ï»¿ WPS6361
Policy Research Working Paper 6361
The Unfairness of (Poverty) Targets
Melanie Allwine
Jamele Rigolini
Luis F. LÃ³pez-Calva
The World Bank
Latin America and the Caribbean Region
Office of the Chief Economist
and Poverty Reduction and Economic Management Unit
February 2013
Policy Research Working Paper 6361
Abstract
Adopted on September 8, 2000, the United Nations significant negative relation between initial average
Millennium Declaration stated as its first goal that income and poverty reduction performance, with the
countries â€œâ€¦[further] resolve to halve, by the year 2015, poorest countries in the sample going from the worst to
the proportion of the worldâ€™s people whose income is the best performers in poverty reduction. The analysis
less than one dollar a day and the proportion of people also quantifies how much poorer countries would have
who suffer from hungerâ€¦â€? Each country committed scored better, had they had the same level of initial
to achieve the stated goal, regardless of their initial average income as wealthier countries. The results suggest
conditions in terms of poverty and inequality levels. a remarkable change in poverty reduction performance,
This paper presents a framework to quantify how much in addition to the reversal of ranks from worst to best
initial conditions affect poverty reduction, given a performers. The application of this framework goes
level of â€œeffortâ€? (growth). The framework used in the beyond poverty targets and the Millennium Development
analysis allows for the growth elasticity of poverty to Goals. Given the widespread use of targets to determine
vary according to changes in the income distribution resource allocation in education, health, or decentralized
along the dynamic path of growth and redistribution, social expenditures, it constitutes a helpful tool to
unlike previous examples in the literature where this is measure policy performance toward all kinds of goals.
assumed to be constant. While wealthier countries did The proposed framework can be useful to evaluate the
perform better in reducing poverty in the last decade and importance of initial conditions on outcomes, for a wide
a half (1995â€“2008), assuming equal initial conditions, array of policies.
the situation reverses: the paper finds a statistically
This paper is a product of the office of the Chief Economist and Poverty Reduction and Economic Management Unit, Latin
America and the Caribbean Region. It is part of a larger effort by the World Bank to provide open access to its research
and make a contribution to development policy discussions around the world. Policy Research Working Papers are also
posted on the Web at http://econ.worldbank.org. The author may be contacted at jrigolini@worldbank.org.
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Produced by the Research Support Team
The Unfairness of (Poverty) Targets
Melanie Allwine Jamele Rigolini Luis F. LÃ³pez-Calva 1
George Washington University World Bank World Bank
1
We would like to thank James Foster, Norman Loayza, Pedro Olinto, Renos Vakis and seminar participants at George
Washington University for useful early discussions. 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.
Introduction
Adopted on September 8 of 2000, the United Nations Millennium Declaration stated as its first goal that
countries â€œâ€¦[further] resolve to halve, by the year 2015, the proportion of the worldâ€™s people whose
income is less than one dollar a day and the proportion of people who suffer from hungerâ€¦â€? (General
Assembly resolution 55/2). The resolution was adopted by all 189 Member States of the United Nations,
over 140 of them represented directly by their Head of State. Each country committed to achieve the
stated goal, regardless of their initial conditions in terms of poverty and inequality levels.
The target of halving extreme poverty between 1990 and 2015 is on track to being met. The proportion
of people living on less than $1.25 a day in purchasing power parity (PPP) terms has already declined
from 47 percent in 1990 to 24 percent in 2008, a reduction of more than 2 billion to less than 1.4 billion
(United Nations, 2012). Yet, the UN progress report shows enormous differences across continents. The
target will be met mainly due to the impressive poverty reduction achievements in Southeast Asia.
Between 1990 and 2008, Southeast Asia reduced extreme poverty from 45 percent of the population to
17 percent. Looking at China alone, the progress is even more remarkable, with extreme poverty falling
from 60 percent in 1990 to 13 percent in 2008. On the other extreme, during the same period, Sub-
Saharan Africa reduced its extreme poverty by a modest 9 percentage points, from 56 to 47 percent.
Heterogeneity in achievements tends to be associated with differences in the rates of economic growth.
Poverty plummeted alongside Chinaâ€™s stunning economic performance of the past three decades; while
the more modest growth performance of Sub-Saharan countries led to less impressive rates of poverty
reduction. Now that growth has picked up in Sub-Saharan Africa, the widespread expectation is that
poverty reduction will show more dramatic results.
This paper does not question the fundamental role that growth has played, and always will as a
necessary condition, in achieving sustainable poverty reduction. It shows, however, that poverty
achievements are substantially affected by the way that poverty is measured and targets are set. While
the Millennium Declaration encouraged all developing countries to pursue the goal of cutting poverty in
half, it did not account for the fact that countries start off with different initial conditions, specifically in
terms of where the poverty line is located with respect to the initial distribution of income. In fact,
countries are evaluated according to the same overarching target, independently of where they started.
But the same policies and growth rates could have a dramatically different impact on poverty reduction,
depending on where the poverty line stands with respect to the distribution of income. This paper
2
brings forth two elements to prior analysis on the subject of how initial conditions matter. It quantifies
the effect of initial conditions (that is, average income per capita and inequality in the reference year)
on outcomes given a level of â€œeffortâ€? (i.e. growth in per capita income). The underlying motivation
supporting such a distinction is that, in any benchmarking exercise, countries and policymakers cannot
be â€œheld accountableâ€? for the initial distribution of income, while they should (at least to some extent)
be held accountable for the evolution of the income distribution. This evolution is characterized, in our
exercise, by the growth in per capita income. 2
The framework used in the analysis allows for the growth elasticity of poverty to vary according to
changes along the dynamic path of growth and redistribution, unlike previous mechanical examples in
the literature where this is assumed to be constant. The framework is also important in the context of
aid allocation decisions (both domestic and international) that aim at rewarding â€œeffortâ€? and maximizing
the poverty impact of transfers.
Specifically, the analysis quantifies how much poorer countries would have scored better, had they had
the same level of initial per capita income as wealthier countries, by attributing to each country the
median initial per capita income measured across countries. The results suggest a remarkable change in
poverty reduction performance: while wealthier countries did perform better in reducing poverty in the
last decade and the half (1995-2008), assuming equal initial conditions, we find the poorest countries in
the sample going from the worse to the best performers in poverty reduction. The reversal of the
relationship, and the large magnitude of such reversal extends to other dimensions of poverty beyond
the headcount index, such as the poverty gap.
To measure whether the MDG targets may have been markedly unfavorable towards the poorest
countries, the simulations are repeated for those countries which in the benchmark year of the MDGs
(1990) had rates of extreme poverty above 10 percent. Interestingly, the difference between actual
poverty reduction rates and the counterfactual ones becomes less striking, a result of this subgroup of
countries being more similar in initial per capita income. Nonetheless, substantial differences subsist.
For example, if China would have had in the 1990s the same per capita income as Ghana, with the same
growth performance, it would have faced half of a percentage point less in annual poverty reduction.
Accumulating this difference over 20 years, China would have had in 2008 extreme poverty rates almost
10 percentage points higher than the actual ones. As such, initial conditions appear to be an important
2
While changes in inequality can also be characterized as â€œeffort,â€? they impact less poverty reduction in the long
run, and remain more challenging to simulate (see the discussion in the methodology section).
3
factor in the success of the poverty reduction efforts. This is particularly relevant given that, by
disentangling policymakersâ€™ efforts from initial conditions out of their control, achievements can be
benchmarked more accurately.
With the elements presented in this paper, we aim to contribute to the discussion of the new definition
of objectives in the post-2015 MDG agenda. Moreover, the application of such a framework goes
beyond the Millennium Development Goals. Given the widespread use of targets to determine resource
allocation, in education, health, or decentralized social expenditures, it can constitute a helpful tool to
measure policy performance towards all kinds of goals. The proposed framework can be useful to
evaluate the importance of initial conditions on outcomes, for a wide array of policies. While the
present analysis focuses on poverty, the methodology can be used as a principle to evaluate
performance indicators in a more general context.
Basic Framework and Related Literature
The fact that initial per capita income and inequality levels matter in poverty reduction for a given a
level of growth has been discussed previously in the literature. It has been first analyzed by Bourguignon
(2002), who looks, similarly to us, at the growth elasticity of poverty to explain heterogeneity in poverty
reduction across countries. The research analyzes the identity that links poverty reduction, mean
income growth, and distributional change. The growth elasticity of poverty is found to be a decreasing
function of the development level of a country and of the degree of inequality of the income
distribution, under the assumption that income is log-normally distributed. Per the basic identity
analyzed, a permanent redistribution of income plays two roles in poverty reduction: an instantaneous
reduction through the distribution effect; and (as it contributes to a permanent increase in the growth
elasticity of poverty) an acceleration of poverty reduction for a given rate of economic growth. Initial
conditions thus, can play a crucial role in the transformation of growth into poverty reduction, with
poorer countries having in general a lower elasticity of poverty to growth. The paper, however, fails to
quantify these effects, and also to allow the elasticity of poverty reduction to vary along the growth and
inequality path.
Ravallion (2012) analyzes the implications of initial distribution and high initial poverty to explain why
poverty convergence is not observed across countries. Using a recent dataset for 100 developing
countries, his findings indicate that, despite the so-called advantages of backwardness and growth in
the development process, countries starting with higher poverty rates do not see higher proportionate
4
rates of poverty reduction. Two â€œpoverty effectsâ€? work against mean-convergence. A high level of initial
poverty has an unfavorable impact on growth, such that countries with a higher initial incidence of
poverty are likely to experience lower rates of growth, controlling for the initial mean. At the same time,
a high poverty rate makes economic growth less effective in reducing poverty. In other words, the
advantage of starting out with a low mean income is wasted for many poor countries, given their high
poverty rates. His analysis, however, focuses more on causal links, and less on the â€œmechanicalâ€? impact
of initial conditions. Moreover, by using regression methodologies, the analysis only accounts for a
fraction of the impact of initial conditions, and fails to capture nonlinear effects.
Along this line of reasoning, Easterly (2008) emphasizes how initial conditions have slowed down
poverty reduction in Africa. He makes the point that the continentâ€˜s progress towards the poverty
reduction Millennium Development Goal is unfairly benchmarked against that of other regions, given
Africaâ€˜s poorer initial conditions. Given a log-normal distribution of income, low poverty elasticity of
growth is found in a country with a low initial per capita income. In this sense, a higher growth of mean
income would be required to achieve the same percentage reduction in poverty than in a country with a
higher per capita income. By having the lowest per capita income of any region, Africa is disadvantaged
in its goal of cutting poverty in half. The continent would need a higher economic growth than other
regionsâ€™ to attain the goal, in order to compensate for its low-poverty elasticity. The argument lies at the
heart of our analysis. The discussion in Easterly (2008), however, remains theoretical, as he makes no
attempt to quantify the extent to which Sub-Saharan African countries may remain disadvantaged. In
fact, in the context of the MDGs, initial conditions seem to matter less than one would have thought.
In order to analyze this process, the dynamic links between initial conditions, growth and poverty
reduction should be understood. Figure 1 summarizes the main theoretical argument on which the
present analysis is based. Consider two countries with identical distributions of income, but with
different means: average income in country A is lower than average income in country B. Because the
two distributions intersect the poverty line at different places, given equal growth rates in average
income, poverty reduction rates will look differently as different numbers of people will â€œcrossâ€? the
poverty line. In our example, the poverty line crosses both distributions on the right of their apex (i.e.,
we consider the case of two countries where a majority of the population is poor), and therefore, for
equal growth rates poverty achievements will be more marked in the richer country. But, as Figure 1
shows, this does not always have to be the case, in particular if we compare countries with large
differences in average income.
5
Figure 1: Distribution of income and poverty reduction
A
Poverty line
B
Density
Income
In setting universal targets of poverty reduction, many countries could thus be penalized simply due to
their initial distribution of incomes. Uganda, for instance, had in 1990 a GDP per capita of 563 dollars (in
PPP terms), less than half the GDP per capita of China (1,100 dollars). Because of these different initial
conditions, in order to achieve the same poverty reduction rates as China, Ugandaâ€™s growth rates would
have had to be even higher than the Chinese ones. Setting equal poverty reduction targets for China and
Uganda may thus not be fair.
As discussed, we are not the first to make the argument, but previous paper did not quantify these
effects. Moreover, as Figure 1 shows, the growth elasticity of poverty varies along the growth path,
hence by just considering initial elasticities (or average ones, as in Ravallion, 2012)), cross-country
differences could be grossly over- or under-estimated. In this paper we aim at relax the constant
elasticity assumption, and look for the actual growth realizations, to find out how much have actual
poverty reduction achievements been affected by differences in initial conditions.
6
Data and Methodology
The main data source of our analysis is the World Bankâ€™s Povcal database
(http://iresearch.worldbank.org/PovcalNet). The Povcal database provides information on mean
expenditures or income (in 2005 USD PPP terms) and the Lorenz curve of expenditures/income
distributions for specific countries and years, both of which are estimated from nationally representative
household surveys. The Povcal database includes data on 131 countries. Since no parametric
information is provided for Latin America, we also draw information directly from nationally
representative household surveys for 18 Latin American countries.
The Povcal database reports Lorenz curves using two approximation methods: the General Quadratic
and the Beta Lorenz curve approximations. Combined with information on mean expenditures or
income, these approximations allow estimating poverty headcount and gap indexes quite accurately, as
well as Gini coefficients (see Datt, 1998, and the Appendix for details on all approximations used in the
analysis). For each country in Povcal, we use the approximation that best fits the actual poverty
headcounts (see below). Additionally, for the 18 Latin American countries, we first convert income into
2005 USD PPP using conversion factors reported by the International Comparison Program, and then
estimate ourselves the best parametric approximation of the income distribution.
To adequately represent poverty in all the countries analyzed, we use a 2.5 dollars a day poverty line.
We base our estimation for each country either on income or expenditures, as reported by PovCal or the
surveys (given the focus of the analysis on poverty change within countries, we do not apply any form of
correction for harmonizing income and expenditure data). The timeframe we consider in our simulations
is circa 1995 to 2008, which allows us to capture the largest number of countries. To correct for the fact
that, for some countries, the time frame slightly diverts from these two reference years, we report
results in terms of average annual changes. Moreover, to achieve comparability with the Millennium
Development Goals (MDG) reports, we also repeat the analysis for a subset of countries with high initial
rates of extreme poverty (more than 10 percent of the population with income/expenditures below 1.25
dollars a day), and look at changes between 1990 and 2008.
In our main analysis, out of the 84 countries available in the Povcal (including the Latin American ones),
we exclude those that had poverty rates below 5 percent in 1995, since our approximation would
capture with high errors the actual poverty changes in these countries. We also exclude Georgia from
the analysis, as with poverty rates more than doubling in the period, it is a clear outlier. We are thus left
with 74 countries (see Table 1). For the same reason, in the analysis focusing on the MDGs, we only
7
consider countries that had a rate of extreme poverty in 1990 above 10 percent. For this simulation we
are left with 38 countries (see Table 2).
Accuracy of the parametric approximation
The core of our analysis is based on numerical simulations that apply changes to parameterized income
distributions. The accuracy of such numerical approximations is therefore of central importance.
Accordingly, in what follows, we compare the accuracy of each parametric approach using data for Latin
America, where we can use the survey data to obtain direct estimates of the poverty rates.
Figure 2: Actual vs. Predicted headcount ratios (Latin America, 1995)
0.7
Actual
0.6 Best fit of General Quadratic and Beta Lorenz
Lognormal
0.5 Nonparametric
0.4
0.3
0.2
0.1
0
Figure 2 presents four different estimates of the headcount ratio (at a threshold of 2.5 dollars a day in
PPP terms). The first is the actual poverty headcount ratio estimated directly from household surveys
(our estimate of reference for the comparison). The second estimate represents the headcount ratio
computed from the best fit between the General Quadratic and the Beta Lorenz approximations. The
third estimate is the headcount ratio computed from an income distribution parameterized with a
lognormal approximation. Finally, the last estimate is the headcount ratio computed from a fully
nonparametric approximation of the shape of the income distribution. Among the three
parameterizations, the one that deviates significantly from the actual poverty estimate is the lognormal
8
approximation. This is because the lognormal relies only on one parameter (the standard deviation) to
capture the shape of the income distribution, while the other approximations rely on two or more, and
are thus better able to capture the tails of the distributions. Whenever possible, we will therefore use
the best fit of the General Quadratic and the Beta Lorenz curves (as reported by the PovCal database,
and, for LAC by the comparison with actual data), and leave the use of the lognormal approximation
only for simulations that will not allow for this.
Simulations
The heart of our analysis lies in simulating counterfactual poverty changes between 1995 and 2008,
where we keep country-specific â€œperformances,â€? but attribute equal â€œinitial conditionsâ€? to countries.
These two concepts remain specific to the way we structure the simulations, and deserve some
attention.
Let us consider a space of income distributions f(y,Âµ,G) characterized by two parameters: average
income Âµ, and an inequality index G. We then consider a poverty index H(z,Âµ,G) that can be mapped to
each income distribution as follows:
âˆž
H ( z , Âµ , G ) = âˆ« h( y, z ) f ( y, Âµ , G )dy (1)
0
where z can be thought of as an absolute reference threshold, which we shall denote as the poverty line,
and h(y,z) a function that can be integrated. Observe that common indexes, such as the poverty
headcount and the poverty gap, can all be expressed in a form compatible with (1).
Consider then, two income distributions, í µí±“(í µí±¦, í µí¼‡0 , í µí°º0 ), and í µí±“(í µí±¦, í µí¼‡1 , í µí°º1 ). Think of the first set of
parameters as characterizing the distribution of income in a given country in period zero (i.e., 1995 for
our main simulations), and of the second set as characterizing the distribution of income in period one
(i.e., 2008). For a poverty line z, changes in poverty between the two periods can then be expressed as:
H ( Âµ1 , G1 ) âˆ’ H ( Âµ 0 , G0 ) .
Observe that, up to this point, we could run the comparison using actual poverty rates. The purpose of
the parameterization, however, is to allow us to simulate counterfactual changes in poverty, so we
proceed as follows. In the simulation, we aim to distinguish between initial conditions (í µí¼‡0 , í µí°º0 ) and
policy performance (âˆ†í µí¼‡â?„í µí¼‡0 , âˆ†í µí°º â?„í µí°º0 ). The underlying motivation supporting such a distinction is that, in
9
any benchmarking exercise, countries and policymakers cannot be â€œheld accountableâ€? for the initial
distribution of income, while they should (at least to some extent) be held accountable for the evolution
of the income distribution. This evolution is characterized, in our exercise, by the growth in mean
income and changes in inequality.
The idea is then to isolate performance from initial conditions in poverty reduction. To do this, in
Ì… ) of the
computing counterfactual poverty changes, we attribute to each country the median (í µí¼‡Ì…0 , í µí°º0
initial parameters in our sample of countries, but keep, for each country, the country-specific
í µí±? í µí±? í µí±? í µí±?
performance ï¿½í µí¼€í µí¼‡ = âˆ†í µí¼‡í µí±? â?„í µí¼‡0 , í µí¼€í µí°º = âˆ†í µí°º í µí±? â?„í µí°º0 ï¿½. Our counterfactual simulated poverty change can
therefore be expressed as follows:
âˆ†H ( ) 0 (
Ë† c = H Îµ c â‹… Âµ ,Îµ c â‹… G âˆ’ H Âµ M ,G M
Âµ 0 G 0 0 ) (2)
The simulated poverty change considers therefore the country-specific improvements in per capita
í µí±? í µí±?
income and inequality ï¿½í µí¼€í µí¼‡ , í µí¼€í µí°º ï¿½ that the countries actually experienced, but applies them to median
initial conditions, which are kept equal across countries.
We conclude with an observation about the alternative parameterizations that drive our simulations.
Observe that our simulations are based on the assumption that it is possible to capture the shape of the
income distribution â€“ which drives inequality â€“ by a single inequality parameter, G. This is a rather
restrictive assumption, which may hamper the quality of the numerical approximation. For instance, the
lognormal approximation, which summarizes an income distribution based on its average and standard
deviation, tends to approximate poorly the tails of a distribution, which can generate substantial errors
in approximating poverty rates (Figure 2), and even worse errors when approximating changes in
poverty (Figure 3). This is the reason why both the General Quadratic and the Beta approximations used
in Povcal, as mentioned above, are based on multiple parameters that are calibrated to achieve a good
fit of the tails.
Accuracy notwithstanding, from the point of view of our simulations, the problem with using more than
one parameter in approximating the shape of the income distribution is that it breaks the bijective
relation between the parameter â€œGâ€? in the poverty index (1), and any economic index of inequality that
has economic meaning. Consider, for instance, measuring progress in inequality by the Gini coefficient,
10
and the case of the General Quadratic approximation. Under such an approximation, the shape of an
income distribution is approximated by three parameters, a, b and c (see the Appendix). Any change in
the Gini coefficient can thus be associated with infinite combinations of changes in these three
parameters; but these combinations have different impacts on the ultimate change in poverty, as they
affect the shape of the income distribution differently. With any approximation that uses two or more
parameters to approximate the shape of an income distribution, it is therefore not possible to conduct
counterfactual simulations as in (2).
In running the simulations we are thus faced with two choices: either we run the simulation using the
lognormal approximation â€“ where we can simulate changes both in average per capita income and in
the Gini coefficient â€“ or we only simulate changes in average per capita income (keeping inequality
constant at its initial level) using, however, a more accurate approximation. Neither approach is perfect,
hence, the decision of which one to use ultimately boils down to minimizing approximation errors. For
these purposes, in Figure 3 we compare for Latin America (where we have the actual household surveys)
the actual changes in poverty against three simulation alternatives that all use the lognormal
parameterization: 3 the first approach takes into consideration the actual changes in per capita income,
but keeps income inequality at its initial level; the second approach takes into consideration actual
changes in inequality, but keeps per capita income at its initial level; and, finally, the third approach
takes into consideration both actual changes in per capita income and inequality.
3
The simulations use the first and second moments of the actual income distributions as the parameters for the
lognormal approximation.
11
Figure 3: Contribution to poverty reduction of mean income growth vs. changes in inequality, Latin America, ca. 1995-2008
0.05
0
Reduction in headcount ratio
-0.05
-0.1
-0.15
Actual
-0.2 Simulations with respect to mean only
Simulation with respect to gini
Simulations with respect to mean and gini
-0.25
Two facts emerge. First, while the lognormal is already a poor parameterization to approximate poverty
levels, regarding approximating poverty changes, its performance is even worse: in some cases, the
approximation misrepresents changes in poverty by up to 30 percent of the actual value. The use of the
lognormal approximation should thus be avoided, when possible. Second, for countries where poverty
declined significantly between 1995 and 2008, the difference between considering both changes in per
capita income and inequality, and only changes in per capita income, remains relatively small. This
finding suggests that, for countries that faced large reductions in poverty, changes in per capita income
preponderantly capture the variation in poverty. Given our focus on long term changes in poverty, for
the sake of greater accuracy, in what follows we thus use the more accurate General Quadratic and Beta
approximations, and run all the simulations maintaining income inequality constant at its country-
specific initial level.
Main Findings
Our first exercise is to assess how ranks in poverty reduction performance are affected by the
assumption of equal initial conditions across countries. The simulated ranks present the great advantage
of depending only on the assumption of giving equal initial levels of average income í µí¼‡0 to all countries,
12
and not on the specific value of the initial average per capita income that we choose for the simulation.
The reason is simple. Observe that the poverty index (1) decreases monotonically with average income.
Thus, once initial conditions have been equalized, there is a bijective relation between growth in
average income, and the actual poverty reduction performance. Hence, in the simulations, only growth
but not the value chosen as an equalizing initial condition, affects the rank.
Figure 4 shows the actual ranks in poverty reduction performance (measured as the annual percentage
change in poverty between 1995 and 2008), and the simulated rank under the assumption of equal
initial average income across countries. The counterfactual picture we obtain changes dramatically the
performance of poorer vs. richer countries. Before assuming initial conditions, we observe an increasing
(and statistically significant) relation between initial average income, and the rank in poverty reduction
performance (measured as percentage change in poverty). The poorest countries in the sample, such as
Malawi, Madagascar and Burundi score around the worse in the sample, while many of the wealthiest
countries, such as Russia, Chile and Malaysia score high. It would be tempting to conclude that richer
countries are better managed to reduce poverty, but the conclusion would be misleading. When we give
each country equal initial conditions, the picture reverses: we observe a statistically significant negative
relation between initial average income, and poverty reduction performance. For instance, Mali, Nepal,
and Niger â€“ which previously performed poorly â€“ are now among the best performers in poverty
reduction.
13
Figure 4: Actual and simulated poverty reduction (rank; annual percentage change), ca. 1995-2008
80
Rank in poverty reduction (annual % change)
KEN
KEN MKD
MDG ZMB
MDG DOM
DOM
BDI NGA ZMB
NGA CIV
MKD
MWI MRT KGZ
TZA
TZA BGDETH BGR
PRY
60
CIV GTMBOL COL
CAF KHM NAM GTM
MLI IND SLV ARG
MOZ GNB ZAF PAN
BDI PHL BOL COL
NER LAO BRA
HND ALBKGZ
UGA KHM PHL NAM CRI
CHL
ZAF ECU ARG
PAK
SEN NIC VENPER
40
NPL EGY SLV MYS
BGD INDIDN PRY
SWZ GNB LAO GHA HND TUR
EGY
NIC MAR MEX
VNM SEN
CAF GHA LKA ALB VEN
IDN MDV PAN
BRA
MOZ UGA CMR CRI
CMR TUN ECU
PER
CHN JOR
20
SWZ MAR TUR RUS
CHN ARM BGR
ETH JAM MEX MYS
MLI MWI GMB MNG
NPL
VNMTJK TUN
MNG ROM JOR CHL
NER ROM JAM UKR
PAK MDA
AZE
MDA POL UKR RUS
TJK POL
KAZ
KAZ
GMB MDV
0
1 5 10 15
Average daily income per capita, 1995
Actual Equal initial conditions
Note: the shaded area represents the 95 percent confidence bands.
Figure 4 suggests that initial conditions affect poverty reduction performances significantly.
Nonetheless, we may be presenting a biased picture in considering only percentage changes in poverty
reduction, given that richer countries â€“ having lower initial levels of poverty â€“ may find it easier to
reduce poverty in relative terms. Figure 5 therefore presents the same simulation but ranking absolute
changes in poverty (i.e. annual percentage points of poverty reduction). Although actual poverty
reduction performance no longer seems to vary with per capita income in a statistically significant
manner, the fundamental results continue to hold: after attributing equal initial conditions to countries,
poorer countries score better â€“ and their higher score is statistically significant at the 5 percent level.
14
Figure 5: Actual and simulated poverty reduction (rank; annual percentage points), ca. 1995-2008
80
Rank in poverty reduction (annual % points)
KEN MKD
MDG ZMB DOM
BDI NGA ZMB MKD
NGA MRT
CIV
MRT
MWI GTM
KGZ BGR
TZA
TZA PRY ARG
60
BOL COL
BGR
COL
ETH CIV ALB GTMBOL
BDI BGD SLV
KHM NAM KGZ ARG
ALB
ZAF SLV PAN
CAF PHL PRY
CRI
IND KHM EGY CRI
MLI MOZ HND BRA
GNB IND ZAF CHL
BGD IDN PHL VENTUR
40
NER LAO CHL
NAM ECU MYS
LKA VENPER PAN
MYS
NIC MAR BRA
UGA LAO ROM HND RUS
SEN GHA EGY PER
TUR
CHN UKR
ECU
UGA PAKSEN JOR
GNB PAK JOR
TUN
MOZ GHA CMR NIC
20
NPL MNG MAR
JAM
NPL IDN MEX
TUN MEX
SWZVNM RUS
ETH LKA ARM POL
CAF CMRAZE ARMJAM
VNMTJK POL
CHN KAZ MDV
UKR
NER MDA
ROM
MNG MDA
SWZ AZE KAZ
MLI MWI GMB
GMB
TJK
MDV
0
1 5 10 15
Average daily income per capita, 1995
Actual Equal initial conditions
Note: the shaded area represents the 95 percent confidence bands.
Looking only at ranks, however, only provides a partial picture. Ranks do not allow quantifying how
much poorer countries would have scored better, had they had the same level of initial average income
as wealthier countries. To quantify the change in the performance of poorer countries, we attribute next
to each country the median initial income per capita in our sample, which happens to be that of Jamaica
(4.1 dollars in 2005 PPP terms). Figure 6 shows the results. Not only do ranks get reversed (and both
actual and simulated relationships with initial per capita income remain significant), but the magnitude
of change in poverty reduction performance is quite impressive. If Mali, Nepal, Niger and Tajikistan
would have had the initial per capita income of Jamaica in 1995, they would have virtually eradicated
poverty by now. On the other side of the spectrum, Chile and Russia, which by now have poverty levels
below 1 percent, would have been facing in 2008 poverty rates of 39 and 17 percent, respectively.
15
Figure 6: Actual and simulated annual poverty changes, ca. 1995-2008
15
Annual percentage change in poverty
GMB
10
TJK
MDV
KAZ
POL RUS
PAK MDA
AZE UKR
NER AZE UKR
VNM MNG MDA
ROM
ROM
MLI MWI NPL ETH JAM JOR CHL
TJK CHN ARM TUN
5
SWZ GMB MNG JOR RUS
MOZ UGA IDN
CMR ARM MAR TUN MEX TUR
MDV
MYS
BGR
CAF SEN GHA
GNB LAO LKA MAR MEX
TUR
BGD IND CHN CMR EGY PER
ECU CRI
MYSBRA
LKA NIC ALB VEN PAN
VNM NIC
EGY VENPER
ECU
SWZ
NPL IDN GHA
KHM PHL HND ALB SLV CRI
PRY
BRA CHL
BDI UGA PAKSEN PHL NAMZAF SLV PAN ARG
ARG
MLI NER
MOZ
CAF MWI GNB LAO
IND KHM NAM KGZGTMBOL
BOL COL
BDI TZA BGD
NGA ETH CIV
MRT GTM
KGZ PRY
BGR
0
NGA ZMB
MDG CIV
MKD DOM
MDG KEN
ZMB
KEN MKD
-5
1 5 10 15
Average daily income per capita, 1995
Actual Equal initial conditions
Figure 7 shows that the reversal of the relationship, and the large magnitude of such reversal, extends
to other dimensions of poverty, such as the poverty gap. Again, had Mali, Nepal, Niger and Tajikistan
had the same initial per capita income as Jamaica in 1995, they would have literally closed the poverty
gap by now. In Chile, on the other hand, due to its high income inequality, a significant amount of the
population would still remain well below the poverty line (i.e., a poverty gap of 0.14).
16
Figure 7: Actual and simulated annual poverty gap changes, ca. 1995-2008
GMB
15
Annual percentage change in poverty gap
10 MDV
TJK
MDV
MWI GMB PAK MDA KAZ
POL RUS
NER TJK
ETH MNG MDA
AZE UKR
UKR
AZE JOR
MLI VNM
NPL ARMJAM
ROM RUS CHL
SWZ CHN MNG
CMR ARM TUN
JOR BGR
MEX MYS
UGA
GNB IDN LKA MAR TUN
5
CAF MOZ BGD LAO GHA MEX
ECU CRI
PAN
VNM IND
SEN CHN CMR NIC TUR
PER
TUR
BDI KHM LKA EGY
NIC MYSBRA
SWZ
NPL PAK ECU
PER
NER IDN GHA
SEN EGY NAM KGZ
NAM SLV
VEN PRY
CRI
PAN CHL
MLI UGA ETH PHL ALB BRA
BGR
MOZ
CAF MWI GNB LAO
IND KHM PHL ZAF
KGZ
ZAF SLV
BGD HND COL
PRY ARG
ARG
MKDBOL DOM COL
BDI TZA NGA MRT GTM
0
MDG CIV GTM DOM
NGA ZMB MKD
CIV
MDG
ZMB KEN
-5
KEN
1 5 10 15
Average daily income per capita, 1995
Actual Equal initial conditions
Do initial conditions matter for the Millennium Development Goals?
The previous analysis comprised a quite heterogeneous group of countries, from low to middle income
ones. The equalization of initial average income across all countries can thus imply, for the poorest and
richest countries, major shifts of where the poverty thresholds crosses the income distribution.
It is then natural to ask whether the drastic reversals of trends observed in the previous figures would
hold for more homogenous groups of countries. Such a question is essential in gauging whether the
MDG targets may have been excessively unfavorable towards the poorest countries.
Towards this purpose, we repeat the simulations for those countries in our database that in 1990 (the
benchmark year of the MDGs) had rates of extreme poverty above 10 percent (that is, 10 percent or
more of the population in 1990 living on less than 1.25 dollars a day). To capture the largest time span,
we run the simulations for the period 1990-2008, but since the timeframe varies significantly across
countries at times, we report average annual changes.
17
Figure 8: Actual and simulated annual changes in extreme poverty, ca. 1990-2008
1.8
East Asia & Pacific (excl. China)
1.54
2.62
China
2.07
1.97
South Asia (excl. India)
1.82
1.22
India
.9
.58
Sub-Saharan Africa
.59
.33
Latin America & Caribbean
.79
0 .5 1 1.5 2 2.5
Annual change in poverty (% points)
Actual Equal initial conditions
Note: poverty figures are population weighted.
Among the subset of countries, we have chosen as initial condition the median average income in 1990,
which corresponds to the daily per capita income of Ghana (1.6 dollars a day) for the simulation. For the
sake of clarity, and to enhance similarities with the MDG reports, we present in Figure 8 the regional
poverty reduction averages. 4
Because this subgroup of countries is more similar in initial average income than the larger group of
countries from the previous analysis, the difference between actual poverty reduction rates and the
ones that countries would have experienced under equal initial conditions is less striking. Nonetheless,
substantial differences remain. If China would have had in the 1990s the same average income than
Ghana, with the same growth performance, it would have faced half of a percentage point less in annual
poverty reduction. Given that these differences accumulate over a time span of almost twenty years,
4
To obtain regional averages, we weighted the annual poverty reduction in each country in our sample by the
initial population of that country. Our comparison does not mirror exactly the MDG reports for several reasons: (i)
the sample of countries differ; (ii) we chose to show poverty reduction in India and China separately; and (iii)
because of limited data availability for some regions, we have grouped countries differently (see Table 2).
18
China would have thus had in 2008, extreme poverty rates almost 10 percentage points higher than the
actual ones.
In full similarity with China, India would have also faced lower annual rates of extreme poverty
reduction by a third of a percentage point under these conditions. Observe, however, that even if India
and China would have had the same initial conditions as Ghana, because of their stunning growth
performance the countries would still be facing much higher rates of extreme poverty reduction: almost
twice (for India) and four times (for China) the average of Sub-Saharan African countries.
On the other side of the spectrum, had Latin American countries had the same average income as
Ghana, their rates of extreme poverty reduction would have been more than double the actual ones.
This is because Latin American countries (and in particular Brazil) started, on average, with a much
higher income per capita. Thus, in the 1990s, the extreme poverty line was already crossing the income
distribution past its mode, at a point where the income-growth elasticity of poverty reduction was
already relatively low.
Finally, observe that, on average, the simulations suggest that poverty reduction would not have
changed much for Sub-Saharan Africa if all its countries would have had the same average income as
Ghana. This result, however, depends very much on the choice of the level of the initial conditions, and
on the fact that for most Sub-Saharan African countries in the sample the initial level of average income
remained relatively close to that of Ghana (relative to, say, that of China or Latin American countries), so
that poverty reduction rates for Sub-Saharan countries remain relatively unaffected by the change in
initial conditions.
Implications and Conclusions
This paper quantifies how much initial conditions â€“ the position of the poverty line with respect to the
distribution of income â€“ affect poverty reduction, given a level of â€œeffortâ€? measured in terms of growth
in per capita income. While wealthier countries did perform better in reducing poverty in the last
decade and a half (1995-2008), assuming equal initial conditions, the situation reverses: we find a
statistically significant negative relation between initial average income and poverty reduction
performance, with the poorest countries in the sample going from being the worst to the best
performers in poverty reduction. The fundamental result also holds if we consider absolute changes in
poverty reduction vis-Ã -vis percentage ones.
19
Initial conditions appear to be an important factor in the success of the poverty reduction efforts. While
policymakers can influence changes in the mean and shape of the income distribution, they cannot
affect initial conditions. This information is particularly relevant given that, by disentangling
policymakersâ€™ efforts from initial conditions out of their control, achievements can be benchmarked
more accurately. Collier and Dollar (2002), for example, have analyzed the distribution of aid that would
maximize poverty reduction considering the quality of policies and the initial levels of poverty. Their
framework, however, fails to fully disentangle the effect of initial conditions to assess effort and it does
not incorporate the dynamics in a way this framework does by allowing for a change of the growth
elasticity of poverty over time.
While our analysis focuses on poverty, the proposed framework remains relevant for indicators beyond
poverty â€“ it can be applied to any indicator that divides an underlying distribution. Given the importance
of target-based mechanisms for different purposes, including allocating resources, the methodology
presented can be key to correctly assess the importance of initial conditions in evaluating policy
performance towards different goals.
20
References
Bourguignon, FranÃ§ois (2002) "The growth elasticity of poverty reduction: explaining heterogeneity
across countries and time periods," DELTA Working Papers 2002-03, DELTA (Ecole normale supÃ©rieure).
Collier, P. and D. Dollar (2002), â€œAid Allocation and Poverty Reductionâ€?, European Economic Review 46,
1475-1500.
Datt, Gaurav, (1998) "Computational tools for poverty measurement and analysis," FCND discussion
papers 50, International Food Policy Research Institute (IFPRI).
Easterly, William (2009) â€œHow the Millennium Development Goals are Unfair to Africa,â€? World
Development, Vol. 37, No. 1, pp. 26-35.
Kakwani, N. (1980). â€œOn a class of poverty measures.â€? Econometrica. 48(2). 437â€“446.
Ravallion, Martin (2012) "Why Don't We See Poverty Convergence?," American Economic Review,
102(1): 504â€“23.
United Nations (2012) â€œThe Millennium Development Goals Report 2012â€?. New York: UN.
United Nations General Assembly (2000). United Nations Millennium Declaration. GA Res. 55/2, UN
GAOR, 55th session Sess., Supp. 2, U.N. Doc. A/RES/55/2 (2000) 5.
Villasenor, J. and B. C. Arnold (1989). â€œElliptical Lorenz curves.â€? Journal of Econometrics. 40(2). 327â€“338.
21
Table 1: Poverty trends (USD 2.5 a day)
Period 1 Period 2
Average monthly per Poverty Average monthly per Poverty
Country Year Gini Year Gini
capita income (USD PPP) headcount capita income (USD PPP) headcount
Albania 1997 151 0.16 0.29 2008 174 0.11 0.35
Argentina 1995 378 0.11 0.49 2008 413 0.09 0.46
Armenia 1996 108 0.51 0.44 2008 127 0.25 0.31
Azerbaijan 1995 87 0.54 0.35 2008 201 0.07 0.34
Bangladesh 1995 42 0.91 0.33 2010 52 0.86 0.32
Bolivia 1997 214 0.35 0.53 2008 214 0.32 0.52
Brazil 1995 297 0.29 0.60 2008 356 0.17 0.51
Bulgaria 1995 282 0.05 0.31 2007 274 0.02 0.28
Burundi 1992 26 0.97 0.33 2006 29 0.96 0.33
Cambodia 1994 65 0.77 0.46 2007 78 0.71 0.40
Cameroon 1996 82 0.65 0.41 2007 115 0.43 0.39
Central African 1993 25 0.94 0.61 2008 51 0.86 0.56
Chile 1994 378 0.12 0.50 2009 494 0.01 0.42
China 1996 60 0.76 0.32 2008 110 0.47 0.35
Colombia 1996 303 0.22 0.46 2008 343 0.20 0.51
Costa Rica 1995 260 0.17 0.46 2008 372 0.10 0.50
Cote dâ€™Ivoire 1995 81 0.62 0.37 2008 88 0.59 0.42
Croatia 1998 513 0.00 0.27 2008 88 0.59 0.42
Dominican Rep 1996 252 0.18 0.47 2008 246 0.19 0.49
Ecuador 1995 198 0.36 0.54 2008 254 0.21 0.51
Egypt 1995 98 0.46 0.30 2008 114 0.32 0.31
El Salvador 1995 194 0.28 0.47 2008 214 0.22 0.47
Ethiopia 1995 45 0.90 0.40 2005 51 0.88 0.30
Ghana 1998 63 0.74 0.41 2005 80 0.64 0.43
Guatemala 2000 186 0.35 0.45 2006 200 0.35 0.47
Guinea 1994 42 0.88 0.45 2007 57 0.80 0.39
Honduras 1995 121 0.54 0.53 2008 225 0.42 0.33
India 1993 47 0.90 0.36 2010 60 0.81 0.33
Indonesia 1996 52 0.87 0.31 2008 72 0.69 0.33
Jamaica 1993 125 0.32 0.36 2004 279 0.11 0.46
Jordan 1997 152 0.22 0.36 2008 200 0.08 0.34
Kazakhstan 1996 137 0.28 0.35 2008 333 0.00 0.26
Kenya 1994 78 0.66 0.42 2005 65 0.77 0.46
Kyrgyz Republic 1993 173 0.37 0.54 2008 134 0.31 0.37
Lao PDR 1997 49 0.88 0.35 2008 63 0.78 0.37
Macedonia 1998 192 0.07 0.28 2008 294 0.09 0.44
Madagascar 1993 36 0.93 0.46 2010 28 0.95 0.44
Malawi 1997 30 0.95 0.50 2004 34 0.94 0.39
Malaysia 1995 263 0.17 0.49 2009 400 0.06 0.46
Maldives 1998 205 0.43 0.63 2004 176 0.20 0.37
Mali 1994 24 0.96 0.51 2010 46 0.87 0.33
Mauritania 1995 79 0.62 0.37 2008 84 0.61 0.40
Mexico 1996 193 0.35 0.51 2008 312 0.15 0.48
Moldova 1997 95 0.50 0.37 2008 183 0.14 0.35
Mongolia 1995 81 0.58 0.33 2007 150 0.25 0.37
Morocco 1999 130 0.36 0.39 2007 161 0.24 0.41
Mozambique 1996 30 0.95 0.44 2008 47 0.88 0.46
Namibia 1993 147 0.68 0.74 2004 146 0.59 0.64
Nepal 1995 38 0.94 0.35 2010 68 0.72 0.33
Nicaragua 1993 107 0.59 0.57 2005 148 0.42 0.46
Niger 1994 30 0.95 0.42 2007 53 0.85 0.35
Nigeria 1996 39 0.91 0.47 2010 40 0.90 0.49
Pakistan 1996 47 0.91 0.29 2007 66 0.76 0.30
Panama 1995 275 0.28 0.58 2009 325 0.17 0.52
Paraguay 1995 286 0.27 0.52 2008 265 0.21 0.48
Peru 1997 203 0.32 0.54 2008 251 0.21 0.49
Philippines 1994 83 0.64 0.43 2009 104 0.53 0.43
Poland 1996 138 0.23 0.33 2008 370 0.01 0.34
Romania 1994 99 0.39 0.28 2008 227 0.04 0.31
Russia 1996 287 0.12 0.46 2008 471 0.00 0.42
Senegal 1994 50 0.87 0.41 2005 67 0.72 0.39
South Africa 1995 158 0.49 0.57 2009 257 0.40 0.63
Sri Lanka 1995 83 0.62 0.35 2006 119 0.43 0.40
Swaziland 1995 34 0.93 0.35 2009 80 0.69 0.51
Tajikistan 1999 43 0.92 0.29 2009 100 0.42 0.31
Tanzania 1991 33 0.95 0.34 2007 37 0.93 0.38
The Gambia 1998 42 0.87 0.50 2003 82 0.66 0.47
Tunisia 1995 154 0.30 0.42 2005 218 0.15 0.41
Turkey 1994 204 0.17 0.42 2008 305 0.07 0.39
Uganda 1996 40 0.91 0.37 2009 68 0.76 0.40
Ukraine 1995 205 0.14 0.39 2008 324 0.00 0.28
Venezuela 1995 177 0.29 0.47 2006 213 0.21 0.45
Vietnam 1993 40 0.91 0.36 2008 85 0.58 0.36
Zambia 1996 46 0.87 0.48 2006 42 0.87 0.55
22
Table 2: Poverty trends (USD 1.25 a day)
Period 1 Period 2
Average monthly per Poverty Average monthly per Poverty
Country Year Gini Year Gini
capita income (USD PPP) headcount capita income (USD PPP) headcount
Bangladesh 1991 34 0.70 0.28 2010 52 0.43 0.32
Brazil 1990 300 0.13 0.61 2008 356 0.08 0.55
Burundi 1992 26 0.84 0.33 2006 29 0.81 0.33
Cambodia 1994 65 0.44 0.46 2007 78 0.32 0.40
Central African 1993 25 0.83 0.61 2008 51 0.63 0.56
China 1990 41 0.60 0.29 2008 133 0.13 0.38
Cote dâ€™Ivoire 1988 99 0.14 0.37 2008 88 0.24 0.42
Ecuador 1994 182 0.16 0.47 2008 254 0.08 0.51
El Salvador 1991 168 0.18 0.52 2008 214 0.07 0.47
Ethiopia 1995 45 0.61 0.40 2005 51 0.39 0.30
Ghana 1991 49 0.51 0.38 2005 80 0.28 0.43
Guatemala 2000 186 0.13 0.45 2006 200 0.14 0.47
Honduras 1991 91 0.35 0.51 2008 225 0.24 0.33
India 1988 45 0.54 0.31 2010 60 0.26 0.33
Indonesia 1990 43 0.54 0.29 2008 72 0.23 0.33
Kenya 1992 93 0.38 0.57 2005 65 0.44 0.46
Lao PDR 1992 43 0.56 0.30 2008 63 0.34 0.37
Madagascar 1993 36 0.72 0.46 2010 28 0.81 0.44
Mali 1994 24 0.86 0.51 2010 46 0.50 0.33
Mauritania 1993 71 0.43 0.50 2008 84 0.23 0.40
Mongolia 1995 81 0.19 0.33 2007 150 0.00 0.37
Namibia 1993 147 0.49 0.74 2004 146 0.32 0.64
Nepal 1995 38 0.68 0.35 2010 68 0.25 0.33
Nicaragua 1993 132 0.18 0.50 2005 148 0.17 0.46
Niger 1992 34 0.73 0.36 2007 53 0.44 0.35
Nigeria 1992 40 0.62 0.45 2010 40 0.68 0.49
Pakistan 1990 38 0.65 0.33 2007 66 0.21 0.30
Panama 1991 215 0.21 0.58 2009 325 0.04 0.52
Peru 1997 203 0.14 0.54 2008 251 0.06 0.49
Philippines 1991 81 0.31 0.44 2009 104 0.18 0.43
Senegal 1991 45 0.66 0.54 2005 67 0.34 0.39
South Africa 1993 172 0.24 0.59 2009 257 0.14 0.63
Sri Lanka 1991 76 0.15 0.32 2006 119 0.07 0.40
Swaziland 1995 34 0.79 0.35 2009 80 0.41 0.51
Tanzania 1991 33 0.73 0.34 2007 37 0.68 0.38
Uganda 1989 37 0.69 0.44 2009 68 0.38 0.40
Vietnam 1993 40 0.64 0.36 2008 85 0.17 0.36
Zambia 1991 47 0.63 0.60 2006 42 0.69 0.55
23
Appendix. Description of Parameterizations
The lognormal approximation assumes that the income distribution can be approximated by a lognormal
(lnxâˆ’Âµ)2
1 âˆ’
distribution of the following form fx = xÏƒ e 2Ïƒ2 , where x is income, fx is the probability density
âˆš2Ï€
function, Âµ is the mean and Ïƒ is the standard deviation. The lognormal distribution is thus fully
described by its mean and standard deviation, implying that one only needs to estimate the mean and
standard deviation of income to fully parameterize the distribution.
The General Quadratic approach of Villasenor and Arnold (1989) uses a general quadratic form to
estimate the Lorenz curve. The general quadratic form of a Lorenz curve can be rewritten as L(1 âˆ’ L) =
a(P2 âˆ’ L) + bL(P âˆ’ 1) + c(P âˆ’ L) where P is the cumulative proportion of the population and L is the
cumulative share in aggregate consumption or income. To calculate the parameters a, b, and c we
rename t = L(1 âˆ’ L) , u = (P 2 âˆ’ L), v = L(P âˆ’ 1), and w = (P âˆ’ L) and run the ordinary least-squares
regression of t on u , v, and w with no intercept. Datt (1998) provides details on parameter restrictions,
and on how to derive Gini and poverty headcount coefficients.
The Beta Lorenz approach of Kakwani (1980) uses a beta distribution to approximate the Lorenze curve.
This implies that the Lorenz curve can be written as L = P âˆ’ Î¸P Î³ (1 âˆ’ P)Î´ , where again P is the
cumulative proportion of the population and L is the cumulative share in aggregate consumption. Then
to estimate the parameters Î¸, Î³, and Î´ one must simply rearrange and take the logarithm of the above
in order to get ln(L âˆ’ P) = ln(âˆ’Î¸) + Î³ ln(P) + Î´ln(1 âˆ’ P), rename t = ln(L âˆ’ P), u = ln(P),
v = ln(1 âˆ’ P) and then estimate the parameters by running the ordinary least squares regression of
t on u and v. See also Datt (1998).
For the nonparametric approximation we use a nonparametric form to approximate the income
distribution. In practice, we use a univariate kernel density estimation, where the density of x can be
1 n wi xâˆ’x
Ì‚K (x; h) =
estimated as: f âˆ‘ K( h i) where W = âˆ‘n
i=1 wi , K(z) is a kernel function, and h is the
W i=1 h
smoothing parameter (ie the bandwidth). We use an Epanechnikov kernel function.
The goodness of fit criterion defined in Datt (1998) is used to determine which of the parameterizations
of the Lorenz curve (the General Quadratic or the Beta Lorenz) is the best fit. This goodness of fit
criterion is simply the sum of the squared errors between the actual Lorenz and projected Lorenz from
the respective approximation up to the estimated head count ratio. Because the Lorenz curve is
24
continuous, the percentage of the population is separated into 1,000 bins. Datt (1998) describes this
mathematically as follows:
k
ï¿½Ä± âˆ’ Li )2
SSE = ï¿½(L
i=1
where Li is the estimated Lorenz curve from the specified parameterization, Li is the actual Lorenz
curve, and k = ï¿½kï¿½âˆ‘kâˆ’1 ï¿½ k
i=1 pi â‰¤ HÄ± â‰¤ âˆ‘i=1 pi ï¿½ where Hi is the estimated Head Count Ratio. The sum of
squared errors (SSE) is calculated for each specification and the parameterization with the lowest SSE is
used.
25