WPS3814
A Normal Relationship?
Poverty, Growth, and Inequality
J. Humberto Lopez and Luis Servén*
The World Bank
Abstract
Using a large crosscountry income distribution dataset spanning close to 800 country
year observations from industrial and developing countries, this paper shows that the size
distribution of per capita income is very well approximated empirically by a lognormal
density. Indeed, the null hypothesis that per capita income follows a lognormal
distribution cannot be rejected  although the same hypothesis is unambiguously rejected
when applied to per capita consumption. The paper shows that lognormality of per capita
income has important implications for the relative roles of income growth and inequality
changes in poverty reduction. When poverty reduction is the overriding policy objective,
poorer and relatively equal countries may be willing to tolerate modest increases in
income inequality in exchange for faster growth  more so than richer and highly
unequal countries.
JEL classification codes: C12, C23, D31, I32.
World Bank Policy Research Working Paper 3814, January 2006
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the
exchange of ideas about development issues. An objective of the series is to get the findings out quickly,
even if the presentations are less than fully polished. The papers carry the names of the authors and should
be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely
those of the authors. They do not necessarily represent the view of the World Bank, its Executive Directors,
or the countries they represent. Policy Research Working Papers are available online at
http://econ.worldbank.org.
*This paper was prepared for the World Bank's Latin America Regional Studies program. We thank Lant
Pritchett for comments and Patricia Macchi for excellent research assistance.
I. Introduction
In recent years, poverty reduction has been formally enshrined as the goal of
development policy worldwide, and a rapidly expanding analytical and empirical
literature has sought to clarify whether poverty changes are driven mainly by growth in
aggregate income or by growth in the relative incomes of the poor. The question is of
more than scholarly interest, because it has major implications for the roles of growth
promoting and inequalityreducing policies in the poverty reduction process. For
example, if trends in relative incomes were found to account for the lion's share of
poverty changes, policy makers might face a tradeoff between fast growth and rapid
poverty reduction.1
In a recent contribution, Kraay (2005) decomposes poverty changes into three
ingredients: (i) growth in average incomes; (ii) the sensitivity of poverty to growth; and
(iii) changes in the distribution of income. In a large crosscountry sample, he finds that
growth in average incomes accounts for some 70 percent of the variation in (headcount)
poverty changes in the short run, and over 95 percent in the medium to long run. In
contrast, crosscountry differences in the sensitivity of poverty to growth play a minimal
role. Together, these results suggest that growthoriented policies hold the key to poverty
reduction.
In turn, Ravallion (1997, 2004) presents an empirical model relating poverty
changes to the distributioncorrected rate of growth.2 His estimates underscore the key
role of initial inequality: depending on its level, a onepercent increase in income levels
reduces poverty by as much as 4.3 percent (in very low inequality countries) or as little as
0.6 percent (in high inequality countries). This suggests that fast poverty reduction will
be hard to achieve without declines in inequality, especially in very unequal countries.3
This paper reassesses the roles of growth and inequality for poverty reduction
from a different perspective, based on the use of a parametric approach to model the size
distribution of income. Thus, the paper follows an abundant literature spanning over a
century  from Pareto (1897) to Gibrat (1931), Kalecki (1945), Rutherford (1955),
Metcalf (1969), Singh and Maddala (1976) and Bourguignon (2003)  that has attempted
to approximate the distribution of income using a variety of functional forms.
Specifically, the paper uses a large crosscountry database including both industrial and
developing countries and spanning almost 40 years to test the null hypothesis that the size
distribution of per capita income can be described by a lognormal density.
1This, of course, need not always be the case, since many policies are likely to be both growthpromoting
and equalityenhancing. But some empirical evidence suggests that not all policies have this feature (Barro
2000, Lundberg and Squire 2003, Lopez 2004), and some may force policy makers to face a trade off
between faster growth and increasing inequality.
2 Specifically, Ravallion (1997) interacts the growth rate with one minus the initial Gini coefficient,
whereas Ravallion (2004) considers a distributional term of the form (1Gini) with >1, to incorporate
possible nonlinear interaction effects between the growth elasticity of poverty and initial inequality.
3A similar point is made by Bourguignon (2004), who presents a simulation model calibrated on Mexican
data. When no change takes place in the distribution of income, the model shows that a per capita growth
rate of 3 percent per year over a 10year period lowers the poverty rate by about 7 percentage points. When
the same growth rate is accompanied by declining inequality (a reduction in the Gini coefficient by 10
percentage points), poverty falls twice as much  over 15 percentage points.
A parametric approach offers a number of advantages over empirical variance
decompositions and reducedform regressions. First, it allows a systematic assessment of
the role of countryspecific initial conditions for the povertyreducing effects of growth
and distributional change. While the literature has stressed how initial inequality shapes
the growth elasticity of poverty, little if any attention has been paid to the roles of the
initial level of development or the poverty line itself for the choice of povertyreducing
policies. For example, should the balance between progrowth and prodistributional
policies be the same in Zambia and El Salvador, which have similar Gini coefficients,
despite their wide disparity in terms of per capita income (US$361 and US$2,200
respectively in 2002)? The approach taken in this paper allows a rigorous answer to this
kind of question.
Second, the parametric approach permits removing the straightjacket of a
common poverty line across countries, which otherwise is virtually unavoidable in cross
country empirical work. Most of the available studies use poverty statistics based on the
international PPP US$1 a day poverty line. However, such definition is of little interest
for many middle income developing countries (not to mention industrial economies). As
an example, in Argentina the headcount poverty was in 2002 close to 60 percent when
calculated on the basis of the nationallydefined poverty line, while internationally
comparable poverty indicators based on a dollaraday poverty line would place the
poverty rate around 3 percent. One implication of these diverging assessments is that the
findings from empirical work using crosscountry poverty databases, based on a common
poverty line, tend to give more weight to countries where the US$1 a day poverty
measure makes sense  i.e., lowincome countries where our results below suggest that
growth should be expected to dominate distributional change from the viewpoint of
poverty reduction.
Of course, the advantages of the parametric approach matter only if the chosen
parameterization fits the data well. We find that the null hypothesis of lognormality
cannot be rejected when applied to the distribution of per capita income, regardless of
whether income is measured in gross terms (i.e. before taxes and transfers) or net terms
(after taxes and transfers). However, the same null hypothesis is unambiguously rejected
when applied to per capita consumption data. We conjecture that this rejection may be
due to consumption smoothing, under which the log of consumption may not be normally
distributed even if the log of income is.
The paper derives some implications of this result for the relative roles of growth
and inequality in poverty reduction under alternative initial conditions. We highlight four
main points: (i) inequality hampers poverty reduction, both because of its negative impact
on the growth elasticity of poverty (as stressed in the literature) but, in most scenarios,
also because of its negative impact on the inequality elasticity of poverty; (ii) for a given
poverty line, the impact of growth on poverty is stronger in richer than in poorer
countries, and hence the latter will find it harder than the former to achieve fast poverty
reduction; (iii) the share of the variance of poverty changes attributable to growth should
be generally lower in richer and more unequal countries; and (iv) given the initial levels
of development and inequality, the relative povertyreduction effectiveness of growth and
inequality changes depends on the poverty line  the higher the poverty line, the bigger
the role of growth and the smaller the role of distributional change.
2
The rest of the paper is organized as follows. In Section II we describe the test for
lognormality and the dataset we employ. Section III reports the empirical results. Section
IV derives the implications of lognormality for the poverty reduction roles of growth and
inequality changes. Section V offers some concluding remarks.
II. Testing for lognormality of the size distribution of per capita income
Attempts to model the size distribution of per capita income have a long tradition
in the economics literature, dating back to Pareto (1897). The use of the lognormal
function in this context was pioneered by Gibrat (1931), who found that if offered a good
empirical fit to the observed distribution of income, and provided a theoretical
justification based on a model in which individual incomes are subject to random
proportionate changes.4
Gibrat's work was followed by a large literature extending his basic framework
and offering additional empirical evidence. Kalecki (1945) modified Gibrat's original
setup making negative income changes less likely at low income levels than at high ones,
to account for the fact that the variance of log income remained relatively constant over
time. Rutherford (1955) expanded Gibrat's model introducing birth and death
considerations. He also presented empirical experiments based on the comparison of
theoretical and observed quantiles of the distribution of income, searching for a
functional form that would improve upon the lognormal.5
On the theoretical front, other subsequent papers developed rigorous models that
under fairly general conditions yield lognormal distributions of earnings and/or wealth
(Sargan 1957, Pestieau and Posen 1979). In turn, on the empirical front, the fit of the
lognormal function to the observed distribution of income was found to be somewhat less
satisfactory at the upper end of the distribution (Hill 1959, Cowell 1977), specifically the
top 34 percentiles (Airth 1985). This prompted attempts to fit more complex functional
forms  displaced and/or truncated versions of the lognormal density (Metcalf 1969,
Salem and Mount 1974) or alternative functional specifications (Fisk 1961, Salem and
Mount 1974, Singh and Maddala 1976, McDonald 1984). Both strategies pose a tradeoff
between goodnessoffit and analytical tractability (Metcalf 1969), as well as
interpretability of the parameters (Lawrence 1988), which explains the continuing
popularity of the lognormal specification.6
More recently, Bourguignon (2003) has offered an indirect reassessment of the
empirical validity of the lognormal approximation. He reports OLS regression estimates
in a framework explaining the observed change in a selected poverty measure on the
4Specifically, Gibrat (1931) argued that the good empirical performance of the lognormal density could be
rationalized under the following three conditions: (i) in each period the distribution of income is derived
from that of the previous period by assuming that the variable corresponding to each member of the
distribution is affected by a small proportionate change; (ii) such proportions differ for different members
of the distribution; and (iii) these differences are determined randomly according to a given frequency
distribution.
5Rutherford performed singlecountry estimations using data for the UK in 1949; USA in 1947 and 1948;
Canada in 1947, Australia in 1951, Sri Lanka (Ceylon at the time the paper was written) in 1950 and
Bohemia (modern Slovakia and the Czech Republic) in 1932.
6For example, Dollar and Kraay (2002) resort to the assumption of lognormality in order to complete their
data sample by generating quintile shares from Gini coefficients for those observations for which only the
latter is available.
3
basis of two regressors: a "growth effect", given by average income growth times the
theoretical growth elasticity of poverty calculated under the lognormality assumption,
and an "inequality effect", given by the change in inequality (as measured by the standard
deviation of log income) times the theoretical inequality elasticity of poverty as derived
under the lognormality assumption as well. Under the null of lognormality, both
regressors should carry coefficients equal to unity.
When this empirical approach is implemented on a sample of developingcountry
growth spells, with the change in headcount poverty as dependent variable, Bourguignon
(2003) rejects the null of lognormality but still finds that the lognormal specification
provides a good empirical approximation to actual poverty changes. When the dependent
variable is instead the change in the poverty gap, the null hypothesis is still rejected, and
in addition the fit of the regression is quite poor.
There are, however, two problems with this approach. The first one, noted by
Bourguignon, is that the elasticitybased approach is valid only for infinitesimal changes
in poverty and its determinants. Applying it to discrete changes can result in large
approximation errors, especially given the long duration (ten years and over) of some of
the spells in Bourguignon's sample. The second problem is the implementation of the
approach using poverty databases, which tend to be relatively small, typically include a
considerable number of outliers,7 and often involve substantial measurement error. The
latter is further exacerbated by firstdifferencing the data for the regressions, which raises
the noisetosignal ratio. Below we present a new test of lognormality of the distribution
of income that is not subject to these concerns.
II.1 Empirical approach
In spirit, our approach is closest to that employed by Rutherford (1955). As noted
above, for several countries (one at a time) he compared the observed quintiles of the
distribution of income with their theoretical counterparts derived under the null
hypothesis of lognormality. Formally, we exploit the onetoone mapping that arises
under lognormality between the Gini coefficient and the Lorenz curve L(p) that describes
the relative income distribution.8 Letting G and respectively denote the Gini
coefficient, and the standard deviation of log income, Aitchison and Brown (1966) show
that lognormality implies
= 2 1(1+2 ), G
(1)
and
L(p) = 1( p)  ,
( ) (2)
where(.)denotes the cumulative normal distribution. Hence a change in the Gini
coefficient, and thus in , must be reflected in a matching change in the Lorenz curve
7Kraay (2005) uses a filter to eliminate extreme observations from his poverty dataset. This results in the
loss of over onethird of his original sample.
8Recall that L(p) is the aggregate income share of the bottom 100p percent of the population. Thus, L(0)=0
and L(1)=1.
4
(Aitchison and Brown 1966, chapter 11). Likewise, changes in the Lorenz curve itself
can be mapped into changes in the Gini coefficient.
On a crosscountry basis, what is usually available to the researcher is some
summary information on the shape of the Lorenz curve. One such summary is provided
by the income shares of the different quintiles of the population:
Q20 = L(.2 j)  L(.2( j 1)) for j = 1,2,3,4. (3)
j
Given the onetoone mapping between the Gini coefficient and the Lorenz curve
that follows from (1) and (2), under lognormality there must be also a onetoone
mapping between the Gini coefficient and the quintile shares (3). Thus, a test of the null
hypothesis of lognormality can be based on the comparison of the empirical quintiles, say
E20j, with their Ginibased theoretical counterpartsQ20 . Following this approach, a
j
formal lognormality test can be performed on the basis of the regression model:
E20j = + Q20 j + vitj ,
it it (4)
where j=1,2,3,4 denotes the income quintile; i=1,2,...,N is a country index, and
t=1,2,...,Ti denotes the date of each income (or expenditure) survey available for country
i. In general Ti will differ across countries, resulting in an unbalanced sample. In (4), the
theoretical quintiles Qit are constructed on the basis of the observed Gini coefficients
20 j
Git, as implied by (1)(3):
Qit
20 j= 1(.2j) 2 1
1+ Git
2  1(.2( j 1)) 2 1 1+ Git . (5)
2
Testing for lognormality in (4) is equivalent to testing the joint null hypothesis:
=0; =1. (6)
We should note that the precise null hypotheses entertained here is that the size
distribution of income is described by a twoparameter lognormal function. Strictly
speaking, rejection of (6) does not quite amount to rejection of lognormality more
generally, since the distribution of income might still be characterized by a three
parameter lognormal density. This could happen, for example, if per capita income
follows a displaced lognormal distribution  i.e., it is lognormal over the range above
some unknown minimum level (where is expressed as a ratio to average per capita
income). In such case,9 the variance of log income, and hence the Gini coefficient, remain
unaffected, but from Aitchison and Brown (1969, p.15) it follows that
L( p) = p + (1 ) 1( p)  .
( ) (7)
9Explored, for example, by Metcalf (1969), who fits a displaced lognormal to U.S. personal income.
5
Expression (3) can still be used to compute the theoretical quintiles that
correspond to any given Gini coefficient and shift . However, a regression like (4)
projecting observed income shares on their counterparts constructed under the null of
lognormality, ignoring the shift of the distribution (that is, assuming = 0 when in reality
>0), will result in a positive intercept and a slope less than 1 under the null. The bigger
the shift , the larger the constant and the smaller the slope.
Finally, even if the true income distribution is characterized by the conventional
twoparameter lognormal function, the observed distribution may follow a more complex
form if the availability of data is limited. Data might be completely unavailable outside
some income range, like in the textbook truncation case, or availability could vary in
some systematic fashion with the level of income, like in the model of survey non
compliance examined by Deaton (2004). Depending on the particulars, a number of
possibilities arise regarding the distribution of observed income. Under some special
assumptions (illustrated by Deaton 2004), it might still be described by a twoparameter
lognormal, although both its mean and variance could differ from those of the true
distribution, but under more general conditions it may be characterized by more complex
truncated lognormal distributions, in which case the simple relations (1)(2) break down
and inference based on (4) will reject the null hypotheses (6).10
II.2 Estimation Issues
The choice of estimation technique for (4) is dictated by the properties of the
residual termvitj . If the residuals are i.i.d., OLS suffices to test the null of lognormality.
However, there are two reasons why the assumption of independence may not hold. First,
the residuals for a given country may be correlated across different surveys. In this
regard, Lopez (2004) finds that the Gini coefficient shows significant persistence over
time, and this suggests that the discrepancy between observed quintile shares and
nonlinear transformations of the Gini as in (5) may also show persistence. Second, for
any given survey all four theoretical quintiles are derived from the same Gini coefficient,
and hence the residuals of the four regression observations that result may be mutually
correlated.
In these circumstances, OLS estimates of and will still be consistent but
inference based on the usual OLS covariance matrix will be inappropriate. Of course,
valid inference can be performed using a robust estimator of the covariance matrix of the
OLS coefficients that takes into account the lack of independence of the residuals, as well
as their potential heteroskedasticity. However, under appropriate assumptions about the
structure of the residual covariance matrix, more efficient inference may be possible.
Specifically, assume that the disturbance term follows an errorcomponents model:
vitj = µi + itj , (8)
or the more general
10For example, if the sample is truncated from below (i.e., lowincome observations are lost) then it can be
shown that linear regressions like (4) will yield negative intercepts and slopes above unity.
6
vitj = µi +i + itj ,
t (9)
where µi is an unobservable countryspecific effect, assumed to be i.i.d. with zero mean
and variance µ ; i denotes an effect specific to the tth survey for the ith country, also
2 t
assumed i.i.d. with zero mean and variance , and itj denotes the residual disturbance,
2
assumed i.i.d. with zero mean and variance . 2 11 The µi's, i 's, and the itj 's are
t
assumed mutually independent. Under these assumptions, the covariance structure of the
error term is:
µ + +
2 2 2 if i = l,s = t, j = k
E(vitj ,vk ) =
ls µ +
2 2 if i = l,s = t, j k
(10)
µ 2 if i = l,s t
0 if i l.
These expressions define a twoway errorcomponents model in which the survey
specific effect i is nested in the countryspecific effect µi. If = 0, (9) reduces to the
t 2
standard errorcomponents model (8).
The parameters of the nested error components model given by (4) and (9)(10)
can be estimated in a variety of ways, ranging from ANOVAtype to minimum quadratic
norm and maximum likelihood estimation.12 On the whole, Monte Carlo evidence
reported by Baltagi et al. (2001) suggests that the method chosen makes little difference
for the estimates of the regression coefficients in (4). However, when  like in our case 
one is also interested in the standard errors of the parameter estimates, as well as in the
variance components themselves, maximum likelihood estimation offers the best
performance, especially if the sample is severely unbalanced.13
II.3 Data
We implement the empirical approach described above using the Dollar and
Kraay (2002) dataset, which comprises 794 countryyear observations for which both the
Gini coefficient and the quintile shares are available.14 The dataset combines observations
for which both the Gini coefficient and the quintile shares refer to gross (i.e. before taxes
and transfers) income (47 percent of the observations); net (i.e. after taxes and transfers)
income (29 percent of the observations); and expenditure (24 percent).
Table 1 presents some descriptive statistics. The average Gini coefficient is
roughly the same in the income and expenditure subsamples  0.37 for income and 0.38
for expenditure. This is somewhat surprising since on the basis of conventional
11These assumptions can be further relaxed to allow for heteroskedasticity of itj .
12See Baltagi et al. (2001) and Davis (2002) for discussion.
13See RabeHesketh et al. (2004) on the computational aspects of ML estimation in this context.
14We discard the 158 extra observations for which Dollar and Kraay construct the quintile shares from the
Gini coefficient using equations (1)(3), and thus assuming that the distribution of income is lognormal 
which is precisely what our regressions aim to test.
7
smoothing arguments one would expect less dispersion in the expenditurebased
surveys.15 One might wonder if in our sample the result is driven by the fact that the
panel is unbalanced. But even if we give equal weight to each country's average Gini
coefficient we find a similar picture: the resulting overall means are 0.40 for income
based and 0.42 for expenditurebased observations, respectively. The latter, however, are
more concentrated around their mean. In other words, there is a lower frequency of
countries with extreme (whether high or low) inequality in the expenditurebased
subsample than in the incomebased subsample.
On the other hand, there is a noticeable difference between the average Gini
coefficients of the subsamples based on gross and net income. Gross incomebased Gini
coefficients average 0.40, whereas those based on net income average 0.33. Although one
should be careful in attaching any particular economic interpretation to these figures, it is
tempting to view them as reflecting the effect of government interventions that lead to
income redistribution.
III Empirical results
We turn to the empirical implementation of the lognormality tests. We perform
them on the full sample as well as different subsamples defined by type of data  i.e.,
according to whether the observations are based on expenditure or income surveys and, in
the latter case, whether income is measured on a net or gross basis. Apart from allowing
some robustness checks, this differentiation is also of interest for two other reasons. First,
if the distribution of income is lognormal, and households engage in consumption
smoothing, the distribution of consumption will not be lognormal in general.16 Hence the
common practice of pooling together income and expenditurebased observations in
applied work could yield misleading test results in our case. Second, gross income could
be lognormally distributed while net income is not  for example if taxes and transfers
are lumpsum rather than proportional to income.
To be specific, in addition to (i) the full sample we also consider subsamples of
(ii) incomebased observations only (labeled "Income" for short); (iii) expenditurebased
observations only ("Expenditure"); (iv) gross incomebased observations only ("Gross
Income"); (v) a combination of expenditure and net incomebased observations ("Net");
and finally (vi) net incomebased observations only ("Net Income").
For the full sample, Figure 1.(a) shows a scatter plot of the observed quintile
shares (vertical axis) against their theoretical counterparts, as computed under the null
hypothesis of lognormality (horizontal axis). The data points cluster along the 45degree
line, suggesting that the lognormal distribution provides a fairly close approximation to
15For this reason, Forbes (2000) and Deininger and Squire (1996) raise expenditurebased Gini coefficients
by 0.066 to make them comparable with incomebased ones.
16Under consumption smoothing, current consumption will depend on some weighted sum of current and
future anticipated income. In the textbook permanentincome model, the consumption level of household h
in period t would take the form Ct = At + (1+ r)s
h h t Et xs
[ ]
h , where x is income, A denotes
s=t
financial wealth, r is the real interest rate, is a parameter, and Et denotes the conditional expectation.
Even if xs is lognormally distributed, the infinite sum in the square brackets will not be in general, a simple
consequence of the fact that the sum of lognormallydistributed variables is not itself lognormally
distributed.
8
the size distribution of per capita income / expenditure as summarized by the quintile
shares. Figures 1.(b)1.(f) present similar plots for the various subsamples.17
Table 2 presents the results of OLSbased lognormality tests for the full sample
and the various subsamples described above. The standard errors of the estimates are
computed using a clustering procedure to allow for residual dependence; they also allow
for heteroskedasticity.18
The first thing that stands out in the table is the excellent fit of the regressions in
all the samples considered, with R2 ranging from 0.95 to 0.98. For a crosscountry sample
of this magnitude, such fit is remarkable. In turn, the regression slopes and intercepts are
very close to their expected values under the null of one and zero, respectively. It is worth
noting that in the samples including expenditure observations (the first, third and fifth
columns) the estimated slopes are slightly below one, while the opposite happens in the
regressions including only incomebased observations. Formally, we can reject the null of
unit slope in the Expenditure and Net subsamples (third and fifth columns). In turn, the
estimated intercepts are positive in the samples including expenditurebased observations
and negative in those including only incomebased observations. Like with the slopes, in
the Expenditure and Net subsamples we can also reject the null of zero intercept.
The bottom of Table 2 reports Wald tests of the null hypothesis of lognormality
(6). Under the null, the test statistic follows a chisquare distribution with two degrees of
freedom. As would be expected in the light of the point estimates, the null can be rejected
at the 5 percent level in the two samples in which expenditurebased observations
represent a sizeable share of the total number of data points. In contrast, the samples
containing only incomebased observations show little evidence against the null  the p
values range from 0.62 to 0.93. In the full sample, in which expenditurebased
observations represent only about 20 percent of the total, we also fail to reject the null,
with a pvalue of 0.35.
On the whole, the lognormality tests based on the OLS estimates suggest that the
size distribution of income and expenditure follow significantly different patterns. In
contrast, the distinction between gross and net income seems to be of little consequence.
Table 3 repeats the same tests of Table 2 now based on ML estimation of the nested
errorcomponent model given by (4) and (9)(10). In addition to the information
contained in the preceding table, Table 3 also reports the estimated standard deviations of
the error components in (9), and the results of tests of their individual and joint
significance.
Inspection of Table 3 reveals a picture very similar to the one emerging from
Table 2, in terms of both point estimates and standard errors. The pattern of signs and
magnitudes of the point estimates across samples is the same as before. The middle block
of the table reports the estimated standard deviations of the error components. The
17Figures 1(e) and 1(f) show an apparent outlying observation (corresponding to Q2 in Norway 1989),
which might be viewed as a candidate for removal from the sample. This, however, would be of no
consequence for any of the empirical results in the paper.
18The clustering is done by country. Doing it instead by survey yields slightly larger standard errors but
does not cause any qualitative changes on the results of the hypothesis tests.
9
standard deviation of the surveyspecific effect is in all cases quite small  indeed,
much smaller much than that of the countryspecific effect µ .
The lognormality tests yield the same qualitative conclusion as before: the
evidence from the full sample and the three incomebased samples is consistent with the
null hypothesis of lognormality, as reflected in pvalues ranging from 0.41 in the full
sample to 0.92 in the Gross Income subsample. In contrast, the null is rejected at the 5
percent level in the Expenditure and Net subsamples.
The last three rows of Table 3 report tests of significance of the error components.
The null hypothesis that the variances of the two error components are jointly zero ( µ = 2
= 0) is rejected in all cases. As for the nested surveyspecific component, its variance
2
is insignificant in the Income and Expenditure subsamples; in addition, it falls just
2
short of 5 percent significance in the Gross Income and Net Income subsamples. In turn,
the variance of the countryspecific component µ is significant at the 5 percent level in
2
four of the six samples, and at the 10 percent level in the other two  the Expenditure and
Gross Income subsamples.
On the whole, we may view these tests as generally supportive of the nested error
component specification, except in the Income and Expenditure subsamples, where the
test results suggest that a oneway model might be sufficient. To investigate this further,
Table 4 reports the results of estimating the standard random effects model given by (4)
and (8) for the Income and Expenditure subsamples. The results in the table continue to
support the same basic message as before. The parameter estimates are almost identical to
those in the preceding table, as are the results of the lognormality tests: strong rejection of
the null in the expenditurebased sample, and failure to reject it in the incomebased
sample.
In summary, the empirical tests reported in this section show that in a large cross
country sample the observed distribution of per capita income is consistent with the
hypothesis of lognormality  regardless of whether income is measured before or after
taxes and transfers. In contrast, the same tests reject lognormality of the distribution of
per capita expenditure  although the lognormal specification can still account for a very
high proportion of the observed variation in quintile shares even in the expenditure data.
IV. Growth, inequality and poverty
The finding that per capita income follows a lognormal distribution has important
practical implications for assessing the respective contributions of growth and inequality
to poverty changes. The reason is that under lognormality we can derive simple closed
form expressions for these contributions, which depend only on the prevailing degree of
inequality, and on the poverty line relative to mean per capita income.
For concreteness, let us focus on the FosterGreerThorbecke (1984) [henceforth
FGT] class of poverty measures, which includes those most widely used in applied work.
They are given by the general expression:
10
z
P = z  x
z
o f (x)dx, (11)
where {0,1,2} is a parameter of inequality aversion, z is the poverty line, x is income,
and f(.) is the density function of income. When = 0, (11) reduces to the familiar
headcount ratio, which measures the share of the population below the poverty line z. For
= 1, we get the FGT measure P1, known as the poverty gap, which weighs each poor
individual by his / her distance to the poverty line  heuristically, it provides a measure
of the depth of poverty. Finally, for = 2, we have the squared poverty gap P2 , which
weights each poor individual by the square of his/her income shortfall; thus larger
shortfalls are weighted more than proportionately.
Denoting average per capita income by (i.e., E(x)=), the appendix shows that
under lognormality we can write
P = P (z / v,G) . (12)
Thus, poverty depends only on the Gini coefficient and the poverty line relative to
mean income. Equation (12) provides a starting point to analyze the relative contributions
of growth and changes in inequality to poverty reduction. For = 0, Figure 2 plots a set
of isopoverty curves (i.e., level sets of equation (12)); each of them depicts combinations
of Gini coefficients and mean per capita income / poverty line ratios (/z) that yield a
constant poverty headcount P0 .19 Curves to the Northeast of the graph correspond to
higher levels of the poverty rate.
The slope of these curves depicts the changing tradeoff between growth and
redistribution. The steeper the slope, the bigger the decline in the Gini coefficient
required to keep poverty constant in the face of a given decline in the ratio of mean
income to the poverty line. The curves become increasingly steep, and closer to one
another, as we move downward along them. In other words, the more equal and the
poorer the economy (as reflected, respectively, by a lower Gini coefficient and a lower
mean income / poverty line ratio), the bigger the change in the Gini coefficient required
to offset a given change in mean income relative to the poverty line  i.e., the more
effective growth will be relative to redistribution in attacking poverty. As the economy
becomes richer and more unequal  i.e., as we move to the Northwest of the graph  the
curves become less steep, and therefore a smaller change in the Gini coefficient is now
needed to offset a given change in mean income relative to the poverty line  i.e.,
distributional change now plays a relatively larger role in poverty changes.
We can gauge better the relative roles of growth and distributional change by
evaluating numerically the elasticity of poverty with respect to each of them. From (12),
for a given poverty line we can write
dP = d +G dG
. (13)
P G
19Of course, similar curves can be drawn for {1,2}.
11
Here and G respectively are the elasticities of P with respect to growth20 and
inequality. The appendix derives their exact expressions under the assumption of
lognormality, and shows that they depend only on z/v and G.
Tables 5 to 7 report, for the three FGT poverty measures, the values of and
G that result from various combinations of the Gini coefficient G and the ratio of per
capita income to the poverty line (/z). In the tables, G runs from 0.3 to 0.6 and /z from
1 to 6.21
Inspection of these tables confirms the wellknown result (e.g., Ravallion 1997,
2004; Bourguignon 2003) that the growth elasticity of the various FGT measures is
smaller (in absolute value) the higher the level of inequality. Thus, inequality hampers
the povertyreducing effect of growth, as stressed in the literature. In addition, however,
the tables show that poverty itself (as measured by low per capita income) is another
barrier to poverty reduction: in all three tables, for a given Gini coefficient, the growth
elasticity of poverty declines rapidly (in absolute value) as average income declines in
relation to the poverty line. This suggests a triple povertyreducing effect of growth: first,
the direct effect of income growth on the average level of income; second, the indirect
impact that arises from higher average income via the correspondingly higher growth
elasticity of poverty; and third, the indirect impact that arises from the higher average
income via the correspondingly higher growth elasticity of poverty.
Under most scenarios, inequality itself also has a doubly deterrent effect on
poverty reduction. In addition to lessening the growth elasticity of poverty, as just noted,
higher inequality also lessens the impact of progressive distributional change itself on
poverty  i.e., in tables 57 the inequality elasticity falls as inequality rises, for a given
value of average income relative to the poverty line. However, the relationship is highly
nonlinear, and at very low levels of development (captured in the tables by values of (v/z)
close to one) its sign is reversed, so that a higher Gini coefficient is associated with a
higher inequality elasticity, as shown in the last line of Tables 5 7.
The implication is that, given a common poverty line, poorer and more equal
countries may be in a position to afford some growthinequality tradeoffs. For low values
of (v/z), the povertyreducing effects of growth outweigh the povertyraising effects of a
worsening distribution of income. In other words, very poor countries may be willing to
tolerate modest deteriorations in income equality in exchange for faster growth. Such
tradeoff is much more problematic in richer and highly unequal countries, where small
20Strictly speaking, is the elasticity of poverty with respect to mean income, rather than growth, but we
follow the standard practice in the literature and use the term "growth elasticity" to refer to the change in
poverty that results when income increases by one percent, at given inequality.
21The range from 0.30 to 0.50 amounts roughly to the mean Gini plus/minus one standard deviation in the
overall sample (see Table 1). For the simulations, we raise the upper end to 0.60, a value reminiscent of the
inequality encountered in some Latin American countries. As for the ratio of mean income to the poverty
line, with poverty defined by the standard US$1 per person per day, the range from 1 to 6 is equivalent to
annual per capita income levels between US$365 and US$2140  which correspond roughly to those of
Zambia and El Salvador respectively. Alternatively, with poverty defined by a US$2 per person per day
poverty line, the range for (/z) amounts to per capita income levels from US$730 to US$4280 
approximately the income levels of Indonesia and Uruguay, respectively.
12
inequality increases have a much larger povertyraising effect, and hence policy makers
may be more willing to accept a modest growth decline in exchange for a reduction in
inequality.
Another way to gauge the relative importance of growth and redistribution from
the poverty reduction viewpoint is to measure the respective contributions of growth and
inequality shocks to the observed variation in poverty. This is the approach followed by
Kraay (2005). Under lognormality, it follows from (13) that the variance of poverty
changes can be approximated as
var(dP ) = ( ) var(d ) + (G ) var(dG) + 2 G cov(d
2 2 dG) . (14)
P G G
In the general case of nonzero covariance between growth and inequality changes,
absent any information on their mutual causal precedence, we can use the simplifying
assumption that half the covariance can be attributed to growth, and the other half to
inequality changes. In such case the share of the total variance attributable to the growth
component is:
var(dP ) = ( ) var(d ) + g cov(d
2 dG)
G . (15)
P var(dP )
P
However, to implement (15) numerically we need three more ingredients, namely
the variance of growth, the variance of inequality changes, and the covariance of growth
with inequality changes. Table 8 reports these statistics computed on the basis of two
alternative datasets: the one used thus far in this paper (i.e., Dollar and Kraay 2002), and
the Povmonitor22 database. Figure 3 presents the respective scatter plots. Apart from
coverage (much more limited in Povmonitor), the main difference between both
databases is that the Dollar and Kraay income data is based on National Accounts,
whereas that in Povmonitor is based on household surveys.
Inspection of Table 8 suggests that the different coverage of the two databases is
of little consequence for the volatility of changes in the (log) Gini coefficient: in both
cases the standard deviation is about 0.05. However, the volatility of growth differs more
markedly across the two databases: in the surveybased Povmonitor data, the standard
deviation of growth (0.06) is slightly higher than that of inequality changes, while the
opposite happens in the National Accountsbased Dollar and Kraay data, in which the
standard deviation of growth (0.04) is lower than that of inequality changes. On the other
hand, Table 8 also shows that growth and changes in inequality are nearly uncorrelated in
both databases  a conclusion confirmed by the scatter plots in Figure 3. Although the
correlation coefficients in the table are of opposite signs (negative in the Dollar and
Kraay data, and positive in the Povmonitor data), both are insignificantly different from
zero.23
22http://www.worldbank.org/povmonitor.
23Similar evidence is provided by Deininger and Squire (1996), Chen and Ravallion (1997) and Easterly
(1999).
13
Using these results, Tables 9 and 10 report the simulated values of expression (15)
for alternative values of the Gini coefficient and the mean income / poverty line ratio,
using the variance and covariance patterns of growth and inequality changes shown in the
preceding table. Table 9 uses the values from the Dollar and Kraay sample, while Table
10 uses the values from Povmonitor. In both tables, a number close to 1 means that in the
scenario in question changes in the poverty measure of interest are mainly driven by
growth, whereas a number close to zero means that they are mainly driven by changes in
inequality.24
The simulations in Tables 9 and 10 suggest that, consistent with the previous
discussion, in poorer and more equal countries growth accounts for a larger share of the
variance of poverty changes. Conversely, inequality changes tend to play a more
prominent role in richer and/or more unequal countries. This again suggests that in the
former countries growth should be expected to be the main driver of poverty reduction,
while the opposite would happen in the latter countries.
Notice also that, for any given configuration of Gini coefficient and per capita
income / poverty line ratio, the relative contribution of growth to the overall variance of
poverty changes declines as the poverty measure of interest varies from headcount
poverty to the poverty gap and then its square. As noted by Kraay (2005), this is a natural
consequence of the fact that more bottomsensitive poverty measures place more weight
on changes in the distribution of income than on changes in average income.
So far we have implicitly viewed alternative values of (/z) as reflecting different
levels of average per capita income with a given poverty line. But they could also be
interpreted the other way around, namely reflecting alternative poverty lines with a given
level of average per capita income. In this view, the numerical results above imply that as
the relevant poverty line z becomes more generous  i.e., as (/z) declines  the relative
role of growth in the overall variation of poverty changes must go up as well, and other
things equal this offers a rationale for shifting poverty reduction priorities in favor of
growthoriented policies. In the limit, as (/z) falls to zero so that the poverty rate
approaches 1 distributional change becomes completely ineffective for poverty
reduction. In other words, given the choice of poverty measure P and the initial
conditions in terms of average income and inequality, the location of the poverty line is a
key determinant of the relative effectiveness of growth and redistribution for poverty
reduction.
V. Conclusions
The focus on poverty reduction as the key objective of development policy has
opened a debate on the relative merits of aggregate growth and distributional change as
antipoverty strategies, and the conditions under which one may be more effective than
the other. In this paper we have reexamined that question using a parametric approach to
model the distribution of per capita income. The parametric approach has a long tradition
in the literature, going back over a century. One of its key advantages is that it allows a
24Note that in this calculation we are implicitly assuming that the same variance / covariance pattern applies
regardless of the particular configuration of Gini coefficient and mean income/ poverty line ratio.
14
systematic assessment of how countryspecific initial conditions  inequality, level of
development and poverty line definition  affect the povertyreduction effectiveness of
growth and income redistribution.
The paper's approach is based on the use of a lognormal approximation to the size
distribution of per capita income. We implement this approach in two stages. First, we
perform empirical tests of lognormality of the observed distribution of income, using a
large crosscountry income and expenditure distribution dataset covering over 100
countries and 40 years. Our testing strategy is based on the comparison of the observed
quintile shares of the distribution with their theoretical counterparts under the null
hypothesis of lognormality.
The empirical tests, performed on the full data sample as well as a variety of
subsamples, are very supportive of the lognormal approximation to the distribution of per
capita income, but less so for per capita expenditure. In the former case, the null of
lognormality cannot be rejected in any of the samples considered; in the latter case, it is
consistently rejected. In both cases, however, the lognormal specification yields a very
close approximation to the observed distribution.
Lognormality of the distribution of income allows us to derive, at the second stage
of the analysis, some qualitative and quantitative implications for the relative roles of
growth and inequality in poverty reduction under alternative initial conditions, using a
variety of poverty measures.
Our conclusions can be summarized in four main points. First, inequality hampers
poverty reduction, not only because of its negative impact on the growth elasticity of
poverty (as stressed in the literature) but, in most scenarios, also because of its negative
impact on the inequality elasticity of poverty. Second, for a given poverty line, the impact
of growth on poverty is stronger in richer than in poorer countries, and hence the latter
will find it harder than the former to achieve fast poverty reduction. Third, the share of
the overall variance of poverty changes attributable to growth should be generally lower
in richer and more unequal countries. And fourth, for given initial levels of development
and inequality, the relative povertyreduction effectiveness of growth and inequality
changes depends on the poverty line  the higher the poverty line, the bigger the role of
growth and the smaller the role of distributional change.
15
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18
Table 1. Descriptive statistics, by sample
Gini
Coefficient Q1 Q2 Q3 Q4
All (3,176 observations)
Mean 0.37 0.07 0.11 0.16 0.22
Median 0.35 0.07 0.12 0.16 0.22
Standard Deviation 0.10 0.02 0.03 0.03 0.02
Income (2,420 observations)
Mean 0.37 0.06 0.11 0.16 0.22
Median 0.35 0.06 0.12 0.17 0.23
Standard Deviation 0.10 0.02 0.03 0.03 0.03
Gross Income (1,472 observations) a/
Mean 0.40 0.06 0.11 0.15 0.22
Median 0.37 0.05 0.11 0.16 0.23
Standard Deviation 0.11 0.02 0.03 0.03 0.03
Net Income (892 observations) a/
Mean 0.33 0.07 0.12 0.17 0.23
Median 0.31 0.08 0.13 0.17 0.23
Standard Deviation 0.92 0.02 0.02 0.02 0.02
Expenditure (756 observations)
Mean 0.38 0.07 0.11 0.15 0.21
Median 0.36 0.07 0.12 0.16 0.22
Standard Deviation 0.87 0.02 0.02 0.02 0.01
Note: (a/) The total number of observations of the Gross Income and Net Income
subsamples does not equal to the number of observations of the Income subsample because
56 observations are not classified.
19
Table 2. Lognormality tests
Pooled OLS
Sample
Gross Net
All Income Expenditure Income Net Income
0.983 1.009 0.897 * 1.014 0.961 * 1.006
s.e. 0.014 0.015 0.012 0.021 0.016 0.017
0.002 0.001 0.013 ** 0.001 0.005 ** 0.001
s.e. 0.002 0.002 0.002 0.002 0.002 0.002
R2 0.96 0.96 0.98 0.95 0.98 0.98
# Observations 3,176 2,420 756 1,472 1,484 892
# Countries 130 98 65 75 97 55
Test of the joint hypothesis
Ho: =0; =1
(pvalue) 0.35 0.76 0.00 0.62 0.05 0.93
Notes: The table reports regression results with the observed quintile as dependent variable and the theoretical quintile as
explanatory variable. All regressions include a constant. Robust standard errors using a clustering procedure are reported
below the coefficients.
(*) Ho: =1 rejected at the 5%.
(**) Ho: =0 rejected at the 5%.
Table 3. Lognormality tests
Nested Error Component Model
Sample
Gross Net
All Income Expenditure Income Net Income
0.980 1.007 0.894 * 1.009 0.960 * 1.009
s.e. 0.015 0.016 0.012 0.023 0.016 0.017
0.002 0.001 0.013 ** 0.001 0.005 ** 0.001
s.e. 0.002 0.002 0.002 0.003 0.002 0.002
# Observations 3,176 2,420 756 1,472 1,484 892
# Countries 130 98 65 75 97 55
1.00E02 1.24E02 7.35E03 1.41E02 2.59E02 8.62E03
3.24E07 3.67E12 1.24E08 3.51E07 6.55E07 4.69E08
µ 2.68E03 3.42E03 2.10E03 5.20E03 1.90E03 1.92E03
Hypothesis tests
(pvalues)
Ho: =0; =1 0.41 0.90 0.00 0.92 0.05 0.80
Ho: =µ=0 0.00 0.00 0.02 0.00 0.00 0.00
Ho: =0 0.04 0.50 0.50 0.08 0.03 0.07
Ho: µ =0 0.00 0.03 0.08 0.08 0.00 0.01
Notes: The table reports regression results with the observed quintile as dependent variable and the theoretical quintile as
explanatory variable. All regressions include a constant. Robust standard errors are reported below the coefficients.
(*) Ho: =1 rejected at the 5%.
(**) Ho: =0 rejected at the 5%.
20
Table 4. Lognormality tests
Random Effects Model
Sample
Income Expenditure
1.007 0.894 *
s.e. 0.016 0.012
0.001 0.013 **
s.e. 0.002 0.002
# Observations 2,420 756
# Countries 98 65
Test of the joint hypothesis
Ho: =0; =1
(pvalue) 0.90 0.00
Notes: The table reports regression results with the observed quintile as
dependent variable and the theoretical quintile as explanatory variable. All
regressions include a constant. Robust standard errors are reported below
the coefficients.
(*) Ho: =1 rejected at the 5%.
(**) Ho: =0 rejected at the 5%.
21
Table 5. Headcount: Theoretical elasticities under lognormalty
Growth Elasticity
Gini Coefficient
Mean Income /
Poverty Line 0.30 0.40 0.50 0.60
6 6.05 3.25 1.95 1.22
3 3.94 2.18 1.33 0.86
2 2.80 1.60 1.01 0.66
1.5 2.06 1.23 0.80 0.54
1 1.16 0.78 0.55 0.39
Inequality Elasticity
Gini Coefficient
Mean Income /
Poverty Line 0.30 0.40 0.50 0.60
6 12.34 7.38 5.10 3.89
3 5.17 3.28 2.42 1.97
2 2.48 1.70 1.35 1.18
1.5 1.20 0.92 0.81 0.77
1 0.18 0.24 0.29 0.35
Note: The table reports the theoretical growth and inequality
elasticities, computed under the assumption of log normality, as a
function of the ratio of mean income/poverty line and the Gini
coefficient.
Table 6. Poverty gap: Theoretical elasticities under lognormalty
Growth Elasticity
Gini Coefficient
Mean Income /
Poverty Line 0.30 0.40 0.50 0.60
6 6.45 3.59 2.22 1.44
3 4.45 2.57 1.64 1.09
2 3.37 2.02 1.32 0.90
1.5 2.68 1.67 1.12 0.77
1 1.83 1.23 0.86 0.62
Inequality Elasticity
Gini Coefficient
Mean Income /
Poverty Line 0.30 0.40 0.50 0.60
6 14.08 9.01 6.64 5.38
3 6.70 4.69 3.73 3.22
2 3.82 2.92 2.49 2.27
1.5 2.36 1.98 1.81 1.73
1 1.03 1.05 1.08 1.13
Note: The table reports the theoretical growth and inequality
elasticities, computed under the assumption of log normality, as a
function of the ratio of mean income/poverty line and the Gini
coefficient.
22
Table 7. Squared poverty gap: Theoretical elasticities under lognormalty
Growth Elasticity
Gini Coefficient
Mean Income /
Poverty Line 0.30 0.40 0.50 0.60
6 6.79 3.85 2.41 1.59
3 4.84 2.85 1.84 1.24
2 3.80 2.32 1.53 1.05
1.5 3.12 1.97 1.33 0.92
1 2.27 1.52 1.06 0.76
Inequality Elasticity
Gini Coefficient
Mean Income /
Poverty Line 0.30 0.40 0.50 0.60
6 15.58 10.34 7.83 6.46
3 7.98 5.80 4.71 4.11
2 4.92 3.88 3.34 3.05
1.5 3.31 2.81 2.55 2.42
1 1.73 1.69 1.68 1.70
Note: The table reports the theoretical growth and inequality
elasticities, computed under the assumption of log normality, as a
function of the ratio of mean income/poverty line and the Gini
coefficient.
Table 8. GDP growth and changes in inequality: Descriptive statistics
Standard Deviation Correlation between
Change in GDP Change in Inequality and
Source Inequality Growth GDP Growth
Dollar and Kraay (2002) a/ 0.054 0.037 0.020
Povmonitor b/ 0.049 0.063 0.070
Notes:
(a/) Based on PWT national accounts.
(b/) Based on survey data (http://www.worldbank.org/povmonitor).
23
Table 9. Share of variance in poverty changes due to growth a/
(Based on Dollar and Kraay (2002) database)
Headcount
Gini Coefficient
Mean Income /
Poverty Line 0.30 0.40 0.50 0.60
6 0.28 0.24 0.19 0.14
3 0.48 0.41 0.32 0.23
2 0.66 0.58 0.47 0.33
1.5 0.82 0.73 0.60 0.44
1 0.98 0.94 0.84 0.66
Poverty Gap
Gini Coefficient
Mean Income /
Poverty Line 0.30 0.40 0.50 0.60
6 0.25 0.20 0.15 0.11
3 0.41 0.32 0.24 0.16
2 0.55 0.43 0.31 0.20
1.5 0.67 0.53 0.38 0.24
1 0.83 0.68 0.50 0.32
Squared Poverty Gap
Gini Coefficient
Mean Income /
Poverty Line 0.30 0.40 0.50 0.60
6 0.23 0.18 0.13 0.09
3 0.37 0.28 0.20 0.13
2 0.48 0.36 0.25 0.16
1.5 0.58 0.44 0.30 0.19
1 0.73 0.56 0.39 0.24
Notes: The table reports the share of the overall variance of the
poverty measures, computed under the assumption of
lognormality, attributable to income growth as a function of the
ratio of mean income/poverty line and the Gini coefficient.
(a/) Calculated using the variances and covariance of growth and
changes in inequality from the Dollar and Kraay (2002) database.
24
Table 10. Share of variance in poverty changes due to growth a/
(Based on Povmonitor database)
Headcount
Gini Coefficient
Mean Income /
Poverty Line 0.30 0.40 0.50 0.60
6 0.12 0.10 0.07 0.05
3 0.27 0.22 0.16 0.10
2 0.47 0.37 0.27 0.16
1.5 0.68 0.56 0.40 0.24
1 0.98 0.90 0.73 0.46
Poverty Gap
Gini Coefficient
Mean Income /
Poverty Line 0.30 0.40 0.50 0.60
6 0.11 0.08 0.06 0.03
3 0.22 0.15 0.10 0.06
2 0.34 0.23 0.14 0.08
1.5 0.47 0.32 0.19 0.10
1 0.70 0.49 0.29 0.15
Squared Poverty Gap
Gini Coefficient
Mean Income /
Poverty Line 0.30 0.40 0.50 0.60
6 0.10 0.07 0.05 0.03
3 0.19 0.13 0.08 0.04
2 0.28 0.18 0.11 0.06
1.5 0.37 0.24 0.14 0.07
1 0.55 0.35 0.20 0.10
Notes: The table reports the share of the overall variance of the
poverty measures, computed under the assumption of
lognormality, attributable to income growth as a function of the
ratio of mean income/poverty line and the Gini coefficient.
(a/) Calculated using the variances and covariance of growth and
changes in inequality from the Povmonitor database
(http://www.worldbank.org/povmonitor).
25
Figure 1. Empirical and Theoretical Quintiles
(a) Full Sample (b) Income
s 0.40
s 0.40
ilet 0.30 ilet 0.30
quinlaci 0.20 quinlaci 0.20
pir 0.10 pir 0.10
mE 0.00 mE
0.00
0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25
Theoretical quintiles Theoretical quintiles
(c) Expenditure (d) Gross Income
s 0.40 0.40
ile iles
int 0.30 0.30
qula 0.20 intuqla 0.20
rici 0.10 rici 0.10
Emp 0.00 Emp
0.00
0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25
Theoretical quintiles Theoretical quintiles
(e) Net (f) Net Income
s 0.40
s 0.40
ilet 0.30 ilet 0.30
quinlaci 0.20 quinlaci 0.20
pir 0.10 pir 0.10
mE0.00 mE
0.00
0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25
Theoretical quintiles Theoretical quintiles
26
Figure 2. IsoPoverty Curves for the Headcount Ratio
0.70
P0 = 0.7
0.65
P0 = 0.6
P0 = 0.5 0.60
P0 = 0.4 0.55
Gi
P0 = 0.3 n
P0 = 0.2 0.50 iCoeffi
0.45
P0 = 0.1 ci
0.40 ent
0.35
0.30
0.25
0.20
6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0
Mean Income / Poverty Line
Figure 3. Income Growth and Changes in Inequality
(a) Dollar and Kraay (2002) Database
y 0.4
0.2
inequalit 0.0
in 0.25 0.20 0.15 0.10 0.05 0.20.00 0.05 0.10 0.15 0.20
angehC 0.4
0.6
Growth in mean survey income
(b) Povmonitora/ Database
y 0.4
0.2
Inequalit 0.0
in 0.25 0.20 0.15 0.10 0.05 0.20.00 0.05 0.10 0.15 0.20
0.4
Changes 0.6
Growth in mean survey income
Note: (a/) http://www.worldbank.org/povmonitor.
27
Appendix
In this appendix we derive the growth and inequality elasticities of the FGT family of
poverty measures, given by
z
P = z  x
z
o f (x)dx (A1)
where {0,1,2}. When log x is normally distributed with mean µ and variance 2 we
can denote f(x) dx by d(x/µ,2), so that we can express the different FGT measures as:
z
P0 = d(x / µ, )
2 (A2)
o
z
P1 = z  x z z
z xz 2 (A3)
o d (x/µ, ) = d(x/µ, ) d(x/µ, )=
2 2
o o
z
P2 = z  x2d 2
z
o (x/µ, ) =
(A4)
z z z
d(x / µ, )  2
2 xzd(x / µ, ) + 2 zx2d (x/µ, ).
2
o o o
In order to derive the growth and inequality elasticities of the FGT family under
lognormality we make use of the following result:
z
x jd(x / µ, ) = e jµ+12 j2 2 z
d (x/µ + j ,)dx,
2 (A5)
o o
which follows from Theorem 2.6 in Aitchison and Brown (1966). Using (A5) we can
express (A2A4) respectively as:
z
P0 = d(x / µ, ) ,
2 (A6)
o
z
P1 = d(x / µ, ) 
2 z
d(x / µ + , ) ,
2 2 (A7)
z
o o
z
P2 = d(x / µ, )  2 2 z
2 2 v 2 z
d(x / µ + 2 , ) .(A8)
2 2
z
o o d(x / µ + , ) + e2
z o
Combining these expressions with the relationship linking the normal and lognormal
distributions
28
z
d(x / µ, ) = P(x < z) = P(log x < log z) = ((log z  µ)/ ),
2
o
where (.) is the standard normal cumulative density function; and using also the identity
linking average per capita income to the mean and variance of log income, log =
µ+2/2, (A6)  (A8) can be rewritten as
P0 = (log(z / )
(A9)
+ ),
2
P1 = (log(z / ) log(z/)
(A10)
+ ) (  ),
2 z 2
P2 = (log( z / ) + ) 2 (
log(z/) 2 z / )  3
) . (A11)
2 z  )+ e2(log(
2 z 2
which shows that P = P (z /v, ), and inverting equation (1) in the text we can further
write P = P (z / v,G) .
Growth elasticities
For P0 , the growth elasticity can be found in Bourguignon (2003). It is given by
=0 1 dP0 =  [log(z/
1 ) + ]
1
(A12)
d log( ) P0 2
where (.) = (.)/ (.) , and (.) denotes the standard normal density. In turn, for
{1,2} we can use a result by Kakwani (1990):
= (P  P1) . (A13)
P
Inequality elasticities.
For P0 , the elasticity of poverty with respect to the standard deviation of log per capita
income is given by:
P0 ) + ][(12 
1
P0 = [log(z/
log(z / ))] (A14)
2
In turn, from (1) in the text
dG = 2( )d (A15)
2
and hence the Gini elasticity of poverty is given by
29
G =
0 G P0 = [log(z/ ) + ][(12 
1 log(z / ))] /[ 2( / 2) /G] (A16)
P0 G 2
2
The sign of this expression depends on that of log(z/); it is positive for > z 
2
i.e., when mean income is above the poverty line  but becomes negative as( / z) 0 .
In order to derive the expressions for the Gini elasticity of P1 and P2 we define:
a log(z / )
(A17)
+
2
b a  = log(z / )
(A18)

2
c a + = log(z / ) 3
. (A19)
+
2
First, we consider the elasticities of P1 and P2 with respect to the standard deviation of
log per capita income, which follow from (A10) and (A11):
P1 (a)(b)z(b)(a)
P1 = (A20)
P1
2 2 2 )  ))
3
P2 z 2
P2 = (a)(b)2z(b)(a)+ e (2 (c)+(c)(log(z/ .(A21)
P2
Using again (A15) we finally get:
G = (
1 (a)(b)z(b)(a)) /( 2(
(A22)
P1 2 ) G),
and
G = (
2 (a)(b)2z(b)(a)+ e (2 (c)+(c)(log(z/) 3)))
2 2 2
z /( 2(
P2 2 ) G).
(A23)
30