personal eï¬€ort. This amounts to saying
that once the vector of circumstances has been deï¬?ned, on the basis of normative grounds and
observability constraints, all other factors are implicitly classiï¬?ed 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 circumstances that determine a personâ€™s outcome. Whenever
the dimension of the observed vector c is less than the dimension of the â€œtrueâ€? vector, then we
9 For example, LeFranc et al. (2008) and Peragine and Serlenga (2008) use stochastic dominance conditions to
compare the diï¬€erent 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 individualâ€™s type, that is Âµi (1 âˆ’ Gi ),
where the type mean income Âµi and (1 âˆ’ Gi ) represent, respectively, the return component and the risk component,
with Gi denoting the Gini inequality index within type i. See also Oâ€™Neill et al. (2000) and Nilsson (2005) for
empirical analyses that attempt to provide alternative evaluations of opportunity sets using parametric estimates.
10 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 diï¬€erent countries, making an international comparison of inequality of opportunity
estimates across the world possible.
10
obtain lower-bound estimators of true inequality of opportunity; that is, the inequality that would
be captured by observing the full vector of circumstances. The implication is that the empirical
estimates obtained using this model should be interpreted as lower-bound estimates of IOp.11 Sim-
ilarly, 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 diï¬€erent set of variables
could lead to diï¬€erent 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 individual OGIC and the type OGIC, capture two diï¬€erent intuitions about
the relationship between growth and EOp. The ï¬?rst focuses on the impact of growth on the
distribution of opportunities. The second focuses on the relationship between overall economic
growth and type-speciï¬?c growth.
Given an initial distribution of income Yt and the corresponding smoothed distribution Yts
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 deï¬?ned as follows:
o j Âµt
j
+1
gY s = âˆ’ 1, âˆ€j âˆˆ {1, ..., N } . (3)
N Âµtj
o j
gY s
N measures the proportionate change in the value of opportunities of the individuals ranked
j o j o j
N in the smoothed distributions. Obviously, gY s
N â‰¥ 0 (gY s
N < 0) means that there has
been positive (negative) growth in the value of the opportunity set given to the individuals ranked
11 For a discussion of this issue with reference to a non deterministic, parametric model of EOp, see Ferreira and
Gignoux (2011) and Luongo (2011).
11
j
N respectively in Yts and in12 Yts
+1 .
The individual OGIC provides information on the impact of growth on IOp. Consider the Lorenz
curve of Yts :
j
Âµt
k
j k=1
LYts = N
, âˆ€k âˆˆ {1, ..., N } , âˆ€t âˆˆ {1, ..., T } . (4)
N
Âµt
k
k=1
The individual OGIC deï¬?ned in eq. (3) can be decomposed in such a way that it becomes a function
of the Lorenz curve deï¬?ned in eq. (4), as follows:
j
o j âˆ†LYts
+1 N
gY s = j
(Î³ + 1) âˆ’ 1, âˆ€j âˆˆ {1, ..., N } , (5)
N âˆ†LYts N
j Âµt
j j j Âµ(yt+1 )
where âˆ†LYts N = Âµ(yt ) is the ï¬?rst derivative of LYts N with respect to N, and Î³ = Âµ(yt ) âˆ’1
is the overall mean income growth rate.
j
(N
âˆ†LY s
t+1
)
Thus, when growth is proportional, it does not have any impact on the level of IOp: âˆ†LY s ( N
=
t
i
)
o j o j
1, and gY s
N will just be an horizontal line, with gY s
N = Î³ for all j . On the contrary, when
growth is progressive (regressive) in terms of opportunity, growth acts by reducing (worsening) IOp:
j
âˆ†LY s ( N )
t+1 o j
âˆ†LY s ( N
= 1, and gY s
N will be a decreasing (increasing) curve.
t
i
)
The main aspect that distinguishes the individual OGIC from the standard GIC is represented
by the distributions used to construct that curve. This variation 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 according 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
12 Note j
that, given the assumption of anonymity implicit in this framework, the individuals ranked N
in t can be
j
diï¬€erent from those ranked N in t + 1.
12
type-speciï¬?c mean income but also on the type-speciï¬?c population share and the reranking of types
taking place during the growth process. Now, although these features are desirable 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 suï¬€ers a deterioration of its condition 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 speciï¬?c issue and to investigate the relationship between overall economic growth
and type-speciï¬?c growth, we introduce a second version of the OGIC, which we label the type OGIC.
Letting YÂµt = (Âµ1 (yt ) , ..., Âµn (yt )) be the distribution of type mean income at time t, where
ËœÂµt+1 =
types are ordered increasingly according to their mean, i.e., Âµ1 (yt ) â‰¤ ... â‰¤ Âµn (yt ), and Y
Âµ1 (yt+1 ) , ..., Âµ
(Ëœ Ëœn (yt+1 ))is the distribution of type mean income at time t + 1, where types are
ordered according to their position at time13 t, we deï¬?ne the type OGIC as follows:
i Ëœi (yt+1 ) âˆ’ Âµi (yt )
Âµ
Ëœo
g = , âˆ€i âˆˆ {1, ..., n} . (6)
n Âµi (yt )
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
i
Ëœo
these groups are deï¬?ned on the basis of initial circumstances. g n is horizontal if each type
beneï¬?ts (loses) in the same measure from growth. It is negatively (positively) sloped if the initially
disadvantaged types get higher (lower) beneï¬?t from growth than those initially advantaged.14
The type OGIC diï¬€ers from the standard GIC in two aspects. The ï¬?rst is represented 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 opportunity sets. The second is represented by the weakening of the
anonymity assumption for types. Thus, the type OGIC, tracking the same type over time, provides
13 Note that we track the same type but do not track the same individuals.
14 Note that the type OGIC is a generalization of the idea underlying the ï¬?rst component of Roemerâ€™s (2011) index
of development, that is, â€œhow well the most disadvantaged type is doingâ€?.
13
information on the temporal evolution of the opportunity set.
The OGIC, in both the individual and the type versions, can be used to rank diï¬€erent growth
episodes. Analogously with the literature on the standard GIC, we can apply ï¬?rst-order dominance
criteria based on the OGIC.15 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 diï¬€erent ranking of growth processes. In fact, beyond their interpretation and the
fact that they can be used to investigate diï¬€erent aspects of the relationship between economic
growth and EOp, the diï¬€erences between the individual and the type OGIC are mainly due to
demographic and reranking issues. The following remark makes this point clear.
Remark 1. Let YtA and YtB be two initial distributions of income, and let GA and GB be two
diï¬€erent growth processes taking place, respectively, on YtA and YtB and generating, respectively,
two ï¬?nal distributions of income, YtA B
+1 and Yt+1 . Moreover, let nA and nB be the number of
types, respectively, in YtA and YtB and mAi and mBi be the number of individuals in each type
i = 1, ..., n, respectively, in YtA and YtB . If (i) nAt = nBt , âˆ€t = 1, ..., T , (ii) mAit = mBit
i i
ËœAo
âˆ€i âˆˆ {1, ..., n} , âˆ€t = 1, ..., T , (iii) no reranking of types, then g n ËœBo
g n âˆ€i âˆˆ {1, ..., n} if
Ao j Bo j
and only if gY s
N gY s
N âˆ€j âˆˆ {1, ..., N }.
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-speciï¬?c population size, and when (iii) types keep
their relative position in the type mean income distribution 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 typesâ€™ demography and
condition (iii) imposes restrictions on the rank of the types, it is clear that possible diï¬€erences
in the ordering provided by the two OGICs are determined by variations in the typeâ€™s population
shares, between the two distributions and the two periods compared, and by the reranking of types
over time.
15 For a normative justiï¬?cation of these dominance conditions based on a rank-dependent social welfare function,
see the working paper version of the paper: Peragine et al. (2011).
14
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 analyses16 and, in general, in the case of
an evaluation of policy interventions.
The Cumulative OGIC
So far, we have focused on ï¬?rst-order OGIC dominance, which is a strong condition that is
rarely veriï¬?ed with real data. A weaker condition is obtained by second-order dominance. This
order of dominance builds on the deï¬?nition of the cumulative17 OGIC.
To obtain the cumulative OGIC, one should look at the proportionate diï¬€erence between the
generalized Lorenz curves applied to the smoothed distribution at time t and t + 1, which, after
rearranging, gives the following expression for the individual version:
j
o k
gY s Âµt
k j
j k=1
N LYts N
Go
Ys = j
= +1
j
(Î³ + 1) âˆ’ 1, âˆ€j âˆˆ {1, ..., N } . (7)
N LYts
Âµt
k
N
k=1
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
j j
among smoothed incomes. Clearly, at N = 1, Go
Ys N equals the overall mean income growth
rate, Î³ .
The above decomposition allows to express the cumulative OGIC as depending on two com-
ponents: the overall mean income change and the variation in the 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.
16 See, inter alia, Sutherland et al. (1999).
17 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.
15
On the other hand, the cumulative type OGIC is deï¬?ned as follows18 :
i
j
Ëœo
g n Âµj (yt )
Ëœo i j =1
GYÂµ = i
, âˆ€i âˆˆ {1, ..., n} (8)
n
Âµj (yt )
j =1
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
Ëœo
sloping, depending on the pattern of growth among types. At i = n, G i
equals the overall
YÂµ n
mean growth rate of YÂµ .
OGIC Indexes
To avoid inconclusive results because of the partiality of the dominance conditions 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:19
N
j o j
v N gY S N
1 j =1
GY S = N
. (9)
N j
v N
j =1
j
Given the assumption of anonymity of the individual OGIC, the weight v N depends on the
relative position of individuals in the smoothed distribution, respectively, in t and t + 1. Thus,
the same weight is given to the value of the opportunity set of individuals ranked the same in the
j
smoothed distribution of the two periods20 . v N represents the social evaluation of the growth in
the opportunity enjoyed by individuals in the same position in t and t + 1.
18 Similar to the cumulative inividual OGIC, the cumulative type OGIC is obtained by rearranging the diï¬€erence
between the Generalized Lorenz curves applied to the type mean distributions Y Âµt and YËœÂµ
t+1 .
19 The approach is close in spirit to Essama-Nssah (2005), reviewed in a previous section. For a normative
justiï¬?cation of the rank-dependent approach to IOp analyses, see Peragine (2002), Aaberge et al. (2011), and
Palmisano (2011)
20 See endnote12.
16
Thus, eq. (9) represents a rank-dependent aggregation of the information provided by each single
j
point of the individual OGIC. In particular, imposing monotonicity, v N â‰¥ 0, âˆ€j âˆˆ {1, ..., N },
j j +1
and opportunity inequality aversion, v N â‰¥ v N , âˆ€j âˆˆ {1, ..., N âˆ’ 1}, 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
j j
lowest opportunities. Using the speciï¬?cation v N =2 1âˆ’ 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 = GY S âˆ’ GY S , (10)
N
1 o j
where GY S = N gY S N . OGY S = 0 if growth is proportional; it is positive (negative) if
j =1
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:
n
i i
w n Ëœo
g n
1
GYÂµ = i=1 n . (11)
n i
w n
i=1
i
The function w n is the social weight associated to type i and depends on the rank of the
i
type in the initial distribution of income. As before, this index satisï¬?es monotonicity : w n â‰¥ 0,
i âˆˆ {1, ..., n} (that is, aggregate growth is not decreasing in each type growth) and opportunity
i i+1
inequality aversion : w n â‰¥w n , i âˆˆ {1, ..., n âˆ’ 1} (that is, more weight is given to the income
growth experienced by the most disadvantaged types).
i
i
Following Aaberge et al. (2011) and choosing w n = 1âˆ’ qjt , a Gini-type index of
j =1
opportunity-sensitive growth results.
17
II. THE EMPIRICAL ANALYSES
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 implications of the most recent economic development experienced
by Brazil in terms of EOp.
For both applications, we ï¬?rst provide an assessment of growth according to the equality of
outcome perspective. We then move to the analysis of growth according to the EOp perspective.21
Opportunity and Growth in Italy: The Data
Italy is the ï¬?rst country considered in this section. This analysis is developed using the Bank of
Italyâ€™s â€œSurvey on Household Income and Wealthâ€? (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, deï¬?ned as all persons sharing the same dwelling. The
individual outcome is, then, measured as the household equivalent income in 2010 euro.22 Income
includes all household earnings, transfers, pensions, and capital incomes, net taxes, and social
security contributions. The richest and poorest 1% of the households in each wave are dropped
to avoid the eï¬€ect 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
21 We calculate conï¬?dence intervals for the diï¬€erence 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 t+1 , are
independent and identically distributed observations of the unknown probability distributions F (y t ), F (y t+1 ). Î³ is
the statistic of interest, and its standard error is Ïƒ (F (y t ), F (y t+1 )) = V arÎ³ Ë† (y t , y t+1 ). Our bootstrap estimate of
the standard error is ÏƒË†=Ïƒ F Ë† (y
Ë† (y ), F
t t +1 Ë† (y ), F
, where F t Ë† (y t +1 ) are the empirical distributions observed. The 95%
conï¬?dence interval is obtained by resampling B = 1, 000 ordinary non parametric bootstrap replications of the two
âˆ— , y âˆ— . The standard error of parameter Î³ B
distributions yt t+1 Ë† is obtained using Ïƒ
Ë†B = Ë† âˆ— (b)
b=1 {Î³ Ë† (.)}2 /(B âˆ’ 1),
âˆ’Î³
B âˆ—
b=1 Î³ (b)
where Î³ Ë† (.) = B
. Ë†B â†’ Ïƒ
We know that Ïƒ Ë† when B â†’ âˆž, and, under the assumption that Î³ is approximately
normally distributed, we calculate conï¬?dence intervals: Î³
Ë† = Î³Ë† Â± z1âˆ’Î±/2 ÏƒË†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.
22 We use the OECD equivalence scale given by the square root of the household size.
18
attainment of her parents (three levels: up to elementary 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 identiï¬?cation of the type is not possible because of
missing information about one or more circumstances are excluded. 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 proï¬?les23 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 particular, although some reranking takes place for types of other
regions, ï¬?ve of the six types from the South of Italy are the lowest-ranked at all times.
To analyze growth, we consider two four-year periods: 2002â€“06 and 2006â€“10. The exercise
is appealing because it compares two periods during which Italy faced two diï¬€erent economic
slowdowns. The former was characterized by the almost total absence of growth in 2002 and 2003.
The latter, triggered by the 2008 ï¬?nancial 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 percentile its speciï¬?c growth rate,
expressed in yearly percentage points.
23 All standard errors are obtained using the sample weights according to the suggestion in Banca dâ€™Italia (2012).
19
Figure 1: Italy 2002â€“2006â€“2010: Growth Incidence Curve
5
yearly % growth
0
âˆ’5
0 20 40 60 80 100
% population
GIC 2002âˆ’2006
GIC 2006âˆ’2010
Source: Authorsâ€™ calculation from SHIW (Bank of Italy)
Table 1: Italy 2002-2006-2010: descriptive statistics and partition in types
Area Education Occupation rank02 sample02 02
qi Âµ02
i rank06 sample06 06
qi Âµ06
i rank10 sample10 10
qi Âµ10
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/Higher Blue c./not in l.f. 7 51 0.0068 20106.76 12 49 0.0082 26077.04 13 60 0.0111 26010.30
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
Source: Authorsâ€™ calculations on SHIW (Banca dâ€™Italia).
Types are ranked in ascending order according to the average income at the beginning of each growth
period.
Two features stand out. First, the GICs for the two periods lie in two diï¬€erent domains: positive
for the ï¬?rst 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
20
Figure 2: Italy 2002â€“2006â€“2010: Individual Opportunity Growth Incidence Curve
4 2
yearly % growth
0 âˆ’2
âˆ’4
0 20 40 60 80 100
% population
Individual OGIC 2002âˆ’2006
Individual OGIC 2006âˆ’2010
Source: Authorsâ€™ calculation from SHIW (Bank of Italy)
1.96% for 2002â€“06 and -0.66% for 2006â€“10. Second, the two growth processes show very diï¬€erent
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
respective aggregate indexes of inequality conï¬?rms this supposition: income inequality decreases
during the ï¬?rst period and increases during the second period24 (see Table 2).
We proceed in our analysis with the assessment of the distributional eï¬€ects of growth in the
space of â€œopportunitiesâ€?. The individual OGIC for the periods considered are reported in Figure
2.
The individual OGIC of 2002â€“06 shows that growth acts by increasing the value of the oppor-
tunities for all quantiles of the smoothed distributions.25 However, the growth rate is not stable
across quantiles. In particular, the slightly increasing pattern of the individual OGIC over the
24 The results for the second period are consistent with other empirical evidence on the eï¬€ect of the last ï¬?nancial
and economic crisis. See, for example, Jenkins et al. (2013).
25 To make the individual OGIC and the type OGIC graphically comparable, we partitioned the smoothed distri-
butions into 18 quantiles.
21
whole distribution demonstrates an opportunity-regressive impact of growth.
The peculiarities of this growth process are conï¬?rmed by the value of the synthetic measures of
growth (see Table 2 ). The ï¬?rst index, measuring the extent of the opportunity-sensitive growth,
is positive, as expected because the individual OGIC lies above 0. The second index, exclusively
capturing the equal opportunity-enhancing eï¬€ect 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 â€œopportunityâ€?. For instance, the diï¬€erent
shapes characterizing the GIC 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 2006â€“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 individ-
uals. 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 individual OGIC for 2006â€“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 2002â€“06 lies always above the individual OGIC of 2006â€“10, and the dominance is statistically
signiï¬?cant at all points of the curves.26 .
Hence, the growth process in 2002â€“06 dominates the growth process in 2006â€“10 when both the
extent of growth and progressivity components are considered. However, if we want to focus exclu-
sively on their opportunity-redistributive impact (that is, on the extent to which these processes act
by increasing or reducing IOp), the dominance is not clear because they both show a regressive pat-
tern. It can be helpful, in this case, to compare the values of their respective opportunity-equalizing
indexes, which show that 2002â€“06 is, with statistical signiï¬?cance, less regressive than 2006â€“10.
We can conclude that both of the income dynamics under scrutiny act by increasing IOp.
However, whereas this trend is consistent with the change in outcome inequality in the second
26 This dominance is conï¬?rmed by the comparison of their cumulative individual OGICs (ï¬?gures and data available
upon request)
22
Figure 3: Italy 2002â€“2006â€“2010: Within-Types Growth Incidence Curve
5
2
4
0
% yearly growth
% yearly growth
2 3
âˆ’2
1
âˆ’4
0
0 20 40 60 80 100 0 20 40 60 80 100
% type population % type population
within poorest types gic â€™02âˆ’â€™06 within poorest types gic â€™06âˆ’â€™10
within richet types gic â€™02âˆ’â€™06 within richet types gic â€™06âˆ’â€™10
avg. growth poorest types â€™02âˆ’â€™06 avg. growth poorest types â€™06âˆ’â€™10
avg. growth richest types â€™02âˆ’â€™06 avg. growth richest types â€™06âˆ’â€™10
Source: Authorsâ€™ calculation from SHIW (Bank of Italy)
period, in the ï¬?rst period, the variation of outcome inequality and the variation of opportunity
inequality are in the opposite direction. This result reveals that a conï¬‚ict may arise in the evaluation
of growth when these two diï¬€erent perspectives are adopted for the assessment of the same growth
process.
It is interesting to examine why such a conï¬‚ict arises. If inequality between types increases
while overall outcome inequality declines, the within-type share of total inequality must necessarily
decline.27 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 progressive in both the poorest and richest types, with an higher average growth in the richest
type.28 This within-type dynamic explains the divergence between the income- and opportunity-
based distributional assessments.
Turning the focus to the type-speciï¬?c growth, the picture changes dramatically. The type
27 Note that in these empirical applications, the inequality measure used is additively decomposable for within
and between groups.
28 We aggregate types to have suï¬ƒcient observations in each quantile of the within-type GIC.
23
Figure 4: Italy 2002â€“2006â€“2010: Type Opportunity Growth Incidence Curve
6.00
4.00
% yearly growth
0.00 2.00
âˆ’2.00
âˆ’4.00
0 5 10 15
types
type OGIC 2002âˆ’2006 (C1)
type OGIC 2006âˆ’2010 (C1)
Source: Authorsâ€™ calculation from SHIW (Bank of Italy)
OGIC for 2002â€“06, reported in Figure 4, does not always lie above zero for the whole distribution;
in particular, the types ranked 3 and 15 experience a loss. Most importantly, the shape of the type
OGIC diï¬€ers signiï¬?cantly from the shape of the individual OGIC. According to this perspective,
growth can no longer be classiï¬?ed as regressive. For the Italian case, this is equivalent to saying
that households whose heads were born in the South grow, on average, less than households with
diï¬€erent geographical origins.29
29 As reported in Table 1, the circumstance â€œhead born in the Southâ€? appears in the ï¬?ve poorest types in 2002
and 2010 and in the four poorest types in 2006.
24
Table 2: Italy: 2002â€“2006â€“2010 dominance conditions
quantiles/types rank GIC type OGIC cum. type OGIC individual OGIC cum. individual 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 ***
Source: Authorsâ€™ calculations on SHIW (Banca dâ€™Italia).
*=90%, **=95%, ***=99% are signiï¬?cance levels for the diï¬€erence between curves obtained from 1,000
bootstrap replications of the statistics.
Table 3: Italy: 2002â€“2006â€“2008 Complete rankings and inequality
2002 2006 2010
Âµ(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)
â€™02-â€™06 â€™06-â€™10
GY s 1.821 (0.0145) -0.9532 (0.0155)
OGY s -0.0946 (0.0080) -2.869 ( 0.0244)
GY Âµ -0.2340 (0.3707) -1.2618 (0.0197)
Source: Authorsâ€™ calculations on SHIW (Banca dâ€™Italia).
mld = mean logarithmic deviation or generalized entropy index with parameter 0, GY s = EOp consistent
aggregate measures of growth (eq. 9), OGY s = EOp consistent aggregate measures of growth progressivity
(eq. 10), GY Âµ = Aggregate measure of between-type inequality of growth (eq. 11); 95% bootstrapped
standard errors are reported in parenthesis.
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 distribu-
tion, constructed to evaluate distributional phenomena 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 diï¬€erent types whenever there is a reranking of types over time. In contrast, the
type OGIC tracks types over time. Hence, types are ranked according to their average income
25
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 2006â€“10 type OGIC shows some similarity to the individual
OGIC of the same period. In particular, most of the types experience 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 OGIC conï¬?rm the negative impact of the crisis in terms of the extent of
opportunity 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 ï¬?nding is consistent with the reduction of between-
region inequality in Italy due to their diï¬€erent 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 negative perfor-
mance 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 beneï¬?t most from growth during the ï¬?rst 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 processes.30 It is possible to obtain an unambiguous ordering by
weakening the dominance conditions and comparing the cumulative type OGICs. We ï¬?nd that the
ï¬?rst process dominates the second and that this dominance is always statistically signiï¬?cant. 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 ï¬?rst period and negative for the second, and
their diï¬€erence is statistically signiï¬?cant (see Table 3).
30 Although the ï¬?rst process is better than the second and the dominance is statistically signiï¬?cant for most of
the types, for type 15, the second process is preferred to the ï¬?rst one with statistical signiï¬?cance.
26
It is not an easy task to understand the driving forces of these transformations. Given that,
by deï¬?nition, the rank of types and income are correlated, it is extremely diï¬ƒcult to disentangle
the changes that may have aï¬€ected, in opposite directions, the distribution of outcome and the
distribution of opportunities.31 However, the trend of the North-South divide and labor market
reforms may be considered among the determinants of redistribution since 2002. First, the diï¬€erent
reforms realized in the recent past to reduce the gap in the opportunities accessible to diï¬€erent
individuals have not been able to fulï¬?ll the desired goal. In particular, as shown by Pavolini
(2011), among others, diï¬€erent public services, particularly diï¬€erent measures and interventions
of the welfare state, are still suï¬€ering 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.
Second, the labor market reforms introduced in 1998 and extended in 2000 and 2003, which
mainly aimed to reduce the labor protection legislation (particularly for temporary workers), have
increased wage ï¬‚exibility and job turnover, increasing the â€œinstabilityâ€? in the opportunity faced
by individuals (Jappelli and Pistaferri, 2009). This instability may explain why growth appears
more opportunity regressive in the second period, a period of crisis. Boeri and Garibaldi (2007)
suggest that although job ï¬‚exibility 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 contracts, are more likely to be ï¬?red and are often excluded from
social security beneï¬?ts. We suggest that such an eï¬€ect has been stronger in the southern regions,
thereby explaining the territorial gradient in the diverging trends of diï¬€erent types.
Opportunity and Growth in Brazil: The Data
Our theoretical framework may be of particular interest in the analysis of developing and emerg-
ing economies that experience lively growth processes with a dramatic impact on poverty and re-
distribution. 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
31 This may be a challenging question for future research.
27
Ä±lios (PNAD), a representative survey of the Brazilian population, are used.
DomicÂ´
The unit of observation is the household, and the individual outcome is measured as the monthly
household equivalent income, expressed in 2008 Brazilian real.32 Household income is computed
as the sum of all household membersâ€™ individual 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 circumstances: region
of birth and race. Region of birth is coded in ï¬?ve categories (North, Northeast, Southeast, South,
Center-west), and race is coded in three categories (white/east Asian, black/mixed race, and in-
digenous). Individuals who were born abroad and those classiï¬?ed as â€œotherâ€? for the variable race
are excluded 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 2008.33 .
The full opportunity proï¬?les for the three waves are reported in Table 4 in the appendix.34 In
this table, it is clear that race is the main determinant of the disparity 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).
To analyze the distributional impact of growth in Brazil according to the EOp perspective,
two three-year period growth processes are considered: 2002â€“05 and 2005â€“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%. In contrast, the latter period was a period
of pronounced growth, with an overall mean income growth of approximately 6.36%.
32 Equivalent income is obtained by dividing total income by the square root of the household size.
33 Again, the richest and poorest 1% of the household distribution in each wave are dropped.
34 All estimates are based on the sample weights according to Silva et al. (2002).
28
Figure 5: Brazil: 2002â€“2005â€“2008 Growth Incidence Curve
10
yearly % growth
0 5
0 20 40 60 80 100
% population
GIC 2002Ã¯2005
GIC 2005Ã¯2008
Ä±stica)
Authorsâ€™ calculation from PNAD (Instituto Brasileiro de Geograï¬?a e EstatÂ´
Opportunity and Growth in Brazil: The Results
Table 4: Brazil: 2002â€“2005â€“2008 descriptive statistics and partition in types
Region Race rank02 sample02 02
qi Âµ02
i rank05 sample05 05
qi Âµ05
i rank08 sample08 08
qi Âµ08
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
Ä±stica).
Source: Authorsâ€™ calculations on PNAD (Instituto Brasileiro de Geograï¬?a e EstatÂ´
Types are ranked in ascending order according to the average income at the beginning of each growth
period.
As in the ï¬?rst 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.
29
Figure 6: Brazil: 2002â€“2005â€“2008 Individual Opportunity Growth Incidence Curve
10
yearly % growth
0 âˆ’5 5
0 20 40 60 80 100
% population
Individual OGIC 2002âˆ’2005
Individual OGIC 2005âˆ’2008
Ä±stica)
Authorsâ€™ calculation from PNAD (Instituto Brasileiro de Geograï¬?a e EstatÂ´
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 diï¬€erence
between the GIC coordinates in the two periods is always statistically signiï¬?cant (see Table 5 ).
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 2002â€“05 growth episode, although the GIC lies almost always
above zero, the individual OGIC is positive only for half of the smoothed distribution. This conï¬‚ict
indicates that although the majority of households experience positive growth, the extent of the
losses borne by the richest 15% is substantial in determining the change in the value of the oppor-
tunity sets. This eï¬€ect 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 2002â€“05.
30
This does not happen during 2005â€“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 ï¬?rst when an opportunity-egalitarian
perspective is adopted, and the dominance is statistically signiï¬?cant (see Table 5 ). The sign of
the dominance is also conï¬?rmed by the plot of the cumulative individual OGIC. The progressivity
of the two growth episodes is clariï¬?ed by the decreasing shape of the two curves. These results
are further supported by the estimation of the synthetic measures of growth. The index capturing
the opportunity-sensitive extent of growth is positive for both the 2002â€“05 and 2005â€“08 processes,
but it is higher for 2005â€“08. In the same way, the value of the index capturing the progressivity
of growth, in terms of equality of opportunity, is positive for both processes. This means that
during the two periods, growth acts by alleviating the disparities in opportunities, but this eï¬€ect
is stronger for the 2005â€“08 process (see Table 6 ).
Similar features characterize the assessment of growth when the focus is on the type-speciï¬?c
growth. Figure 7 reports the type OGICs for the 2002â€“05 and 2005â€“08 periods.
Regarding the ï¬?rst period, it is possible to observe that, consistent with the individual OGIC,
most of the types experience a reduction in the value of their opportunity set. These types partic-
ularly include households with an indigenous head.35 However, the curve does not appear to show
a clear pattern; it is progressive for the lowest part of the distribution up to type 7 and then takes
a clear regressive shape. The unstable trend is conï¬?rmed 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
conï¬?rmed by the type-speciï¬?c OGIC. All types experience an increase in the values of their oppor-
tunity 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 ). Thus, we can conclude
that this growth process is beneï¬?cial in terms of opportunity when both size and distributional
aspects are considered.
35 However, recall that this curve does not take into account the relative size of types. In this speciï¬?c case, in fact,
the types that experience an increase in the value of their opportunity set represent over 90% of the population.
31
Figure 7: Brazil: 2002â€“2005â€“2008 Type Opportunity Growth Incidence Curve
30
20
% yearly growth
0 10
âˆ’10
âˆ’20
0 5 10 15
types
type OGIC 2002âˆ’2005
type OGIC 2005âˆ’2008
Ä±stica)
Authorsâ€™ calculation from PNAD (Instituto Brasileiro de Geograï¬?a e EstatÂ´
As is reasonable to expect, the comparison of the two processes highlights an unambiguous
dominance of the second period growth over the ï¬?rst. The diï¬€erence in the OGIC coordinates is
statistically signiï¬?cant for almost all types, as shown in Table 5 in the appendix. For robustness
purposes, we also test the diï¬€erence of the respective cumulative type OGICs coordinates, which
is clearly statistically signiï¬?cant for all types, and the diï¬€erence, which is again signiï¬?cant, of their
aggregate index of growth (see Table 6 ).
Finally, Figure 8, reporting the within-type GICs, explains how the progressive growth of Brazil
between 2002 and 2005 is the joint eï¬€ect 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.
32
Figure 8: Brazil: 2002â€“2005â€“2008 Within-Types Growth Incidence Curve
0
8
âˆ’1
7
% yearly growth
% yearly growth
âˆ’2
6
âˆ’3
âˆ’4
5
0 20 40 60 80 100 0 20 40 60 80 100
% type population % type population
within poorest types GIC â€™02âˆ’â€™05 within poorest type gic â€™05âˆ’â€™08
within richest types GIC â€™02âˆ’â€™05 within richest type gic â€™05âˆ’â€™08
avg. growth poorest types â€™02âˆ’â€™05 avg. growth poorest type â€™05âˆ’â€™08
avg. growth richest types â€™02âˆ’â€™05 avg. growth richest type â€™05âˆ’â€™08
Ä±stica)
Authorsâ€™ calculation from PNAD (Instituto Brasileiro de Geograï¬?a e EstatÂ´
Table 5: Brazil: 2002â€“2005â€“2008 dominance conditions
quantiles/types rank GIC type OGIC cum. type OGIC individual OGIC cum. individual 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 *** -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 ***
Ä±stica).
Source: Authorsâ€™ calculations on PNAD (Instituto Brasileiro de Geograï¬?a e EstatÂ´
*=90%, **=95%, ***=99% are signiï¬?cance levels for the diï¬€erence between curves obtained by 1,000
bootstrap replications of the statistics.
33
Table 6: Brazil: 2002â€“2005â€“2008 complete rankings and inequality
2002 2005 2008
N 366,388 390,046 372,581
Âµ(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)
â€™02-â€™05 â€™05-â€™08
avg. growth. 0.26% 6.36%
GY s 0.7547 (0.0910) 7.4937 (0.1169)
OGY s 0.4384 (0.0559) 0.7891 (0.0566)
GY -1.0213 (0.6120) 9.5304 (1.0758)
Ä±stica).
Source: Authorsâ€™ calculations on PNAD (Instituto Brasileiro de Geograï¬?a e EstatÂ´
mld = mean logarithmic deviation or generalized entropy index with parameter 0, GY s = EOp consistent
aggregate measures of growth (eq. 9), OGY s = EOp consistent aggregate measures of growth progressivity
(eq. 10), GY Âµ = Aggregate measure of between type inequality of growth (eq. 11), 95% bootstrapped
standard errors are reported in parenthesis.
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 determinants 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 beneï¬?cial in terms of improving opportunities and that Brazil has experienced an im-
pressive increase in the degree of EOp, particularly during the 2002â€“05 period. Our conclusions
complement the ï¬?ndings of Molinas et al. (2011), who look at the development of IOp in Brazil
with a speciï¬?c focus on the opportunities of children.
III. 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 ï¬?nal 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
34
the income dynamics of speciï¬?c groups of the population. 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 perspective 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 understanding of the joint dynamic of income inequality and inequality
of opportunity. Another ï¬?eld of application of our framework is the analysis of tax-beneï¬?t 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 developed in this paper, which allow the evaluation of the IOp reduction, seems a
promising path for future research.
35
APPENDIX
Proof of Remark 1. We start by showing the suï¬ƒciency that the individual OGIC implies
i ËœA
Âµ i (yt+1 )
ËœAo
the type OGIC dominance. Let the two type OGICs be deï¬?ned as follows: g nA = ÂµA
âˆ’1
i (yt )
i ËœB
Âµi (yt+1 )
ËœBo
âˆ€i âˆˆ {1, ..., nA } and g nB = ÂµB
âˆ’ 1 âˆ€i âˆˆ {1, ..., nB }. If (i) holds and there is type OGIC
i (yt )
dominance between the two growth processes GA and GB , we will have the following:
i i ËœA
Âµ ËœB (yt+1 )
i (yt+1 ) Âµ
ËœAo
g ËœBo
â‰¥g â‡?â‡’ A
â‰¥ iB , âˆ€i âˆˆ {1 , ..., n }. (12)
n n Âµi (yt ) Âµi (yt )
If (iii) holds, the type OGIC dominance of the growth processes GA over GB will be
i i ÂµA B
i (yt+1 ) Âµi (yt+1 )
ËœAo
g ËœBo
â‰¥g â‡?â‡’ A
â‰¥ B , âˆ€i âˆˆ {1 , ..., n } , (13)
n n Âµi (yt ) Âµi (yt )
where Âµi (yt+1 ) is the mean income of the type ranked i in the ï¬?nal distribution of the typesâ€™ mean
Ëœi (yt+1 ). Now, let the two individual OGICs be deï¬?ned
income, which, under (iii), corresponds to Âµ
Ao j ÂµAt
j
+1
Bo j ÂµBt
j
+1
as follows: gY s
NA = ÂµjAt âˆ’ 1 âˆ€j = 1, ..., NA and gY s
NB = ÂµjBt âˆ’ 1 âˆ€j = 1, ..., NB . (i)
and (ii) implies NA = NB . Hence, if there is individual OGIC dominance of the growth process
GA over GB , we will have the following:
Ao j Bo j ÂµAt
j
+1
ÂµBt
j
+1
gY s â‰¥ gY s â‡?â‡’ â‰¥ âˆ€j âˆˆ {1 , ..., N } . (14)
N N ÂµAt
j ÂµBt
j
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):
mit+1 mit+1
i i ÂµAt
j
+1
ÂµBt
j
+1
ËœAo
g ËœBo
â‰¥g â‡?â‡’ â‰¥ âˆ€i âˆˆ {1 , ..., n }. (15)
n n j =1
ÂµAt
j j =1
ÂµBt
j
If eq. (14) holds, than it must be the case that the dominance in their type aggregation holds,
providing the dominance in eq. (15). Hence we have proved the suï¬ƒciency of the remark.
We now prove the necessary condition by contradiction.
Suppose that eq. (15) holds. Now, pick a type i âˆˆ {1, ..., n}. Assume that for that type
36
ÂµAt +1
ÂµBt +1
âˆƒk {1, ..., mi } such that k
ÂµAt
< Âµk
Bt , then because all individuals in the same type are given
k k
mi ÂµAt+1 mi ÂµBt+1
j j
the same mean income, ÂµAt
âˆ’ ÂµBt
< 0 for a given type i, contradicting eq. (15). QED
j j
j =1 j =1
37
Notes
1
See Essama-Nssah and Lambert (2009) for a comprehensive survey.
2
Hence, we investigate the relationship between growth and inequality of opportunity using a
â€œmicro approachâ€?; an alternative â€œmacro approachâ€? would also be possible by investigating the
relationship between growth and IOp from a cross-country or longitudinal perspective (see Marrero
and Rodriguez 2010).
3
To obtain this conï¬‚ict between type OGIC and GIC, it is necessary that rich individuals
experiencing losses are spread across the majority of socioeconomic groups.
4
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).
5
For a longitudinal perspective on the evaluation of growth, see Bourguignon (2011) and Jenkins
and Van Kerm (2011).
6
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.
7 Ht
Ravallion and Chen (2003) also propose the RP P GRC = 0
g (p) dp/Ht where Ht is the initial
poverty headcount ratio. RP P GRC measures the proportionate income change of the poorest
individuals.
8
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 diï¬€erent meanings of luck.
9
For example, LeFranc et al. (2008) and Peragine and Serlenga (2008) use stochastic dominance
conditions to compare the diï¬€erent 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 individualâ€™s type, that is Âµi (1 âˆ’ Gi ), where the type mean income Âµi and (1 âˆ’ Gi ) represent,
respectively, the return component and the risk component, with Gi denoting the Gini inequality
38
index within type i. See also Oâ€™Neill et al. (2000) and Nilsson (2005) for empirical analyses that
attempt to provide alternative evaluations of opportunity sets using parametric estimates.
10
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 diï¬€erent countries, making an international
comparison of inequality of opportunity estimates across the world possible.
11
For a discussion of this issue with reference to a non deterministic, parametric model of EOp,
see Ferreira and Gignoux (2011) and Luongo (2011).
12
Note that, given the assumption of anonymity implicit in this framework, the individuals ranked
j j
N in t can be diï¬€erent from those ranked N in t + 1.
13
Note that we track the same type but do not track the same individuals.
14
Note that the type OGIC is a generalization of the idea underlying the ï¬?rst component of
Roemerâ€™s (2011) index of development, that is, â€œhow well the most disadvantaged type is doingâ€?.
15
For a normative justiï¬?cation of these dominance conditions based on a rank-dependent social
welfare function, see the working paper version of the paper: Peragine et al. (2011).
16
See, inter alia, Sutherland et al. (1999).
17
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.
18
Similar to the cumulative inividual OGIC, the cumulative type OGIC is obtained by rearranging
the diï¬€erence between the Generalized Lorenz curves applied to the type mean distributions Y Âµt
ËœÂµ .
and Y t+1
19
The approach is close in spirit to Essama-Nssah (2005), reviewed in a previous section. For
a normative justiï¬?cation of the rank-dependent approach to IOp analyses, see Peragine (2002),
Aaberge et al. (2011), and Palmisano (2011)
20
See endnote12.
21
We calculate conï¬?dence intervals for the diï¬€erence 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 t+1 , are independent and identically distributed observations
39
of the unknown probability distributions F (y t ), F (y t+1 ). Î³ is the statistic of interest, and its
standard error is Ïƒ (F (y t ), F (y t+1 )) = Ë† (y t , y t+1 ). Our bootstrap estimate of the standard
V arÎ³
error is Ïƒ Ë† (y t+1 , where F
Ë† (y t ), F
Ë†=Ïƒ F Ë† (y t+1 ) are the empirical distributions observed. The
Ë† (y t ), F
95% conï¬?dence interval is obtained by resampling B = 1, 000 ordinary non parametric bootstrap
âˆ— âˆ—
replications of the two distributions yt Ë† is obtained using
, yt+1 . The standard error of parameter Î³
B B
Î³ âˆ— (b)
Ïƒ
Ë†B = Ë† âˆ— (b)
b=1 {Î³ Ë† (.)}2 /(B âˆ’ 1), where Î³
âˆ’Î³ Ë† (.) = b=1
B . Ë†B â†’ Ïƒ
We know that Ïƒ Ë† when
B â†’ âˆž, and, under the assumption that Î³ is approximately normally distributed, we calculate
conï¬?dence intervals: Î³ Ë† Â±z1âˆ’Î±/2 Ïƒ
Ë†=Î³ Ë†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.
22
We use the OECD equivalence scale given by the square root of the household size.
23
All standard errors are obtained using the sample weights according to the suggestion in Banca
dâ€™Italia (2012).
24
The results for the second period are consistent with other empirical evidence on the eï¬€ect of
the last ï¬?nancial and economic crisis. See, for example, Jenkins et al. (2013).
25
To make the individual OGIC and the type OGIC graphically comparable, we partitioned the
smoothed distributions into 18 quantiles.
26
This dominance is conï¬?rmed by the comparison of their cumulative individual OGICs (ï¬?gures
and data available upon request)
27
Note that in these empirical applications, the inequality measure used is additively decompos-
able for within and between groups.
28
We aggregate types to have suï¬ƒcient observations in each quantile of the within-type GIC.
29
As reported in Table 1, the circumstance â€œhead born in the Southâ€? appears in the ï¬?ve poorest
types in 2002 and 2010 and in the four poorest types in 2006.
30
Although the ï¬?rst process is better than the second and the dominance is statistically signiï¬?cant
for most of the types, for type 15, the second process is preferred to the ï¬?rst one with statistical
signiï¬?cance.
31
This may be a challenging question for future research.
40
32
Equivalent income is obtained by dividing total income by the square root of the household
size.
33
Again, the richest and poorest 1% of the household distribution in each wave are dropped.
34
All estimates are based on the sample weights according to Silva et al. (2002).
35
However, recall that this curve does not take into account the relative size of types. In this
speciï¬?c case, in fact, the types that experience an increase in the value of their opportunity set
represent over 90% of the population.
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