A Unified Approach to Measuring Poverty and Inequality

A Unified Approach to
Measuring Poverty and

Inequality
OTHER TITLES IN THE ADePT SERIES
Health Equity and Financial Protection: Streamlined Analysis with ADePT

Software (2011) by Adam Wagstaff, Marcel Bilger, Zurab Sajaia, and
Michael Lokshin
Assessing Sector Performance and Inequality in Education: Streamlined

Analysis with ADePT Software (2011) by Emilio Porta, Gustavo Arcia,
Kevin Macdonald, Sergiy Radyakin, and Michael Lokshin
ADePT User Guide (forthcoming) by Michael Lokshin, Zurab Sajaia, and

Sergiy Radyakin
For more information about Streamlined Analysis with ADePT software

and publications, visit www.worldbank.org/adept.
STREAMLINED ANALYSIS WITH ADePT SOFTWARE
A Unified Approach to
Measuring Poverty and
Inequality
Theory and Practice
James Foster
Suman Seth
Michael Lokshin
Zurab Sajaia
© 2013 International Bank for Reconstruction and Development / The World Bank
1818 H Street NW
Washington, DC 20433
Telephone: 202-473-1000
Internet: www.worldbank.org
Some rights reserved
1 2 3 4 16 15 14 13
This work is a product of the staff of The World Bank with external contributions. Note that The World Bank does not necessarily own
each component of the content included in the work. The World Bank therefore does not warrant that the use of the content contained in
the work will not infringe on the rights of third parties. The risk of claims resulting from such infringement rests solely with you.
The findings, interpretations, and conclusions expressed in this work do not necessarily reflect the views of The World Bank, its Board
of Executive Directors, or the governments they represent. The World Bank does not guarantee the accuracy of the data included in this
work. The boundaries, colors, denominations, and other information shown on any map in this work do not imply any judgment on the
part of The World Bank concerning the legal status of any territory or the endorsement or acceptance of such boundaries.
Nothing herein shall constitute or be considered to be a limitation upon or waiver of the privileges and immunities of The
World Bank, all of which are specifically reserved.
Rights and Permissions
This work is available under the Creative Commons Attribution 3.0 Unported license (CC BY 3.0) http://creativecommons.org/licenses/
by/3.0. Under the Creative Commons Attribution license, you are free to copy, distribute, transmit, and adapt this work, including for
commercial purposes, under the following conditions:
Attribution—Please cite the work as follows: Foster, James, Suman Seth, Michael Lokshin, and Zurab Sajaia. 2013. A Unified Approach to
Measuring Poverty and Inequality: Theory and Practice. Washington, DC: World Bank. doi: 10.1596/978-0-8213-8461-9 License: Creative
Commons Attribution CC BY 3.0
Translations—If you create a translation of this work, please add the following disclaimer along with the attribution: This translation was
not created by The World Bank and should not be considered an official World Bank translation. The World Bank shall not be liable for any content
or error in this translation.
All queries on rights and licenses should be addressed to the Office of the Publisher, The World Bank, 1818 H Street NW, Washington,
DC 20433, USA; fax: 202-522-2625; e-mail: pubrights@worldbank.org.
ISBN (paper): 978-0-8213-8461-9

ISBN (electronic): 978-0-8213-9864-7
DOI: 10.1596/978-0-8213-8461-9
Cover photo: Scott Wallace/World Bank (girl and child)

Background image: iStockphoto.com/Olga Altunina
Cover design: Kim Vilov
Library of Congress Cataloging-in-Publication Data

Foster, James E. (James Eric), 1955–
Measuring poverty and inequality : theory and practice / by James Foster, Suman Seth, Michael Lokshin, Zurab Sajaia.
pages cm
Includes bibliographical references and index.
ISBN 978-0-8213-8461-9 — ISBN 978-0-8213-9864-7 (electronic)
1. Poverty. 2. Equality. I. Title.
HC79.P6F67 2013
339.4'6—dc23
2012050221
Contents
Foreword .................................................................................................... xi
Preface ....................................................................................................... xv
Chapter 1
Introduction ................................................................................................ 1
The Income Variable ..........................................................................................4
The Data..............................................................................................................4
Income Standards and Size .................................................................................5
Inequality Measures and Spread .......................................................................13
Poverty Measures and the Base of the Distribution .........................................26
Note ...................................................................................................................44
References..........................................................................................................44
Chapter 2
Income Standards, Inequality, and Poverty ......................................... 45
Basic Concepts ..................................................................................................49
Income Standards ..............................................................................................54
Inequality Measures ...........................................................................................81
Poverty Measures .............................................................................................105
Exercises ..........................................................................................................144
Notes................................................................................................................149
References........................................................................................................151
v
Contents
Chapter 3
How to Interpret ADePT Results ......................................................... 155
Analysis at the National Level and Rural/Urban Decomposition .................157
Analysis at the Subnational Level ..................................................................170
Poverty Analysis across Other Population Subgroups....................................183
Sensitivity Analyses ........................................................................................199
Dominance Analyses .......................................................................................207
Advanced Analysis..........................................................................................216
Note .................................................................................................................223
Reference .........................................................................................................223
Chapter 4
Frontiers of Poverty Measurement ...................................................... 225
Ultra-Poverty ...................................................................................................225
Hybrid Poverty Lines....................................................................................... 226
Categorical and Ordinal Variables .................................................................228
Chronic Poverty ..............................................................................................229
Multidimensional Poverty ...............................................................................230
Multidimensional Standards ...........................................................................234
Inequality of Opportunity ...............................................................................238
Polarization ......................................................................................................240
References........................................................................................................241
Chapter 5
Getting Started with ADePT ................................................................. 245
Conventions Used in This Chapter ...............................................................246
Installing ADePT ............................................................................................246
Launching ADePT ..........................................................................................247
Overview of the Analysis Procedure............................................................... 248
Specify Datasets ...............................................................................................249
Map Variables..................................................................................................252
Select Tables and Graphs ...............................................................................254
Generate the Report .......................................................................................257
Examine the Output........................................................................................258
Working with Variables ..................................................................................258
Setting Parameters ..........................................................................................264
Working with Projects ....................................................................................264
Adding Standard Errors or Frequencies to Outputs .......................................265
vi
Contents
Applying If-Conditions to Outputs ................................................................266

Generating Custom Tables .............................................................................268
Appendix ................................................................................................. 271

Income Standards and Inequality ...................................................................271
Censored Income Standards and Poverty Measures .......................................273
Elasticity of Watts Index, SST Index, and CHUC Index to
Per Capita Consumption Expenditure .........................................................275
Sensitivity of Watts Index, SST Index, and CHUC Index to Poverty Line ..... 277
Decomposition of the Gini Coefficient ..........................................................278
Decomposition of Generalized Entropy Measures ..........................................280
Dynamic Decomposition of Inequality Using the Second Theil Measure ....282
Decomposition of Generalized Entropy Measure by Income Source .............284
Quantile Function ...........................................................................................286
Generalized Lorenz Curve ..............................................................................288
General Mean Curve....................................................................................... 289
Generalized Lorenz Growth Curve .................................................................290
General Mean Growth Curve .........................................................................291
References........................................................................................................292
Index ........................................................................................................ 293
Figures
2.1: Probability Density Function ..................................................................51

2.2: Cumulative Distribution Function .........................................................52
2.3: Quantile Function ...................................................................................53
2.4: Quantile Function and the Quantile Incomes .......................................59
2.5: Quantile Function and the Partial Means ..............................................62
2.6: Generalized Means and Parameter a...................................................... 66
2.7: First-Order Stochastic Dominance Using Quantile Functions and
Cumulative Distribution Functions ........................................................71
2.8: Quantile Function and Generalized Lorenz Curve ................................72
2.9: Generalized Lorenz Curve .......................................................................73
2.10: Growth Incidence Curves .......................................................................77
2.11: Growth Rate of Lower Partial Mean Income .........................................78
2.12: General Mean Growth Curves ...............................................................80
2.13: Lorenz Curve .........................................................................................102
2.14: Poverty Incidence Curve and Headcount Ratio ..................................136
2.15: Poverty Deficit Curve and the Poverty Gap Measure ..........................137
vii
Contents
2.16: Poverty Severity Curve and the Squared Gap Measure .......................139
3.1: Probability Density Function of Urban Georgia ..................................157
3.2: Age-Gender Poverty Pyramid ...............................................................198
3.3: Poverty Incidence Curves in Urban Georgia, 2003 and 2006 .............208
3.4: Poverty Deficit Curves in Urban Georgia, 2003 and 2006 ..................209
3.5: Poverty Severity Curves in Rural Georgia, 2003 and 2006 .................211
3.6: Growth Incidence Curve of Georgia between 2003 and 2006 ............212
3.7: Lorenz Curves of Urban Georgia, 2003 and 2006 ................................214
3.8: Standardized General Mean Curves of Georgia, 2003 and 2006 .........216
A.1: The Quantile Functions of Urban Per Capita Expenditure,
Georgia ..................................................................................................287
A.2: Generalized Lorenz Curve of Urban Per Capita Expenditure,
Georgia ..................................................................................................288
A.3: Generalized Mean Curve of Urban Per Capita Expenditure,
Georgia ..................................................................................................290
A.4: Generalized Lorenz Growth Curve for Urban Per Capita
Expenditure, Georgia ............................................................................291
A.5: General Mean Growth Curve of Urban Per Capita Expenditure,
Georgia ..................................................................................................292
Tables
3.1: Mean and Median Per Capita Consumption Expenditure,

Growth, and the Gini Coefficient ........................................................158
3.2: Overall Poverty ....................................................................................160
3.3: Distribution of Poor in Urban and Rural Areas ...................................162
3.4: Composition of FGT Family of Indices by Geography ........................164
3.5: Quantile PCEs and Quantile Ratios of Per Capita Consumption
Expenditure ...........................................................................................166
3.6: Partial Means and Partial Mean Ratios ................................................168
3.7: Distribution of Population across Quintiles .........................................169
3.8: Mean and Median Per Capita Income, Growth, and the Gini
Coefficient across Subnational Regions ...............................................171
3.9: Headcount Ratio by Subnational Regions, 2003 and 2006 .................172
3.10: Poverty Gap Measure by Subnational Regions ....................................174
3.11: Squared Gap Measure by Subnational Regions....................................175
3.12: Quantile PCE and Quantile Ratio of Per Capita Consumption
Expenditure, 2003 .................................................................................177
viii
Contents
3.13: Partial Means and Partial Mean Ratios for Subnational

Regions, 2003 ........................................................................................178
3.14: Distribution of Population across Quintiles by Subnational
Region, 2003 .........................................................................................180
3.15: Subnational Decomposition of Headcount Ratio, Changes between
2003 and 2006 .......................................................................................181
3.16: Mean and Median Per Capita Consumption Expenditure,
Growth, and Gini Coefficient, by Household Characteristics.............184
3.17: Headcount Ratio by Household Head’s Characteristics ......................185
3.18: Distribution of Population across Quintiles by Household Head’s
Characteristics, 2003 .............................................................................187
3.19: Headcount Ratio by Employment Category .........................................189
3.20: Headcount Ratio by Education Level ...................................................191
3.21: Headcount Ratio by Demographic Composition .................................192
3.22: Headcount Ratio by Landownership ....................................................194
3.23: Headcount Ratio by Age Groups..........................................................196
3.24: Elasticity of FGT Poverty Indices to Per Capita Consumption
Expenditure ...........................................................................................199
3.25: Sensitivity of Poverty Measures to the Choice of Poverty
Line, 2003 ..............................................................................................202
3.26: Other Poverty Measures ........................................................................203
3.27: Atkinson Measures and Generalized Entropy Measures by
Geographic Regions, 2003 ....................................................................205
3.28: Consumption Regressions .....................................................................217
3.29: Changes in the Probability of Being in Poverty ...................................220
3.30: Growth and Redistribution Decomposition of Poverty Changes,
Headcount Ratio ...................................................................................222
A.1: General Means and the Sen Mean ........................................................272
A.2: Censored Income Standards ..................................................................273
A.3: Elasticity of Watts Index, SST Index, and CHUC Index to
Per Capita Consumption Expenditure................................................... 275
A.4: Sensitivity of Watts Index, SST Index, and CHUC Index to the
Choice of Poverty Line, 2003 ................................................................277
A.5: Breakdown of Gini Coefficient by Geography ......................................279
A.6: Decomposition of Generalized Entropy Measures by Geography .........280
A.7: Dynamic Decomposition of Inequality Using the Second
Theil Measure ........................................................................................283
A.8: Decomposition of Generalized Entropy Measure by Income Source........284
ix
Foreword
This book is an introduction to the theory and practice of measuring

poverty and inequality, as well as a user’s guide for readers wanting to ana-
lyze income or consumption distribution for any standard household data-
set using the ADePT program—a free download from the World Bank’s
website.
In the prosaic world of official publications, A Unified Approach to
Measuring Poverty and Inequality: Theory and Practice is sure to stand out. It
is written with a flair and fluency that is rare. For readers with little interest
in the underlying philosophical debates and a desire simply to use ADePT
software for computations, this book is, of course, a must. But even for some-
one with no interest in actually computing numbers but, instead, wanting
to learn the basic theory of poverty and inequality measurement, with its
bewildering plurality of measures and axioms and complex philosophical
debates in the background, this book is an excellent read.
But, of course, the full book is designed for analysts wishing to do hands-
on work, converting raw data into meaningful indices and unearthing regu-
larities in large and often chaotic statistical information. The presentation
is comprehensive, with all relevant concepts defined and explained. On
completing this book, the country expert will be in a position to generate
the analyses needed for a Poverty Reduction Strategy Paper. Researchers
xi
Foreword
can construct macrodata series suitable for empirical analyses. Students can
replicate and check the robustness of published results.
Several recent initiatives have lowered the cost of accessing household
datasets. The goal of this book, then, is to reduce the cost of analyzing data
and sharing findings with interested parties.
This book has two unique aspects. First, the theoretical discussion is
based on a highly accessible, unified treatment of inequality and poverty
in terms of income standards or basic indicators of the overall size of the
income distribution. Examples include the mean, median, and other tradi-
tional ways of summarizing a distribution with one or several representative
indicators. The literature on the measurement of inequality has proliferated
since the 1960s. This book provides an excellent overview of that extensive
literature.
Most poverty measures are built on two pillars. First, the “poverty line”
delineates the income levels that define a poor person, and second, various
measures capture the depths of the incomes of those below the poverty line.
The approach here considers income standards as the basic measurement
building blocks and uses them to construct inequality and poverty measures.
This unified approach provides advantages in interpreting and contrasting
the measures and in understanding the way measures vary over time and
space.
Second, the theoretical presentation is complemented by empirical
examples that ground the discussion, and it provides a practical guide to the
inequality and poverty modules of the ADePT software developed at the
World Bank. By immediately applying the measurement tools, the reader
develops a deeper understanding of what is being measured. A battery of
exercises in chapter 2 also aids the learning process.
The ADePT software enables users to analyze microdata—from sources
such as household surveys—and generate print-ready, standardized tables
and charts. It can also be used to simulate the effect of economic shocks,
farm subsidies, cash transfers, and other policy instruments on poverty,
inequality, and labor. The software automates the analysis, helps minimize
human error, and encourages development of new methods of economic
analysis.
For each run, ADePT produces one output file—containing your selec-
tion of tables and graphs, an optional original data summary, and errors and
notifications—in Microsoft Excel® format. Tables of standard errors and
frequencies can be added to a report, if desired.
xii
Foreword
These two components—a unifying framework for measurement and the

immediate application of measures facilitated by ADePT software—make
this book a unique source for cutting-edge, practical income distribution
analysis.
The book is bound to empower those already engaged in the analysis of
poverty and inequality to do deeper research and plumb greater depths in
searching for regularity in larger and larger datasets. But I am also hopeful
that it will draw new researchers into this important field of inquiry. This
book should also be of help in enriching the discussion and analysis relating
to the World Bank’s recent effort to define new targets and indicators for
promoting work on eradicating poverty and enhancing shared prosperity.
The work on this project was facilitated by the proximity of two key
institutions, the World Bank and the George Washington University. But
as anyone who has contemplated the world knows, proximity does not nec-
essarily lead to cooperation. It is a tribute to the authors that they made use
of this natural advantage and, through their shared willingness to support
collaborative research across institutional boundaries, managed to produce
this very useful monograph. My expectation is that this will be the first of
many such collaborations.
Kaushik Basu
Senior Vice President and Chief Economist
The World Bank
xiii
Preface
This book is made possible by financial support from the Research Support
Budget of the World Bank, the Knowledge for Change Program (KCP), and
the Rapid Social Response (RSR) Program. The KCP is designed to pro-
mote high-quality, cutting-edge research that creates knowledge to support
policies for poverty reduction and sustainable development. KCP is funded
by the generous contributions of Australia, Canada, China, Denmark, the
European Commission, Finland, France, Japan, the Netherlands, Norway,
Singapore, Sweden, Switzerland, the United Kingdom, ABN AMRO
Bank, and the International Fund for Agricultural Development. RSR is
a multidonor endeavor to help the world’s poorest countries build effec-
tive social protection and labor systems that safeguard poor and vulnerable
people against severe shocks and crises. RSR has been generously supported
by Australia, Norway, the Russian Federation, Sweden, and the United
Kingdom.
James Foster is grateful to the Elliott School of International Affairs
and Dean Michael Brown for facilitating research on global poverty and
international development. The Ultra-poverty Initiative of its Institute
for International Economic Policy (IIEP), spearheaded by its former direc-
tor, Stephen Smith, has been a focal point of these efforts. A major gift to
the Elliott School from an anonymous donor significantly enhanced the
research capacity of IIEP and helped make the present project a reality.
xv
Preface
We are grateful to the Oxford Poverty and Human Development

Initiative (OPHI) and its director, Sabina Alkire, for allowing Suman
Seth time away from OPHI’s core efforts on multidimensional measures
of poverty and well-being to work on the unidimensional methods pre-
sented here. Streams of students have helped refine the ideas, and we are
particularly grateful to Chrysanthi Hatzimasoura who organized the weekly
Development Tea at the Elliott School in which many useful conversations
have been held.
The authors thank Bill Creitz for his excellent editorial support and
Denise Bergeron, Mark Ingebretsen, and Stephen McGroarty in the World
Bank Office of the Publisher for managing the production and dissemina-
tion of this volume.
xvi
Chapter 1
Introduction
What is poverty? At its most general level, poverty is the absence of accept-
able choices across a broad range of important life decisions—a severe lack of
freedom to be or to do what one wants. The inevitable outcome of poverty
is insufficiency and deprivation across many of the facets of a fulfilling life:
• Inadequate resources to buy the basic necessities of life

• Frequent bouts of illness and an early death
• Literacy and education levels that undermine adequate functioning
and limit one’s comprehension of the world and oneself
• Living conditions that imperil physical and mental health
• Jobs that are at best unfulfilling and at worst dangerous
• A pronounced absence of dignity, a lack of respect from others
• Exclusion from community affairs.
The presence of poverty commonly leads groups to undertake activities

and policies designed to reduce poverty—responses that take many forms and
that are seen at many levels. A family in India helps pay for the children of
its housekeeper or aiya. Buddhists, Confucians, Christians, and Muslims work
together in Jakarta, Indonesia, to deliver alms to the poor during the fasting
month. The governments of Mexico and Brazil implement PROGRESA
(Programa de Educación, Salud y Alimentación, now called Oportunidades)
and Bolsa Família, conditional cash transfer programs to help the poorest
families invest in their children’s human capital and to break the cycle of pov-
erty. A nongovernmental organization from Bangladesh offers microfinance
loans and education to poor people in Uganda.
1
A Unified Approach to Measuring Poverty and Inequality
At the United Nations Millennium Forum in 2000, 193 countries agreed

on the Millennium Development Goals, which, among other targets, aim
to reduce the proportion of people living on $1.25 a day by half within
15 years. Following the Group of 8 (G-8) Summit in Gleneagles, Scotland,
in 2005, the World Bank, the International Monetary Fund, and the African
Development Bank agreed to a plan of debt relief for the poorest countries.
What reasons underlie efforts to alleviate poverty? Individuals often con-
sider alleviating poverty a personal responsibility that arises from religious
or philosophical convictions. Many see poverty as the outcome of an unfair
system that privileges some and constrains opportunities for others—a fun-
damental injustice that can also lead to social conflict and violence if not
addressed. Others view poverty as a denial of universal rights and human
dignity that requires collective action at a global level.
Political leaders often portray poverty as the enemy of social stability
and good governance. Economists focus on the waste and inefficiency of
allowing a portion of the population to fall significantly below potential.
Many countries include poverty alleviation as an essential component of
their programs for sustainable growth and development. Business leaders are
reevaluating the “bottom of the pyramid” as a substantial untapped market
that can be bolstered through efforts to address poverty.
Measurement is an important tool for the many efforts that are address-
ing poverty. By identifying who the poor are and where they are located,
poverty measurements can help direct resources and focus efforts more effec-
tively. The measurements create a picture of the magnitude of the problem
and the way it varies over space and time. Measurements can help identify
programs that work well in addressing poverty. Civil society groups can use
information on poverty as evidence of unaddressed needs and missing ser-
vices. Governments can be held accountable for their policies. Analysts can
explore the underlying relationships between poverty and other economic
and social variables to obtain a deeper understanding of the phenomenon.
How can poverty be measured? The process has three main steps:
1. Choose the space in which poverty will be assessed. The traditional

space has been income, consumption, or some other welfare indicator
measured in monetary units. This book will focus on the traditional
space (although attention is turning to other dimensions, such as
opportunities and capabilities).
2
Chapter 1: Introduction
2. Identify the poor. This step involves selecting a poverty line

that indicates the minimum acceptable level of income or con-
sumption.
3. Aggregate the data into an overall poverty measure. Headcount
ratio is the most basic measure. It simply calculates the share of
the population that is poor. But following the work of Amartya
Sen, other aggregation methods designed to evaluate the depth
and severity of poverty have become part of the poverty analyst’s
standard toolkit.1
Applying and interpreting poverty measures require understanding the

methods used to assess two other aspects of income distribution: its spread
(as evaluated by an inequality measure like the Gini coefficient) and its
size (as gauged by an “income standard” like the mean or median income).
There are several links between income inequality, poverty, and income
standards. For instance, inequality and poverty often move together—
particularly when growth in the distribution is small and its size is relatively
unchanged.
Other links exist for individual poverty measures. To gauge the depth
of poverty, a poverty measure can assess the size of the income distribution
among the poor—or a poor income standard. Other measures incorporate a
special concern for the poorest of the poor and are sensitive to the income
distribution among the poor. This sensitivity takes the form of including a
measure of inequality among the poor within the measure of poverty. Thus,
to measure and to understand the many dimensions of income poverty,
one must have a clear understanding of income standards and inequality
measures.
This chapter is a conceptual introduction to poverty measurement and
the related distributional analysis tools. It begins with a brief discussion
of the variable and data to be used in poverty assessment. It then discusses
the various income standards commonly used in distributional analysis.
Inequality measures are then introduced, and their meanings in income
standards are presented. The final part of this introduction combines those
elements to obtain the main tools for evaluating poverty.
The second chapter complements this introduction by providing a
detailed outline and more formal analysis of the concepts introduced here,
and follows the composition of this chapter closely. The third chapter and
3
the appendix includes tables and figures that may be useful in understanding
some of the concepts and examples in the first two chapters.
The Income Variable
Our discussion begins with the variable income, which may also represent
consumption expenditure or some other single dimensional outcome vari-
able. Data are typically collected at the household level. So to construct an
income variable at the individual level, one must make certain assumptions
about its allocation within the household. Using these assumptions, house-
hold data are converted into individual data that indicate the equivalent
income level an individual commands, thereby taking account of household
structure and other characteristics.
One simplification is to assume that overall income is spread evenly
across each person in the household. However, many other equivalence scales
can be used. This adjustment enables comparisons to be made symmetri-
cally across people irrespective of household or other characteristics. This
simplification justifies the assumption of symmetry invoked when evaluating
income distributions—whereby switching the (equivalent) income levels
of two people leaves the evaluation unchanged. Additionally, it is assumed
that the resulting variable can be measured with a cardinal scale that allows
comparison of income differences across people.
The Data
Income distribution data can be represented in a variety of ways. The

simplest form is a vector of incomes, one for each person in the specified
population. This format naturally arises when the data are derived from
a population census. The population distribution may be proxied by an
unweighted sample, which yields a vector of incomes, each of which rep-
resents an equal share of the population. It can also be represented by a
weighted sample, which differentiates across observations in the vector in a
prescribed way. For clarity, we will focus on the equal-weighted case here.
Of course, a sample carries less information than does a full census, but
the extent of the loss can be gauged and accounted for via statistical analysis.
One further assumption must be made at this point: the evaluation method is
4
invariant to the population size, in that a replication of the vector (having,

say, k copies of each observation for every original observation) is evaluated in
the same way as the original sample vector. This population invariance assump-
tion ensures that the method can be applied directly to a sample vector when
attempting to evaluate a population. More generally, the method depends on
a distribution function, which normalizes the population size to one.
The second way of representing an income distribution is with a cumu-
lative distribution function (cdf), in which each level of income indicates
the percentage of the population having that income level or lower. A
cdf automatically treats incomes symmetrically or anonymously (in that it
ignores who has what income) and is invariant with respect to the popula-
tion size. It is straightforward to construct the cdf for a particular vector of
incomes as a step function that jumps up by 1/N for each observation in the
vector, where N is the number of observations. For large enough samples,
the income distribution can be approximated by a continuous distribution
having a density function whose integral up to an income level is the value
of the cdf at that income level.
Whereas a cdf is a standard representation, one that is even more intui-
tive in the present context is the quantile function. The quantile function
gives the minimum income necessary to capture a given percentage p of
the population, so that, for example, the quantile at p = 12.5 percent is
the income level above which 87.5 percent of the population lies. For the
case of a strictly increasing and continuous cdf, the quantile function is the
inverse of the cdf found by rotating the axes. If the cdf has flat portions or
jumps up discontinuously, then certain alterations to the rotated function
must be made to obtain the quantile function. Another version of the quan-
tile function is Pen’s Parade, which displays the distribution as an hour-long
parade of incomes from lowest to highest.
Income Standards and Size
Given an income distribution, three separate but related aspects are of inter-
est: the distribution’s size, the distribution’s spread, and the distribution’s
base. We will discuss the size issue here. Subsequent sections deal with the
spread and base concepts.
Distribution size is most often indicated by the mean or per capita income.
For the vector representation, the mean is obtained by dividing total income
5
by the total number of people in the distribution. The mean can also be
viewed as the average height (or, in mathematical terms, the integral) of
the quantile function. It is the income level that all people would achieve if
they were given an equal share of overall resources.
Another size indicator, median income, is the income at the midway point
of the quantile function, with half the incomes below and half above. Most
empirical income distributions are skewed so that the mean (which includes
the largest incomes in the averaging process) exceeds the median income
(which is unaffected by the values of the largest incomes). Still another
measure of size is given by the mean income of the lowest fifth of the popula-
tion, which focuses exclusively on the lower incomes in a distribution. Each
of these indicators is an example of an income standard, which reduces the
overall income distribution to a single income level indicating some aspect
of the distribution’s size.
What Is an Income Standard?
To understand what a measure or index means, explicitly stating a set of

properties that it should satisfy is helpful. In the case of an income standard,
there are several requirements that go beyond the basic symmetry and popu-
lation invariance discussed above:
• Normalization states that if all incomes happen to be the same, then

the income standard must be that commonly held level of income—a
natural property indeed.
• Linear homogeneity requires that if all incomes are scaled up or down
by a common factor, then the income standard must rise or fall by
that same factor.
• Weak monotonicity requires the income standard to rise, or at least not
fall, if any income rises and no other income changes.
These basic requirements ensure that the income standard measures

the size of the income distribution as a “representative” income level that
responds “in the right way” when incomes change (for example, these
requirements rule out envy effects). It is easy to see that the size indicators
discussed in the previous section—mean, median, and mean of the lowest
fifth—conform to these general requirements.
6
Common Examples
Four types of income standards are in common use:
• First are the quantile incomes, such as the income at the 10th per-
centile, the income at the 90th percentile, and the median. Each is
informative about a specific point in the distribution but ignores the
values of the remaining points.
• Next are the (relative) partial means obtained by finding the mean of
the incomes below a specific percentile cutoff (the lower partial means)
or above the cutoff (the upper partial means), such as the mean of the
lowest 20 percent and the mean of the highest 10 percent. Each of
these income standards indicates the size of distribution by focusing
on one or the other side of a given percentile and by measuring the
average income of this range while ignoring the rest. As the cutoff
varies between 0 percent and 100 percent, the lower partial mean
varies between the lowest income and the mean income, whereas the
upper partial mean varies between the mean income and the highest
income.
By focusing on a specific income or a range of incomes, the quantile
incomes and the partial means ignore income changes outside that
range. The remaining two forms of income standard, by contrast, are
monotonic so that the increase in income causes the income standard
to strictly rise.
• The general means take into account all incomes in the distribution,
but emphasize lower or higher incomes depending on the value of
parameter a (that can be any real number). When a is nonzero, the
general mean is found by raising all incomes to the power a, then
by averaging, and finally by taking the result to the power 1/a. This
process of transforming incomes and then untransforming the aver-
age ensures that the income standard is, in fact, measured by income
(or, in income space, as we might say).
In the remaining case of a = 0, the general mean is defined to be
the geometric mean. It is obtained by raising all incomes to the power
1/N, then taking the product. For a < 1, incomes are effectively trans-
formed by a concave function, thus placing greater emphasis on lower
incomes. For a > 1, the transformation is convex, and the general
mean emphasizes higher incomes.
7
As a varies across all possible values, the general mean rises from
minimum income (as a approaches −∞), to the harmonic mean
(a = −1), the geometric mean (a = 0), the mean (a = 1), the Euclidean
mean (a = 2), up to the maximum income (as a approaches ∞).
General means with a < 0 are only defined for positive incomes.
• The final income standard is a step in the direction of a “maximin”
approach, which evaluates a situation by the condition of the least
advantaged person. The usual mean can be reinterpreted as the
expected value of a single income drawn randomly from the popula-
tion. Suppose that instead of a single income, we were to draw two
incomes randomly from the population (with replacement). If we
then evaluated the pair by the lower of the two incomes, this would
lead to the Sen mean, defined as the expectation of the minimum of
two randomly drawn incomes.
Because we are using the minimum of the two, this number can be no
higher than the mean and is generally lower. Consequently, the Sen mean
also emphasizes lower incomes but in a different way to the general means
with a < 1, the lower partial means, or the quantile incomes below the
median.
Calculating the Sen mean for an income vector is straightforward.
Create an N × N matrix that has a cell for every possible pair of incomes,
and place the lower value of the two incomes in the cell. Add up all the
entries and divide by the number of entries (N2) to obtain their mean,
which is the expected value of the lower income. This mean has close ties
to the well-known Gini coefficient measure of inequality.
Welfare
The general means for a < 1 and the Sen mean are also commonly inter-
preted as measures of welfare. The key additional property that allows this
interpretation is the transfer principle, which requires an income transfer
from one person to another who is richer (or equally rich) to lower the
income standard. In other words, a regressive transfer that does not change
the mean income should lower the income standard.
One way to justify this property begins with a utilitarian symmetric
welfare function that views welfare derived from an income distribution
to be the average level of (indirect) utility in society, where it is assumed
8
that everyone’s utility function is identical and strictly increasing. In this

context, the intuitive assumption of diminishing marginal utility (each
additional dollar leads to a higher level, but a lower increment, of utility)
yields the transfer principle, because the loss to the giver is greater than the
gain to the richer receiver.
This form of welfare function was considered by Atkinson (1970), who
then defined a helpful transformation of the welfare function called the
equally distributed equivalent income (ede). The ede is that income level which,
if received by all people, would yield the same welfare level as an original
income distribution. The particular ede he focused on was, in fact, the gen-
eral mean for a < 1. Sen suggested going beyond the utilitarian form. One
key nonutilitarian example is the Sen mean, which can be viewed as both
an ede and a general welfare function and also satisfies the transfer principle.
Applications
Income standards are used to assess a population’s prosperity, the way it

compares to other populations, and the way it progresses through time. The
most common examples are country-level assessments of mean or per capita
income and its associated growth rate. This is a mainstay of the growth lit-
erature, and many interesting economic questions about the determinants of
growth and its effect on other variables of interest have been addressed. In
the recent example of The Growth Report: Strategies for Sustained Growth and
Inclusive Development (Commission on Growth and Development 2008),
countries with high and sustained levels of growth in the mean income were
evaluated to see if the factors that made this possible could be identified.
Imagine undertaking a similar study with a different income standard
to focus on one part of the income distribution or, perhaps, even exam-
ining growth in a different underlying variable. Some studies use the
median income, arguing that it corresponds more naturally to the middle
of the income distribution (see, for example, the report by the Commission
on the Measurement of Economic and Social Progress [2009], also known as
the Sarkozy Report). Other authors have used the mean of the lowest fifth of
the population, or a general mean (with a < 0) as a low-income standard, to
examine how growth in one income standard (the mean) relates to growth
in the incomes of the poor. Because each income standard measures the
distribution’s size in a distinct way, examining several at once can clarify the
quality of growth—including whether it is shared or pro-poor growth.
9
Subgroup Consistency
In certain empirical applications, there is a natural concern for certain iden-

tifiable subgroups of the population as well as for the overall population. For
example, one might be interested in the achievements of the various states
or subregions of a country to understand the spatial dimensions of growth.
When population subgroups are tracked alongside the overall population
value, there is a risk that the income standard could indicate contradictory
or confusing trends.
A natural consistency property for an income standard might be that if
subgroup population sizes are fixed but incomes vary, then when the income
standard rises in one subgroup and does not fall in the rest, the overall
population income standard must rise. This property is known as subgroup
consistency; and using a measure that satisfies it avoids inconsistencies aris-
ing from this sort of multilevel analyses. In fact, several income standards
discussed above do not survive this test and, hence, may need to be avoided
when undertaking regional evaluations or other forms of subgroup analyses.
The mean of the lowest 20 percent is subject to this critique because a
given policy could succeed in raising the mean of the lowest 20 percent in
every region of a given country; yet the mean of the lowest 20 percent in
the overall population could fall. The same is true of the Sen mean or the
median. In contrast, every general mean satisfies the consistency require-
ment. In fact, it can be shown that the general means are the only income
standards that are subgroup consistent while satisfying some additional basic
properties.
Moreover, each of the general means has a simple formula that links
regional levels of the income standard to the overall level. If one were
to go further and specify an additive aggregation formula across subgroup
standards—a requirement that might be called additive decomposability—the
only general mean that would survive is the mean itself. The overall mean
is just the population-weighted sum of subgroup means.
Dominance and Unanimity
One motivation for examining several income standards at the same time is
robustness: Do conclusions about the direction of change in the distribution
size using one income standard (say, the mean) hold for others (say, the
nearby generalized means)? A second reason might be focus or an identified
10
concern with different parts of the distribution: Has rapid growth at the top
(say, the 90th percentile income) been matched by growth at the middle
(say, the median) or the bottom (say, the 10th percentile income)?
We can answer questions of this sort by plotting an entire class of income
standards against percentiles of income distribution. We can then use the
curve to determine if a given comparison is unambiguous (one curve is
above the other) or if it is contingent (the curves cross).
A first curve is given by the quantile function itself, which simultane-
ously depicts incomes from lowest to highest. As income standards, quan-
tiles are somewhat partial and insensitive—yet when they all agree that
one distribution is larger than another, this ensures that every other income
standard must follow their collective judgment.
The quantile function represents first-order stochastic dominance, which
also ensures higher welfare according to every utilitarian welfare function
with identical, increasing utility functions. Thus, on the one hand, the
robustness implied by an unambiguous comparison of quantile functions
extends to all income standards and all symmetric welfare functions for
which “more is better.” On the other hand, if some quantiles rise and others
fall, then the resulting curves will cross and the final judgment is contingent
on which income standard is selected. In this case, the quantile function can
be helpful in identifying winning and losing portions of the distribution.
A second curve of this sort is given by the generalized Lorenz curve, which
graphs the area under the quantile function up to each percent p of the
population. It can be shown that the height of the generalized Lorenz curve
at any p is the lower partial mean times p itself. For example, if the lowest
income of a four-person vector were 280, then the generalized Lorenz curve
value (ordinate) at p = 25 percent would be 70.
A comparison of generalized Lorenz curves conveys information on
lower partial means, with a higher generalized Lorenz curve indicating
agreement among all lower partial means. As income standards, the lower
partial means are insensitive to certain increments and income transfers.
Yet when all these income standards are in agreement, it follows that every
monotonic income standard satisfying the transfer principle would abide by
their judgment.
Indeed, the generalized Lorenz curve represents second-order stochas-
tic dominance, which signals higher welfare according to every utilitarian
welfare function with identical and increasing utility function exhibiting
diminishing marginal utility (Atkinson’s general class of welfare functions).
11
However, if generalized Lorenz curves cross, then the final judgment is

contingent on which monotonic income standard satisfying the transfer
principle is employed.
Notice that when quantile functions can rank two distributions, gen-
eralized Lorenz curves must rank them in the same way, because a higher
quantile function ensures that the area beneath it is also greater. However,
even when quantile functions cross, generalized Lorenz curves may be
able to rank the two distributions. We will use these two curves and their
stochastic dominance rankings later in discussing inequality and poverty
measurement.
A final curve depicts the general mean levels as the parameter a var-
ies. Given the properties of the general means, this curve is increasing in
a and tends to the minimum income for very low a and rises through the
harmonic, geometric, arithmetic, and Euclidean means, tending toward the
maximum income as a becomes very large. A higher quantile function will
raise the general mean curve. A higher generalized Lorenz curve will raise
the general mean curve for a < 1 or the general means that favor the low
incomes. The curve is useful for determining whether a given comparison of
general means is robust, and if not, which of the income standards are higher
or lower. It will also be particularly relevant to discussions of inequality in
later sections.
Growth Curves
Some analyses go beyond the question of which distribution is larger to con-

sider the question of how much larger in percentage terms is one distribution
than another. This question is especially salient when the two distributions
are associated with the same population at two points in time. Then the
next question becomes at what percentage rate did the income standard
grow. Such growth is most often defined by income per capita, or the mean
income. However, the defining properties of an income standard ensure that
its rate of growth is a meaningful number that can be compared with the
growth rates of other income standards, either for robustness purposes or for
an understanding of the quality of growth.
A growth curve depicts the rate of growth across an entire class of income
standards, where the standards are ordered from lowest to highest. Each of
the dominance curves described above suggests an associated growth curve.
12
For the quantile function, the resulting growth curve is called the growth
incidence curve. The height of the curve at p = 50 percent gives the growth
rate of the median income. Varying p allows us to examine whether this
growth rate is robust to the choice of income standard or whether the lower
income standards grew at a different rate than the rest.
The generalized Lorenz growth curve indicates how the lower partial means
are changing over time, so that the height of this curve at p = 20 percent is
the rate at which the mean income of the lowest 20 percent of the popula-
tion changed over time. Finally, the general means growth curve plots the
rate of growth of each general mean against the parameter a. When a = 1,
the height of the curve is the usual growth rate of the mean income; a = 0
yields the rate of growth for the geometric mean, and so forth. As we will
see below, each of these growth curves can be of help in understanding the
link between growth and the evolution of inequality over time.
Inequality Measures and Spread
The second aspect of the distribution—spread—is evaluated using a numeri-

cal inequality measure, which assigns each distribution a number that
indicates its level of inequality. The Gini coefficient is the most commonly
used measure of inequality. It measures the average or expected difference
between pairs of incomes in the distribution, relative to the distribution size,
and also is linked to the well-known Lorenz curve (discussed below). The
Kuznets ratio measures inequality as the share of the income going to the top
fifth divided by the income share of the bottom two-fifths of the population.
Finally, the 90/10 ratio is the income at the 90th percentile divided by the
10th percentile income. It is often used by labor economists as a measure of
earnings inequality. These are just a few of the many inequality measures
used to evaluate income distribution.
What Is an Inequality Measure?
There are two main ways to understand what an income inequality measure
actually gauges. The first way is through the properties it satisfies. The
second makes use of a fundamental link between inequality measures and
income standards. We begin with the first approach.
13
There are four basic properties for inequality measures:
• The first two are symmetry and population invariance properties, which
are analogous to those defined for income standards. They ensure that
inequality depends entirely on income distribution and not on names
or numbers of income recipients.
• The third is scale invariance (or homogeneity of degree zero), which
requires the inequality measure to be unchanged if all incomes are
scaled up or down by a common factor. This ensures that the inequal-
ity being measured is a purely relative concept and is independent of
the distribution size. In contrast, doubling all incomes will double
distribution size as measured by any income standard, thereby reflect-
ing its respective property of linear homogeneity.
• The final property is the weak transfer principle, which in this context
requires income transfer from one person to another who is richer
(or equally rich) to raise inequality or leave it unchanged. In other
words, a regressive transfer cannot decrease inequality. This is an
intuitive property for inequality measures. It is often presented in a
stronger form, known as the transfer principle, which requires a regres-
sive transfer to (strictly) increase inequality.
The Gini coefficient and the Kuznets ratio satisfy all four basic properties
for inequality measures. The 90/10 ratio satisfies the first three but violates the
weak transfer principle: a regressive transfer between people at the 5th percen-
tile and the 10th percentile can raise the 10th percentile income, thus lowering
inequality as measured by the 90/10 ratio. Although this result does not rule
out the use of the intuitive 90/10 ratio as an inequality measure, it does suggest
that conclusions obtained with this measure should be scrutinized.
The four basic properties define the general requirements for inequality
measures. Additional properties help to discern between acceptable mea-
sures. For example, decomposability and subgroup consistency (discussed in a
later section) are helpful in certain applications. Transfer sensitivity ensures
that an inequality measure is more sensitive to changes in the income dis-
tribution at the lower end of the distribution.
A second way of understanding inequality measures relies on an intui-
tive link between inequality measures and pairs of income standards. The
basic structure is perhaps easiest to see in the extreme case where there are
only two people and, hence, only two incomes. Letting a denote the smaller
14
income of the two, and b denote the larger income, it is natural to measure
inequality by the relative distance between a and b, such as I = (b − a)/b,
or some other increasing function of the ratio b/a. Indeed, scale invariance
and the weak transfer principle essentially require this form for the measure.
Suppose that instead of evaluating the inequality between two people, we
want to measure the inequality between two equal-sized groups. A natural
way of proceeding is to represent each group’s income distribution using an
income standard. This yields a pair of representative incomes—one for each
group—that can then be compared. Where a denotes the smaller of these
two incomes and b the larger, it is natural to measure inequality between the
two groups as I = (b − a)/b, or some other increasing function of the ratio
b/a. For example, if the distributions are the earnings of men and women and
the income standard is the mean, then b/a would be the ratio of the aver-
age income for men to the average income for women—a common indica-
tor of inequality between the two groups. As will be discussed below, this
“between-group” approach is useful in decompositions of inequality by popu-
lation subgroup and also in the measurement of inequality of opportunities.
The general idea that inequality depends on two income standards is also
relevant when evaluating the overall inequality in a population’s distribu-
tion of income. But instead of applying the same income standard to two
distributions, we now apply two income standards to the same distribution.
One of the income standards (the upper standard) places greater weight
on higher incomes, and the second (the lower standard) emphasizes lower
incomes; so for any given income distribution, the lower-income standard’s
value is never larger than the upper-income standard’s value.
This is true, for example, when the lower standard is the geometric mean
and the upper is the arithmetic mean or, alternatively, when the lower is
the 10th percentile income and the upper is the 90th percentile income.
Inequality is then seen as the extent to which the two income standards are
spread apart: where a denotes the lower-income standard and b the upper-
income standard, overall inequality is I = (b − a)/b, or some other increasing
function of the ratio b/a.
Common Examples
Virtually all inequality measures in common use are based on twin income
standards. This is easily seen in the case of the 90/10 ratio, and generalizes to
any quantile ratio b/a, where a corresponds to the income at a percentile p of
15
the distribution and b is the income at a higher percentile q of the distribu-

tion. The quantile incomes are relatively insensitive income standards, and
hence they yield inequality measures that are somewhat crude and that dis-
agree with the weak transfer property that is traditionally regarded as a basic
property of inequality measures. Nonetheless, they succeed at conveying
tangible information about the distribution—namely, the extent to which
two quantile incomes differ from one another—and can be informative, if
crude, measures of inequality.
The Kuznets ratio has as its twin income standards the mean of those
from 40 percent downward and the mean of those from 80 percent upward,
respectively. This can be generalized to any ratio of two standards of this form
by varying the cutoffs. The resulting measure, which we call the partial mean
ratio, is given by b/a, where a is the lower partial mean at p and b is the upper
partial mean at q. The case where p = 10 percent and q = 90 percent is often
called the decile ratio. Another related measure is the income share of the top
1 percent, which is a multiple of the partial mean ratio with p = 100 percent
and q = 99 percent. Although each partial mean ratio satisfies four basic
properties of an inequality measure, the component income standards are
still rather crude and focus on only a limited range of incomes. Those falling
outside the range are ignored entirely, while the income distribution within
the range is also not considered. The resulting measure is thus insensitive to
certain transfers. As before, though, the twin standards and their ratio convey
tangible and easily understood information about the income distribution.
The Gini coefficient is defined as the expected (absolute) differ-
ence between two randomly drawn incomes divided by twice the mean.
Calculating the Gini coefficient is therefore straightforward:
1. Create an N × N matrix having a cell for every possible pair of

incomes, and place the absolute value of their difference in the cell.
2. Add all the entries and divide by the number of entries (N2) to
obtain the expected value of the absolute difference between two
randomly drawn incomes.
3. Divide by two times the mean income of the distribution to obtain
the Gini coefficient. It is a natural indicator of how “spread out”
incomes are from one another.
The Gini coefficient has as its twin income standards the mean and the
Sen mean and can be written as I = (b − a)/b, where b is the mean and a is
16
the Sen mean. The expected (absolute) difference between two incomes
can be written as (a′ − a), where a′ is the expectation of their maximum and
a is the expectation of their minimum. Because the mean b can be written
as (a′ + a)/2, the difference (b − a) is half of the expected absolute difference
between incomes, which confirms that (b − a)/b is an equivalent definition
of the Gini coefficient. In other words, the Gini coefficient is the extent to
which the Sen mean falls below the mean as a percentage of the mean.
Atkinson’s class of inequality measures also takes the form I = (b − a)/b,
where the upper-income standard b is also the mean, but now the lower-
income standard a is a general mean with parameter a < 1. This income
standard focuses on lower incomes by raising each income to the a power,
averaging across all the transformed incomes, then converting back to
income space by raising the result to the power 1/a. A lower value of the
parameter a yields an income standard that is more sensitive to lower
incomes and is lower in value. This will be reflected in a higher value for
(b − a)/b, so the percentage loss from the mean is seen to be higher.
The final example is the family of generalized entropy measures, whose
definition and properties vary with a parameter a. There are three distinct
ranges for the parameter: a lower range where a < 1, an upper range where
a > 1, and a limiting case where a = 1.
When a < 1, the generalized entropy measures evaluate inequality in
the same way as the Atkinson class of inequality measures (and, in fact, are
monotonic transformations). For example, when a = 0, the measure is the
mean log deviation or Theil’s second measure given by ln(b/a), where b is the
arithmetic mean and a is the geometric mean. Atkinson’s version is (b − a)/b.
Over the second range where a > 1, the general mean places greater
weight on higher incomes and yields a representative income that is typi-
cally higher than the mean. If all incomes were equal, the general mean
and the mean would be equal. However, when incomes are unequal, the
general mean will rise above the mean. The extent to which this occurs
is used by the measure to evaluate inequality. For example, the inequality
measure obtained when a = 2 is (half) the squared coefficient of variation, that
is, one-half of the variance over the squared mean. The general mean in this
case is the Euclidean mean, which first squares all incomes, then averages
the transformed incomes, and finally returns to income space by taking the
square root. The Euclidean mean and the mean of the two-income distribu-
tion (4, 4) are both 4. Altering the distribution to (1, 7) raises the Euclidean
mean to 5 but leaves its mean at 4.
17
The final case where a = 1 leads to Theil’s first measure, which is one of
the few inequality measures without a natural twin standards representation,
but is, in fact, a limit of such measures.
Inequality and Welfare

The Gini coefficient and Atkinson’s family share a social welfare interpreta-
tion. Both are expressible as I = (b − a)/b, where b is the mean income of
the distribution and a is an income standard that can be viewed as a welfare
function (satisfying the weak transfer principle). Note that the distribution
where everyone has the mean has a level of welfare that is highest among all
distributions with the same total income, and its measured level of welfare
is just the mean itself (by the normalization property of income standards).
The mean b is the maximum value that the welfare function can take
over all income distributions of the same total income. When incomes are
all equal, a = b and inequality is zero. When the actual welfare level a falls
below the maximum welfare level b, the percentage welfare loss I = (b − a)/b
is used as a measure of inequality. This is the welfare interpretation of both
the Gini coefficient and the Atkinson class.
The simple structure of these measures allows us to express the welfare
function by the mean income and the inequality measure. A quick rear-
rangement leads to a = b(1 − I), which can be reinterpreted as the welfare
function a viewed as an inequality-adjusted mean. If there is no inequality
in the distribution, then (1 − I) = 1 and a = b. If the inequality level is
I > 0, then the welfare level is obtained by discounting the mean income
by (1 – I) < 1. For example, if we take I to be the Gini coefficient, the Sen
mean (or Sen welfare function) can be obtained by multiplying the mean by
(1 − I). Similarly, if we take I to be the Atkinson measure with parameter
a = 0, then the welfare function is the geometric mean, and it can be
obtained by multiplying the mean by (1 − I).
Applications
Inequality measures are used to assess the extent to which incomes are
spread apart in a country or region and the way this level changes over time
and space. Of particular interest is the interplay between a population’s aver-
age prosperity, as represented by the mean income, and the income distribu-
tion, as represented by an inequality measure. The positive achievement of
18
a high per capita income can be viewed less favorably if inequality is high,
too. The combined effect on welfare can be evaluated using an inequality-
adjusted mean.
The Kuznets hypothesis postulates that growth in per capita income ini-
tially comes at a cost of a higher level of inequality, but eventually inequal-
ity falls with growth. The resulting Kuznets curve, which depicts per capita
income on the horizontal axis and inequality on the vertical axis, has the
shape of an inverted U. If the hypothesis were true, then a rapidly grow-
ing developing country could have only moderate welfare improvements,
whereas a moderately growing developed country could experience rapid
improvements in welfare, all because of the changing levels of inequality.
An alternative view takes the initial level of inequality as one of the
determinants of income growth. For example, greater inequality might lead
to a higher average savings rate if the richer groups have a greater propen-
sity to save, and this can positively influence long-term growth. Conversely,
high inequality might create political pressure to raise the marginal tax rate
on the rich, which could diminish incentives to invest and grow. These
applications of inequality measures view inequality as a valuable macro
indicator of the health of a country’s economy that influences and is affected
by other macro variables.
Other applications try to assess the origins of inequality in the micro
economy. Could inequality in earned incomes be due to (a) a high return
to education, (b) a decline in union power, (c) increased competition from
abroad, (d) discrimination, or (e) demographic changes such as increased
female labor force participation? Mincer (1974) equations can help trace
earnings inequality to the underlying characteristics of the labor force,
including the level and distribution of human capital. Oaxaca decomposi-
tions (1973) test for the presence of discrimination by sex, race, or other
characteristics and have been adapted to evaluate the contribution of demo-
graphic changes to observed earnings inequality.
Depending on the policy question, it may make sense to move from
an overall inequality measure (that evaluates the spread across the entire
distribution) to a group-based inequality measure (that compares the mean
or other income standard across several groups). The latter, more limited,
notion of inequality can often have greater significance, particularly if
the underlying groups are easy to understand and have social or political
salience. Examples include racial, sex, and ethnic inequality, or the inequal-
ity between urban and rural areas.
19
The techniques for evaluating between-group inequality involve smoothing

incomes within each subgroup to the subgroup mean (or other income stan-
dard) and then applying an inequality measure to the resulting smoothed
distribution. Because the inequality within groups is suppressed, all that is
left is between-group inequality.
Similar techniques have recently been employed to evaluate the inequal-
ity of opportunity in a given country. This exercise begins by identifying
circumstances or the characteristics of a person that are not under the direct
control of that person and arguably should not be systematically linked
to higher or lower levels of income. The population is then divided into
subgroups of people sharing the same circumstances and the distribution is
smoothed to suppress inequalities within subgroups. The inequality of the
smoothed distribution then measures how much inequality is present across
subgroups and, hence, how much is associated with circumstances. It can
be viewed as a measure of the inequality of opportunity (given the posited
circumstances).
The overall inequality in a country could be very high. But if the three
main ethnic groups have more or less the same average levels of income,
inequality of opportunity across the ethnic groups may not be such an
important issue—much of the inequality arises from variations within eth-
nic groups. If the mean incomes vary greatly across ethnic groups so that the
between-group inequality level is also quite high, then a concern for social
stability may lead policy makers to address the high level of inequality of
opportunity.
Analogous discussions might be made for other indicators besides
income. For instance, if we are evaluating the distribution of health, then
the way that health varies across subgroups defined by an indicator of socio-
economic status (SES)—such as occupation, income, education, or education
of the parents—may be more salient than the overall distribution of health.
The strength of the gradient or positive relationship between health and SES
variables is often viewed as a key indicator of the inequity of health and is the
target of policies to affect this particularly objectionable portion of health
inequalities.
Different inequality measures have properties that make them well
suited for certain applications. Decomposability is one such property dis-
cussed below. A second is transfer sensitivity, which ensures that a measure
is especially sensitive to inequalities at the lower end of the distribution
(in that a given transfer of income will have a greater effect the lower the
20
incomes of the giver and the receiver). Transfer sensitive measures include
the Atkinson family of measures, Theil’s two measures, and the “lower half”
of the generalized entropy measures with a < 2.
Note that the coefficient of variation (a monotonic transformation of
the generalized entropy measure with a = 2) is transfer neutral in that a
given transfer has the same equalizing effect up and down the distribution:
a one-unit transfer of income between two rich people has the same effect
on inequality as does a one-unit transfer of income between two poor people
the same initial income distance apart. The upper half of the generalized
entropy measures with a > 2 focuses on inequality among upper incomes.
The Gini coefficient is often considered to be most sensitive to changes
involving incomes at the middle, but this is not entirely accurate. The effect
of a given-sized transfer on the Gini coefficient depends on the number of
people between giver and receiver, not on their respective income levels.
Because, empirically, there tend to be more observations bunched together
in the middle of the distribution, the effect of a transfer near the middle
tends to be larger.
Subgroup Consistency and Decomposability
Although the variance is not itself a measure of relative inequality (it vio-
lates scale invariance and focuses on absolute differences), the analysis of
variance (ANOVA) provides a natural model for decomposition of inequal-
ity measures into a within-group and a between-group term. The motivating
question here is given a collection of population subgroups, how much of
the overall inequality can be attributed to inequality within the subgroups,
and how much can be attributed to inequality across the subgroups.
Answers to this type of question become feasible when an inequality mea-
sure is additively decomposable, in which case the within-group inequality term
is expressible as a weighted sum of the inequality levels within the groups, the
between-group term is the inequality measure applied to the smoothed distri-
bution, and the overall inequality level is just the sum of the within-group and
between-group terms. The contributions of within-group and between-group
inequality (within-group inequality divided by total inequality and between-
group inequality divided by total inequality, respectively) will sum to one.
Decomposition analysis can help clarify the structure of income inequal-
ity across a population. It can identify regions or sectors of the economy
that disproportionally contribute to inequality. And when the subgroups are
21
defined with reference to an underlying variable such as schooling, the anal-

ysis can help identify the extent to which the variable explains inequality.
To analyze changes in inequality over time, one can separate the effect
of changes in population sizes across subgroups (for example, arising from
demographic factors) from the fundamental shifts in subgroup income dis-
tributions. This can be combined with regression analysis to model income
changes and to pinpoint the variables that appear to be driving inequality.
The generalized entropy measures are the only inequality measures sat-
isfying the usual form of additive decomposability, with the Theil measures
(a = 0 and a = 1) and half the squared coefficient of variation (a = 2) being
most commonly used in empirical evaluations. The second Theil measure,
also called the mean log deviation, has a particularly simple decomposition
in which the within-group term is a population-share weighted average of
subgroup inequality levels. This streamlined weighting structure can greatly
simplify interpretation and application of decomposition analyses.
The allied property of subgroup consistency is helpful in ensuring
that regional changes in inequality are appropriately reflected in overall
inequality. Suppose there is no change in the population sizes and mean
income levels of the subgroups. If inequality rose in one subgroup and was
unchanged or rose in each of the other subgroups, it would be natural to
expect that inequality overall would rise. For additively decomposable mea-
sures, this rise in inequality is assured: because the smoothed distribution is
unchanged, the between-group term is unaffected. Because the weights on
subgroup inequality levels are fixed (when subgroup means and population
sizes do not change), the within-group term must rise.
Subgroup consistency is a more lenient requirement, because it does not
specify the functional form that links subgroup inequality levels and overall
inequality. Consequently, on the one hand we find that the Atkinson mea-
sures (which are transformations of the generalized entropy measures) are all
subgroup consistent without being additively decomposable. On the other
hand, the Gini coefficient is not subgroup consistent.
The problem with the Gini coefficient arises when the income ranges
of the subgroup distributions overlap. In that case, the effect of a given dis-
tributional change on subgroup inequality can be opposite to its effect on
overall inequality. The Gini coefficient can be broken into a within-group
term, a between-group term, and an overlap term—and it is the overlap
term that can override the within-group effect to generate subgroup incon-
sistencies.
22
One alternative to numerical inequality measures for making inequality

comparisons is the so-called Lorenz curve and its associated criterion of
Lorenz dominance. The Lorenz curve graphs the share of income received
by the lowest p percent of the population as p varies from 0 percent to
100 percent. A completely equal distribution yields a Lorenz curve where
the lowest p percent receives p percent of the overall income, or the
45 degree line. Inequality results in a Lorenz curve that falls below this line
in accordance with the extent and location of the inequality. When one
compares two distributions, a higher Lorenz curve is associated with lower
inequality. This is the case of Lorenz dominance in which one distribution
is unambiguously less unequal than another. Alternatively, if the two Lorenz
curves cross, no unambiguous determination can be made.
The Lorenz curve is a useful tool for locating pockets of inequality along
the distribution. For example, if a portion of the curve is straight, then there
is no inequality over that slice of the population. If it is very curved, then
there is significant inequality over the relevant population range. It also can
help determine if a given inequality comparison is robust to the choice of
inequality measure.
Indeed, when the Lorenz curve of one distribution dominates the Lorenz
curve of another distribution, it follows that every inequality measure sat-
isfying the four basic properties (symmetry, replication invariance, scale
invariance, and the weak transfer principle) will not go against this judg-
ment, whereas the subsets of measures satisfying the transfer principle are in
strict agreement with the Lorenz judgment (that the first has less inequality
than the second). So these unambiguous judgments are also unanimous
judgments across wide classes of inequality measures.
The Lorenz curve is also the generalized Lorenz curve divided by the mean.
At p = 0 percent, both curves have the value 0 percent; at p = 100 percent,
the Lorenz curve has the value 100 percent, whereas the generalized Lorenz
curve takes the mean as its value. At any percentage of the population p,
the generalized Lorenz curve is p times the associated lower partial mean at p,
and the Lorenz curve is p times the lower partial mean over the mean.
If one recalls the link between second-order stochastic dominance and
the generalized Lorenz curve, it follows that when the means of the two dis-
tributions under comparison are the same, a distribution has greater equality
according to Lorenz dominance exactly when it has higher welfare for the
23
general class of welfare functions considered by Atkinson. This is a very use-

ful result called Atkinson’s Theorem, which provides an interesting welfare
basis for (fixed mean) Lorenz comparisons.
There is a useful link between the points along the Lorenz curve and a
simple class of inequality measures. Consider the partial mean ratios obtained
when p is variable and q is fixed at 100 percent. With q = 100 percent, the
upper partial mean is the mean itself, and the partial mean ratio becomes a
comparison between a lower partial mean (for example, ap) and the overall
mean b.
Now consider the Lorenz curve evaluated at the pth percentile. The verti-
cal distance between the Lorenz curve and the 45-degree line of perfect equal-
ity is simply p times the inequality measure Ip = (b – ap)/b associated with the
partial mean ratio b/ap. Consequently, Lorenz dominance—which ensures
that one of the vertical distances is larger and the rest are no smaller—is
equivalent to the requirement that Ip is larger for some p and no smaller for
every remaining p. The Lorenz curve can thus be viewed as the dominance
curve associated with Ip or, equivalently, the associated partial mean ratios.
Although these measures are crude—evaluating inequality by comparing
the mean of the lowest p of the population to the overall mean—they col-
lectively imply Lorenz dominance and, hence, agreement for the entire set
of inequality measures satisfying the four basic properties.
Each of the three curves generated by a class of income standards—the
quantile curve, the generalized Lorenz curve, and the general means curve—
provides a natural way of depicting a related twin-standard inequality mea-
sure. Identify the two income standards a and b of the measure, and draw a
line segment connecting the associated points along the curve.
Note that the lower standard a is to the left and the higher standard b is
to the right. The relative slope of this line (or the slope relative to the value of
either a or b) is a proxy for the associated inequality level, with a higher rela-
tive slope implying a higher inequality level. For example, the relative slope
of the line connecting the 10th and the 90th percentile incomes along the
quantile curve represents the extent of inequality according to the 90/10 ratio.
Along the generalized Lorenz curve, the relative slope of the line from
p = 20 percent to q = 100 percent is linked to the associated partial mean
ratio discussed previously. Along the general mean curve, the relative slope
of the line from the geometric mean (a = 0) to the mean (a = 1) corresponds
to Theil’s second measure, or the mean log deviation.
24
A similar discussion applies to all the generalized entropy measures, apart

from Theil’s first measure. It is interesting to note that although Theil’s first
measure is not a twin-standard measure, it is represented as the relative slope
of the general mean curve at a = 1. In the extreme case where all incomes are
the same, the quantile and general means curves will be entirely flat, because
all the income standards are the same and correspond to the income level of
everyone. The generalized Lorenz curve is a straight line from 0 to the mean,
and the inequality measure (b − a)/b takes on the value 0 in this case.
Growth and Inequality
The twin-standard view of inequality offers fresh insights on the relation-

ship between growth and inequality. For example, use the Gini coefficient,
with its underlying income standards of the Sen mean a and the (arithme-
tic) mean b, to evaluate the distribution of income at two points in time. If
inequality as measured by the Gini coefficient has risen, then this is equiva-
lent to saying that b grew more between the two periods than a. But the
growth rate of b is precisely the usual income growth rate.
Consequently, to evaluate whether the change in the income distribu-
tion from one period to the next has increased or decreased the Gini coeffi-
cient, one need only calculate the growth rate of the Sen mean and compare
it to the usual growth rate. If the growth rate of the Sen mean is lower than
the usual growth rate, then the Gini coefficient rises. If the Sen growth rate
is larger than the usual growth rate, then the Gini coefficient falls.
An analogous discussion holds for Theil’s second measure, except that
now growth in the geometric mean is compared to the usual growth rate.
In both cases, the mean is the higher income standard, and the same would
be true for the generalized entropy measures below the first Theil measure
(or the Atkinson measures) and for the partial mean ratios underlying the
Lorenz curve.
In contrast, for the upper half of the generalized entropy measures, the
mean is the lower income standard a whereas the general mean is the higher
income standard b, so the growth criterion for inequality is reversed. For
example, the income standards of the squared coefficient of variation are the
mean income and the Euclidean mean. If the Euclidean mean growth rate
exceeds the usual growth rate, then the inequality level, as measured by the
squared coefficient of variation, rises.
25
The growth curves described above can be useful in understanding

the attributes of growth and the effect on inequality. Each depicts growth
rates for a class of income standards, starting with standards favoring lower
incomes to the left and with standards favoring higher incomes to the right.
In the proportional growth case where all incomes rise by the same percent-
age, the growth curves will be constant at that percentage level. If higher
incomes tend to be rising more rapidly, then the growth curve will have a
positive slope, thereby reflecting higher growth rates among the income
standards that emphasize higher incomes. If lower incomes are growing
more, then the growth curve will have a negative slope. The latter case
might be viewed as one form of pro-poor or inclusive growth.
Each growth curve has implications for the inequality measures associ-
ated with its constituent income standards. The growth incidence curve
reveals changes in inequality as measured by the quantile ratios (such as the
90/10 ratio). The generalized Lorenz growth curve provides information on
inequality as measured by its partial mean ratios. And the general means
growth curve reveals how inequality changes for virtually all generalized
entropy measures and for the Atkinson measures.
Poverty Measures and the Base of the Distribution
The final aspect examined here is the base or the bottom of the income
distribution and the main topic of this book: poverty. Evaluation of poverty
begins with an identification step in which the people considered poor are
specified and continues with an aggregation step in which the data of the
poor are combined to obtain a numerical measure. These two steps make up
a methodology for measuring poverty in an income distribution.
The identification step is usually accomplished by selecting a level of
income, called the poverty line, below which a person in a given distribution
is considered poor. In its most general formulation, a poverty line is specified
for every possible income distribution, so that the set of poor people in a pop-
ulation depends on the prevailing living conditions. Finding a proper func-
tional relation between poverty line and income distribution is, of course,
a challenging problem, and one that is subject to much controversy.
Most evaluations of poverty have settled on two very simple approaches:
(a) an absolute approach that takes the poverty line to be a constant and
26
(b) a relative approach that takes the poverty line to be a constant fraction
of an income standard.
Absolute Poverty Line
An absolute poverty line is a fixed cutoff that does not change as the distribu-
tion being evaluated changes. Examples include the following:
• The $1.25-per-day standard of the World Bank that is used to com-

pare poverty across many poor and middle-income countries over
time
• The domestic poverty lines in most developing countries that are
used to compare poverty within the country over time
• The nearly $15-per-day standard in the United States (per person
in a family of four in 2009 dollars) that has been used for almost
50 years.
An absolute poverty line is frequently used for evaluating poverty within

a country over short-to-moderate spans of time or across two countries when
they have roughly similar levels of development. The approach may be
harder to justify over longer periods of time or in a comparison of countries
with very different levels of development.
Absolute poverty lines are often held constant over many periods,
then updated to reflect changing living standards. After updating of lines,
comparisons are typically not made across the two standards. Instead, each
distribution is evaluated at the new, updated poverty line. The U.S. poverty
line has remained fixed (in real terms) since 1965; the nominal poverty
line is adjusted for inflation. A 1995 National Academy of Sciences recom-
mendation to update the line to reflect current living standards has yet to
be implemented. The World Bank’s main poverty standard was updated in
2005, and all income distributions back to 1981 were reevaluated at the
new line.
Absolute poverty lines are by far the most commonly used approach for
identifying the poor over time and space and are universally used in low-
and middle-income countries. They allow transparent comparisons where
the changes in measured poverty can be attributed purely to changes in the
distribution rather than to a moving poverty cutoff.
27
However, there are some practical challenges associated with the con-
struction of absolute poverty lines:
• Several competing methods are available for deriving an absolute

poverty line from a reference set of observations, each of which can
generate a different poverty income cutoff.
• The reference set of observations must be selected, and this reference
set, too, can influence the cutoff.
• To a certain extent, then, the choice of absolute poverty line is arbi-
trary. This arbitrary quality tempers the interpretation of results but
can be partially addressed with the help of variable line robustness
techniques discussed below.
• There is the related question of how frequently to update an absolute
poverty line. But here the trade-offs are clear: it must be fixed long
enough to be able to discern the underlying changes in poverty, and
it must be updated often enough so that the standard is reasonably
consistent with prevailing circumstances.
Relative Poverty Line
A relative poverty line is an explicit function of the income distribution—

namely, a constant fraction of some income standard. One example is the
European Union’s country-level poverty lines, which are set at 60 percent
of a country’s median (disposable) income. The nature of a relative poverty
line dictates that the cutoff below which one is considered to be poor varies
proportionally with its income standard. Indeed, a level of income that is
above the poverty line in one distribution may lie below the poverty line of
a second distribution having a higher income standard.
Relative poverty lines are most often used in countries with higher
incomes, where there is less concern about achieving a minimum absolute
level of living and greater interest in inclusion or relative achievements.
Unlike absolute poverty lines, the endogenous determination of relative lines
also automatically updates the standard over time and space. However, this
determination is done by making a very strong assumption on the functional
form of the link between poverty line and income standard and by choosing
an income standard and a specific fractional cutoff. Those components are
often selected without a great deal of scrutiny or exploration of alternatives.
28
Moreover, with a relative line, the analysis of a change in poverty over

time (or space) is less transparent. There are now two sources of change:
(a) the direct impact of the change in the distribution and (b) the indi-
rect impact through the change in the underlying income standard and,
hence, the poverty standard. This second component is quite important, yet
depends on the assumed functional form of the relative poverty line.
The elasticity of a relative poverty line with respect to its income
standard is 1. If the income standard rises by 1 percent, then the relative
poverty line will rise by 1 percent. In contrast, with an absolute poverty
line, there is no change in the poverty standard when there is a 1 percent
increase in the same income standard; the elasticity is 0 for an absolute
poverty line.
Intermediate poverty lines exist—hybrid or weak relative poverty lines.
They offer a poverty line that is a function of the income distribution, but
with fixed (or weakly rising) elasticity between 0 and 1. The intermediate
poverty lines are a topic of continuing research.
No matter which of these approaches to setting a poverty line is chosen,
the outcome for a given distribution is a specific income cutoff and a subset
of the population identified as being poor. For simplicity and because of the
greater prevalence of absolute lines, we will assume that a fixed poverty line
is given. The next step is to determine how to aggregate the data to obtain
an overall picture of poverty.
What Is a Poverty Measure?
A poverty measure is a way of combining information on income

distribution—especially incomes of the poor—to obtain a number that
represents the poverty level in the distribution given the poverty line. The
most common measures are counting measures, which evaluate poverty by
numbers of people. The best-known counting measure is the headcount
ratio, defined as the percentage of the total population that is poor.
An easy way of expressing a counting measure is to construct the depri-
vation vector, which replaces each poor income with 1 and every nonpoor
income with 0. The headcount ratio is simply the mean of the deprivation
vector or distribution. The headcount ratio is linked to the cumulative dis-
tribution function, which for continuous distributions is simply the graph of
the headcount ratio as the poverty line is varied.
29
Other measures evaluate poverty by the average gap or depth of poverty:
• The normalized gap vector is constructed by replacing income of each

poor individual with the normalized gap (or the gap between the pov-
erty line and the income expressed as a share of the poverty line) and
income of every nonpoor individual with 0. The poverty gap measure
is the mean of the normalized gap vector. It is sensitive to both the
prevalence of poverty in a society and the extent to which the poor
fall below the poverty line.
• Another measure is based on the squared gap vector, which uses
the square of the normalized gap for each poor person. The squar-
ing process emphasizes the larger gaps relative to the smaller gaps.
The squared gap or Foster-Greer-Thorbecke (FGT) measure index is
the mean of the squared gap vector. It is sensitive to the prevalence
of the poor, the extent to which their incomes fall below the poverty
line, and the distribution of their incomes or shortfalls.
All of those measures are members of a parametric family of indices: the

FGT family of poverty indices is derived by taking the mean of an a-gap
vector, which is obtained by raising each positive entry in the normalized
gap vector by a power of a ≥ 0.
There are two main ways of interpreting what a poverty measure is actu-
ally measuring. One way is by examining the properties that the measure
satisfies. The other makes use of income standards in interpreting the mea-
sure. We begin with the axiomatic approach.
Poverty Measure Properties
There are six basic properties for poverty measures:
• The first two are the symmetry and population invariance properties given
above for income standards and inequality measures. They are impor-
tant for ensuring that the measure is based on the anonymous distribu-
tion and not on the income recipients’ names or the population size.
• The third basic property is the focus axiom, which requires the pov-
erty measure to ignore changes in the distribution involving nonpoor
incomes. This approach ensures that the measure focuses on poor
incomes in evaluating poverty.
30
• The fourth property is scale invariance, which requires the poverty

measure to be unchanged if all incomes and the poverty line are
scaled up or down by the same factor. This approach makes sure that
the measure is independent of the unit of measurement of income.
The first four properties are invariance properties, which indicate how
various changes in the distribution should not be taken into account by the
measure. The next two properties are dominance properties that require the
measure to be consistent with certain basic changes in the distribution.
• The fifth property is weak monotonicity, which requires poverty to

rise or be unchanged if the income of a poor person falls—in other
words, a decrement in a poor income cannot decrease poverty. Weak
monotonicity is a central property of a poverty measure and is often
presented in a stronger form, known as monotonicity, which requires
an increment in a poor income to (strictly) decrease poverty.
• The final property considers the effect of a transfer on poverty. The
weak transfer property requires poverty to fall or be unchanged as a
result of a progressive transfer (from richer to poorer) between two poor
people. This property also has a stronger version, known as the transfer
principle, which requires poverty to (strictly) increase as a result of a
regressive transfer (from poorer to richer) between two poor people.
Notice that both the monotonicity axiom and the transfer principle
allow the number of poor to be altered in the process, whereas the weaker
versions do not.
The headcount ratio, the poverty gap measure, and the FGT index satisfy
all six basic axioms. The headcount ratio satisfies weak monotonicity and
the weak transfer principle (because it is unaffected by the distributional
changes specified in the two properties), but it violates the two stronger
versions. The poverty gap measure satisfies the monotonicity axiom, but it
violates the transfer principle (because it is unaffected by a small regressive
transfer). The FGT index satisfies both stronger axioms.
Some additional properties can also be helpful in evaluating poverty
measures. Transfer sensitivity requires a decrement in the income of a poor
person, when combined with an equal-sized increment in the income of a
richer poor person, to raise poverty. It ensures that a given-sized transfer has
a larger poverty-reducing effect at lower poor incomes. Decomposability and
31
subgroup consistency have proved to be very important for regional evalua-

tions of poverty and for targeting. They are discussed below.
Income Standards
Another way of understanding poverty measures makes use of our previous

insights from income standards. Like inequality measures, most poverty
measures are based on a comparison of two income levels. In this case, how-
ever, one of them is the fixed poverty line z, whereas the other is an income
standard applied to a modified distribution that focuses on the poor.
Two forms of modification are employed, leading to two general forms
of poverty measures. The first makes use of a censoring process that ignores
the portion of any income lying above the poverty line z. The censored
distribution x* for a given distribution x replaces all incomes above z with
z itself. Applying an income standard to the censored distribution yields a
poor income standard, which reflects the size of the censored distribution and
is clearly bounded above by z (the maximum value achieved when no one
is poor).
Many poverty measures take the form P = (b − a)/b, or some monotonic
transformation, where a is some poor income standard and b is the poverty
line z. P measures poverty as the shortfall of the poor income standard from
the poverty line as a percentage of the poverty line. For example, if a were
the mean censored income m(x*), then the resulting poverty measure would
be (z − m(x*))/z, which is another way of expressing the poverty gap. Below
we will see other poverty measures that share this general structure but
employ different income standards.
The second form of modification changes the focus from incomes to
income gaps. The gap distribution g* is found by replacing the income x*i in
x* with the income gap z − x*i. The gap will be 0 for anyone who is nonpoor,
and it increases in size as the income of a poor person falls further below z.
Applying an income standard to the gap distribution yields a gap stan-
dard, which measures the overall departure of incomes in x* from z. Many
poverty measures take the form P = a/b, or some monotonic transformation,
where a is some gap standard and b is the poverty line z. P measures poverty
using a gap standard taken as a percentage of the poverty line. For example,
if a were the mean gap m(g*), then the resulting poverty measure would be
m(g*)/z, which is another way of defining the poverty gap. Below we will
32
discuss several other poverty measures that share this structure but use dif-
ferent income standards in constructing the gap standard.
Common Examples
The first general form of poverty measures uses an income standard applied
to the censored distribution. An income standard that puts progressively
greater weight on lower incomes will yield a poverty measure that is sensi-
tive to the distribution of income among the poor. The Sen-Shorrocks-Thon
(SST) index is given by (b − a)/b, where a is the Sen mean applied to x*and
b is the poverty line. This measure inherits its characteristics from the Sen
mean: it satisfies all six basic properties and monotonicity and the transfer
property. Increments and progressive transfers among the poor are reflected
in a strictly higher poor income standard a, and hence a lower poverty level.
The next measure is based on another income standard that emphasizes
lower incomes. The Watts index is defined as ln(b/a), where a is the geomet-
ric mean applied to the censored distribution and b is the poverty line z. It
likewise satisfies the six basic axioms and the strict forms of monotonicity
and the transfer principle. Additionally, the geometric mean has the prop-
erty that a given-sized transfer among the poor has a greater effect at lower
income levels, so the poverty measure satisfies transfer sensitivity.
The Watts index can be expanded to an entire class of measures, each
of which uses a general mean to evaluate the censored distribution. The
Clark-Hemming-Ulph-Chakravarty (CHUC) family of indices compares the
poor income standard a = ma (x*) for a ≤ 1 and the poverty line b = z. There
are two forms of the measure: the original form (b − a)/b and a decompos-
able form obtained by a simple transformation. The measure becomes the
poverty gap at a = 1 and the Watts index (or a transformation) at a = 0.
The properties of the general means ensure that the CHUC measures satisfy
all six basic properties for poverty measures, for monotonicity, and for a < 1
the transfer principle as well as transfer sensitivity.
The second general form of poverty measures uses an income stan-
dard applied to the gap distribution. The key family of measures has a
traditional decomposable version and an alternative version that is only
subgroup consistent.
The FGT family of decomposable poverty indices was defined above
as the mean of the a-gap distribution and includes the headcount ratio for
33
a = 0, the poverty gap measure for a =1, and the FGT or squared gap mea-
sure for a = 2. Alternatively, we can transform each of the measures in the
range a > 0 by raising it to the power 1/a. This yields a subgroup-consistent
measure that compares a gap standard a = ma (g*) to the poverty line b = z
via the formula P = a/b.
The properties for the FGT measures in this range follow from the prop-
erties of the associated general means. The first five properties and mono-
tonicity are immediately satisfied for all a > 0. For the transfer principles,
note that the general means with a < 1 emphasize the smaller entries, those
with a > 1 emphasize the larger entries, and a = 1 ignores the distribution
altogether. Thus, the FGT measures satisfy the weak transfer principle for
a ≥ 1 and the transfer principle for a > 1. In an analogous way, the FGT
index for a = 2 is transfer neutral in that a given-sized progressive transfer
among the poor has the same effect at lower incomes, whereas the FGT
measures with a > 2 satisfy transfer sensitivity.
The above discussion excludes the case a = 0, which corresponds to
the headcount ratio. The simple structure of this poverty measure does not
admit an interpretation of an income standard applied to the censored or
gap distribution. Instead, a second censoring must be applied to obtain a
distribution in which all nonpoor incomes are replaced by z and all poor
incomes are replaced by 0. Let x** denote the resulting doubly censored dis-
tribution. The headcount ratio can be represented as (b − a)/b, where a =
μ(x**) and b = z. In other words, it is the poverty gap of the doubly censored
distribution that converts nonpoor incomes to z and poor incomes to 0.
The first censoring ensures that the measure focuses on incomes of the
poor. The second censoring forces the headcount ratio to ignore the actual
income levels of poor people and violate monotonicity. The headcount
ratio suppresses information that is relevant to poverty (the actual incomes
of the poor) in order to capture one key aspect of poverty (the prevalence
of poverty). Replacing x** with x* in this representation would recover this
information and yield the poverty gap measure.
Poverty, Inequality, and Welfare
Poverty measures satisfying the transfer principle are called distribution sensi-
tive because they account for the inequality of poor incomes in ways that
the headcount ratio or the poverty gap cannot. In fact, each of the above
distribution-sensitive poverty indices is built on a specific income or gap
34
standard that is closely linked to an inequality measure. For the SST index,
it is the Gini coefficient. For the CHUC indices, the Atkinson measures are
used. For the Watts index, the mean log deviation is the inequality measure.
In each case, the inequality measure is applied to the censored distribution
x* with greater censored inequality being reflected in a higher level of poverty
(for a given poverty gap level).
The FGT measures (for a > 1) use generalized entropy measures applied
to the gap distribution g* with greater gap inequality leading to a higher
level of poverty (for a given level of the poverty gap). The focused inequal-
ity measures underlying these distribution-sensitive poverty indices ignore
variations in incomes above the poverty line. Trends in focused inequality
may well be very different from trends in overall inequality.
Certain income standards can be viewed as welfare functions, and this
link can provide yet another lens for interpreting poverty measures. The Sen
mean underlying the SST index and the general means for a ≤ 1 that are
behind the CHUC indices can be interpreted as welfare functions. In each
case, the welfare function is applied to the censored distribution to obtain
the poor income standard a, which is now seen to be a censored welfare func-
tion that takes into account the incomes of the poor and only part of the
incomes of the nonpoor (up to the poverty line).
For these measures, poverty and censored welfare are inversely related.
Every increase in poverty is seen as a decrease in censored welfare. Of
course, the trends in censored welfare may be very different from the trends
in overall welfare, as the latter take into account the actual incomes of the
nonpoor. We will see below another link between welfare and poverty when
we consider poverty comparisons over a range of lines.
Applications
A poverty methodology can be used to identify the poor (through its

identification step) and to evaluate the extent of poverty (through the
aggregation step). The first step by itself allows many interesting analyses
to be conducted, given appropriately rich data. Consider, for example, the
following questions:
• Who are the poor and how do they differ from the nonpoor? A range
of characteristics can be examined—including location, household
size, ethnicity, education indicators, health indicators, housing, and
35
ownership of certain assets—to see what it means to be poor. This is

part of a countrywide poverty profile that relies purely on the identi-
fication step.
• What drives the dynamics of poverty? If panel data are available, one
can explore the factors that seem to be forcing people into poverty
or allowing them to escape. Even if two periods of data are not part
of a panel (and hence not linked at the personal level), one can
investigate how other general factors, such as food prices or economic
conditions, affect the likelihood of being in poverty.
• Is a given poverty program reaching its intended recipients? The leakage
or coverage of poverty programs can be evaluated to gauge the likeli-
hood that a recipient is not poor or that a poor person is a nonrecipient.
• What affects and is affected by the condition of being poor? In some
studies, the deprivation vector, or indicator function for poverty, is a
key outcome variable. In other studies, it is an important dependent
variable.
The aggregation step goes beyond a simple identification of the poor

and provides a quantitative measure of the extent of poverty for any given
population group. A poverty measure can be used to monitor poverty in
a country over time and space. Poverty profiles evaluate the structure of
poverty in a country by considering how poverty varies across an array of
population subgroups.
Other applications include using a poverty measure as a basis for targeting
social programs or for assessing their poverty impact. It is often thought that
chronic poverty is qualitatively different from transient poverty. Panel data
can allow the two to be evaluated in order to discern whether the poverty in
a given region tends to be of one form or the other. Some people currently
not in poverty may, nonetheless, be vulnerable to becoming poor. Poverty
measures can be adapted to create measures of vulnerability to poverty.
Optimal taxation exercises use a welfare function as the objective func-
tion with which to evaluate the competing objectives of a larger pie versus a
more equitable distribution. For many policy exercises, it may make sense to
focus on the poor by using a censored welfare function or a poverty measure:
Are food subsidies more effective in improving poverty than income trans-
fers? This and other questions can be addressed in theory or practice once a
poverty measure has been chosen. The choice of poverty measure will affect
the answers obtained.
36
Subgroup Consistency and Decomposability
Many programs designed to address the needs of the poor are implemented
at the local level. Suppose we are evaluating such a program in a country
with two equal-sized regions. We find that poverty has fallen significantly
in each region, yet when poverty is measured at the country level, it has
increased. This possibility could present significant challenges to the analyst
and could prove rather difficult to explain to policy makers. It turns out that
the inconsistency between regional and national poverty outcomes may be
due entirely to the way poverty is measured.
To ensure that this possibility does not arise, one can require the poverty
measure to satisfy subgroup consistency. This property requires that if poverty
falls in one subgroup and is unchanged in another and both have fixed popu-
lation sizes, then the overall poverty level must likewise fall. The SST index
is not subgroup consistent because of its use of the Sen mean. The FGT and
CHUC measures, which depend on general means, are subgroup consistent
and thus would not be subject to the regional-national dilemma.
Subgroup consistency requires overall poverty to move in the same
direction as an unambiguous change in subgroup poverty levels. A stronger
property provides an explicit formula that makes the link between overall
and subgroup poverty. A poverty measure is said to be (additively) decompos-
able if overall poverty is a population-share weighted average of subgroup
poverty levels. Unlike the case of inequality measures, there is no between-
group term in this decomposition. The reason is that the standard against
which subgroup poverty is evaluated is a fixed poverty line. In contrast, an
inequality measure typically evaluates subgroup inequality relative to sub-
group means, then takes the variation of subgroup means into account as
another source of inequality.
Additively decomposable poverty measures transparently link subgroup
poverty to overall poverty. This approach can be particularly useful in
generating a coherent poverty profile in which a broad array of population
subgroups and their poverty levels can be broken down or reassembled as
needed. Consider these questions:
• Is a given change in overall poverty caused by changes in subgroup

poverty levels, by population shifts across subgroups, or by a combina-
tion of the two effects? A counterfactual approach, which constructs
an artificial intermediate distribution to separate the two, can help
37
quantify the relative impacts of demographic changes and the

changes in subgroup poverty on the overall poverty level.
• What share of overall poverty can be attributed to a particular popu-
lation group? We can define a subgroup’s contribution to overall pov-
erty to be the population share of a subgroup times the poverty level
of the subgroup divided by the overall poverty level. Some subgroups
with low levels of poverty may have large contributions as a result of
their population sizes. Others may have smaller population shares,
but still have large contribution shares because subgroup poverty
levels are high.
For decomposable poverty measures, subgroup contributions must sum

to one.
The above discussion assumes that it is possible to select a correct poverty

line to separate the poor from the nonpoor. Yet it is clear that any cutoff
selected is bound to be arbitrary and that alternative poverty lines could be
chosen with equal justification. Conclusions obtained at the original pov-
erty line may be reversed at some other reasonable standard. They also could
be robust to a change in the poverty line.
To help discern which of these possibilities is true—a reversal or una-
nimity for all poverty lines—we can construct a poverty (value) curve which
graphs the poverty measure as a function of the poverty line over the rel-
evant range of poverty lines. If the original comparison continues to hold
at all poverty lines in the range, then the comparison is robust. This gives
rise to a (variable line) poverty ordering, which ranks one distribution as hav-
ing less poverty than another when its poverty curve is not above (and is
somewhere below) the poverty curve of the other distribution. The range of
poverty lines usually begins at 0 and ends at some highest value z*, although
it is instructive to consider the case where there is no upper bound. Our
discussion begins with the latter case.
Although the general approach can be used with any poverty measure,
it is standard to focus on the three main measures from the FGT family: the
headcount ratio, the poverty gap measure, and the FGT squared gap mea-
sure. The headcount ratio for a given poverty line is the share of the popu-
lation having incomes below the poverty line. Consequently, the poverty
38
curve for the headcount ratio traces the cumulative distribution function
associated with the distribution (except that it takes its limits from the left
rather than the right when it has jumps), so the poverty ordering is first-
order stochastic dominance.
If one recalls the above discussion of stochastic dominance, this poverty
ordering is equivalent to having a higher quantile function and also to
having greater welfare according to every utilitarian welfare function with
identical, increasing utility functions. The poverty curve associated with the
headcount ratio is often called the poverty incidence curve.
The poverty curve for the poverty gap measure is closely linked to the
area beneath (or the integral of) the poverty incidence curve (or the cdf),
which is another way of representing second-order stochastic dominance.
Hence, the poverty ordering for the poverty gap measure is simply second-
order stochastic dominance. By the previous discussion, this means that the
poverty ordering can also be represented by the generalized Lorenz curve,
with a higher generalized Lorenz curve indicating unambiguously lower (or
no higher) poverty according to the poverty gap measure.
In addition, there is a useful welfare interpretation of this poverty order-
ing: it indicates higher welfare according to every utilitarian welfare func-
tion with identical and increasing utility function exhibiting diminishing
marginal utility (Atkinson’s general class of welfare functions). The curve
found by plotting the area beneath the poverty incidence curve for each
income level z is often called the poverty deficit curve.
The FGT index has a poverty curve that is closely linked with the area
beneath the poverty deficit curve (or the double integral of the cdf), and
hence its poverty ordering is linked to a refinement of second-order stochas-
tic dominance called third-order stochastic dominance. This poverty ordering
also has a welfare interpretation: higher welfare according to every utilitar-
ian welfare function with identical and increasing utility function exhibit-
ing diminishing and convex marginal utility.
The final condition on the convexity of marginal utility ensures that
the welfare function is more sensitive to transfers at the lower end of the
distribution—a welfare version of the transfer sensitivity axiom. The curve
found by plotting the area beneath the poverty deficit curve for each income
level z is often called the poverty severity curve.
Notice that the poverty orderings for the three FGT measures are nested
in that if the headcount ratio’s ordering ranks two distributions, then the
poverty gap’s ordering also ranks the distributions in the same way (but not
39
vice versa). Further, the poverty gap’s ordering implies (but is not implied
by) the FGT index’s ordering. Because the poverty deficit curve is found by
taking the area under the poverty incidence curve, a higher poverty inci-
dence curve leads to a higher poverty deficit curve. The same is true for the
poverty deficit and poverty severity curves.
The poverty orderings of the Watts and CHUC indices can also be eas-
ily constructed and lead to another nested set starting with second-order
dominance for the poverty gap measure. The poverty ordering for the Watts
index, for example, is simply generalized Lorenz (or second-order stochastic)
dominance applied to the distributions of log incomes. Each CHUC poverty
ordering likewise applies generalized Lorenz dominance to distributions of
transformed incomes (see Foster and Jin 1998).
Placing an upper limit z* on the range of poverty lines is equivalent
to comparing poverty curves (or the poverty incidence, deficit, or severity
curves) over this limited range or to using censored distributions associated
with z*. For example, the limited range poverty ordering for the poverty gap
is equivalent to comparing the generalized Lorenz curves of the censored
distributions or to comparing censored welfare levels across all utilitarian
welfare functions with identical and increasing utility functions that have
diminishing marginal utility.
In the above example, we varied the poverty line while holding the
poverty measure fixed. We can also vary the poverty measure for a given
poverty line to examine robustness to the choice of measure. For example,
using a five-dimensional vector, one can depict the poverty levels of the
FGT measures for a = 0, 1, and 2; the Watts index; and the SST index.
Vector dominance would then be interpreted as a variable measure poverty
ordering that ranks distributions when all five measures unanimously agree.
An analogous approach using poverty curves can be employed when
using poverty measures indexed by a parameter. Consider a poverty curve
that depicts the CHUC indices (z − ma (x*))/z for a ≤ 1 and the FGT indices
ma (g*))/z for a ≥ 1. We are using the income standard version of each mea-
sure (rather than the decomposable version) because of its nice interpreta-
tion as a normalized average gap. The poverty measure at a = 1 is the usual
poverty gap measure. As a rises, the FGT values progressively rise because
the measures with higher a use a general mean that focuses on the higher
gaps in the gap vector g*.
The extent to which poverty rises as a > 1 rises depends on the gen-
eralized entropy inequality in g* for a. To the left, the CHUC values
40
progressively rise as the measures with lower a use a general mean that
focuses on lower incomes in the censored vector x*. The extent to which
poverty rises as a < 1 falls depends on the generalized entropy inequality in x*
for a. A higher curve would then be interpreted as the variable measure pov-
erty ordering that ranks distributions when all these poverty measures agree.
The above approaches to varying the poverty line and the poverty mea-
sure can be combined to examine the robustness of comparisons to changing
both simultaneously. Interestingly, though, in certain cases it is enough to
examine a variable line poverty ordering. For example, if two distributions
can be ranked by the poverty ordering of the headcount ratio, then they will
also be ranked in the same way by the poverty ordering associated with any
given poverty measure satisfying the basic axioms and monotonicity. This is
also true for certain limited range poverty orderings.
Even in cases lacking a clear ranking for the relevant set of poverty lines
(or measures), a poverty curve can be very useful in identifying ranges of
poverty lines (and measures) where the ranking is unchanged and where the
ranking reverses. This general methodology for checking the robustness of
poverty comparisons is quite powerful.
Growth and Poverty
It is sometimes helpful to determine how fast poverty is falling or rising

over time and to explore the extent to which the growth rate of poverty is
robust to a change in the poverty line or measure. Associated with each of
the above poverty curves is a poverty growth curve that gives the growth rate
of poverty for each poverty line or measure. For example, the variable line
poverty growth curves for the three standard FGT measures are the same
as the growth curves of the poverty incidence, deficit, and severity curves.
Negative rates of growth throughout would indicate that poverty has
fallen, and this conclusion is robust to changing the poverty line. If growth
rates are similar across an entire range of poverty lines, then this suggests the
percentage change in poverty is robust to changing the poverty line. Note,
though, that poverty measures like the CHUC and the FGT measures have
two versions—the decomposable version and the income standard version,
which are monotonic but not direct (proportional) transformations of one
another. Although the two versions will always agree on whether poverty
has risen or fallen (for a given poverty line), the growth rates will, in general,
be different.
41
We have seen above how the trend in inequality can be evaluated by

comparing the growth rates of the two income standards underlying the
inequality measure. The trend in poverty can likewise be evaluated by com-
paring the growth rate of the poverty line to the growth rate of the poor
income standard (or gap standard) associated with the poverty measure. An
absolute poverty line has a growth rate of zero, so poverty will decrease over
time when the poor income standard has a positive growth rate (or the gap
standard has a negative growth rate). If a relative poverty line is used, the
growth rate in the poverty line is the same as the growth rate in the income
standard underlying the relative poverty line.
Relative poverty will thus decrease over time when the overall income
standard grows more slowly than the poor income standard, or more quickly
than the gap standard. For example, suppose the relative poverty line is half
the mean income and the poverty measure is the poverty gap. Then poverty
will decrease over time if the mean income grows more slowly than the
mean censored income. Alternatively, relative poverty will decrease if the
mean income grows faster than the mean gap.
By plotting the growth rates for a range of income standards or gap stan-
dards and comparing them to the economywide growth rate, one can make
robust comparisons of relative poverty. An analogous exercise is possible for
the hybrid or weakly relative poverty lines whose elasticity with respect to the
underlying income standard (called the income elasticity of the poverty line) falls
between 0 (as with absolute lines) and 1 (as with relative lines). The growth
rates of the poor income standards or gap standards are compared to the overall
growth rate of the economy times the income elasticity of the poverty line to
determine whether poverty of this form unambiguously decreases or increases.
A key question related to growth and poverty is whether general eco-
nomic growth translates into elevated incomes for the poor. Is growth
“shared” among all strata of society or are the poor excluded from growth?
To address this question, various approaches to evaluating shared or pro-
poor growth have been advanced.
A first approach compares the growth in the mean income to the growth
in a lower or higher income standard. If the growth rate of a lower income
standard exceeds the general growth rate so that the growth elasticity of the
low-income standard is greater than one, then this rate is seen as evidence of
pro-poor growth. If the growth rate for a high-income standard is lower than
the general growth rate—so that the growth elasticity of the high-income
standard is less than one—then this is also evidence of pro-poor growth.
42
If one uses the twin-standards interpretation of inequality, then this

approach is equivalent to requiring an associated inequality measure to fall. Let
a and b be the two income standards, with a ≤ b, where one of the two is the
mean, and let I be an inequality measure based on these twin standards (so that
I is a monotonic transformation of b/a). Growth is pro-poor if a grows faster
than b, which is equivalent to a falling ratio b/a and, hence, to a decrease in
the associated inequality measure I. For example, one might describe growth as
pro-poor if the Sen mean grew faster than the mean, and hence the Gini coef-
ficient decreased. Or we could note that the Euclidean mean grew slower than
the mean, and hence the coefficient of variation declined. This is basically the
inequality-based approach to pro-poor growth we have discussed above.
A second poverty-based approach compares the actual change in poverty
to the level that might be expected along a counterfactual growth path.
Suppose that the distribution of income changes from x to x′ and that this
leads to a change in measured poverty from P to P′. Construct a counter-
factual income distribution x″ that has the same mean as x′ and the same
relative distribution as x, and let P″ be its level of poverty. The growth
from x to x′ is then said to be pro-poor if the resulting change in poverty
P′ − P exceeds the counterfactual change P″ − P; in other words, the rate of
poverty reduction from actual growth is faster than the counterfactual rate
from perfectly balanced growth. Of course, the relevance of this conclusion
depends on the choice of counterfactual distribution and its assumption that
the relative income distribution should not change.
A related technique is often used to analyze the extent to which a given
change in poverty is primarily due to changes in the mean (the growth effect)
or changes in the relative distribution (the distribution effect). As before, let
x″ be the counterfactual distribution having the same relative distribution
as the initial distribution x and the same mean as the final distribution x′.
The overall difference in poverty P′ − P can be expressed as the sum of the
growth effect P″− P and the distribution effect P′ − P″.
This breakdown first scales up the distribution x to the mean income of
x′ to explore how the uniform growth in all incomes alters poverty. Then
it redistributes the income to obtain x′, and explores how the distributional
change alters poverty. Other breakdowns are possible using a different coun-
terfactual distribution or, indeed, a different order of events (redistribute
first, then grow). However, this version has the advantage of being easy to
interpret and can be expressed as the sum of two component terms without
a troublesome residual term.
43
Note
1. The third step may have two substeps, depending on the type of poverty
measure selected: (a) evaluation of individual poverty and (b) selection
of a method to aggregate individual poverty to obtain overall poverty.
References
Atkinson, A. B. 1970. “On the Measurement of Inequality.” Journal of

Economic Theory 2 (1970): 244–63.
Commission on Growth and Development. 2008. The Growth Report:
Strategies for Sustained Growth and Inclusive Development. Washington,
DC: World Bank and International Bank for Reconstruction and
Development.
Commission on the Measurement of Economic and Social Progress. 2009.
“Report by the Commission on the Measurement of Economic and
Social Progress.” Commission on the Measurement of Economic and
Social Progress, Paris. http://www.stiglitz-sen-fitoussi.fr/en/index.htm.
Foster, J. E., and Y. Jin. 1998. “Poverty Orderings for the Dalton Utility-
Gap Measures.” In The Distribution of Welfare and Household Production:
International Perspectives, edited by S. Jenkins, A. Kapteyn, and B. van
Praag, 268–85. New York: Cambridge University Press.
Mincer, J. 1974. Schooling, Experience, and Earnings. New York: Columbia
University Press.
Oaxaca, R. 1973. “Male-Female Wage Differentials in Urban Labor
Markets.” International Economic Review 14 (3): 693–709.
44
Chapter 2
Income Standards, Inequality,

and Poverty
This chapter complements the introductory chapter by providing a detailed

discussion and more formal analysis of the concepts involved in measuring
income standards, inequality, and poverty. This chapter follows closely the
introduction’s organization. It is divided into four sections. The first sec-
tion introduces notations and basic concepts that will be used throughout
the rest of this chapter. The second and third sections discuss tools and
instruments related to income standards and inequality measures. The
fourth section uses the tools from the second and third sections to construct
poverty measures.
According to Sen’s seminal work (1976a), evaluating poverty within a
society (which may be a country or other geographic region) involves two
steps:
1. Identification, in which individuals are identified as poor or nonpoor

2. Aggregation, in which data about the poor are combined to evaluate
poverty within the society.
However, to identify individuals as poor or nonpoor, we need to select a

space on which their welfare level is to be assessed. The welfare indicator is the
variable for assessing an individual’s welfare level. Thus, evaluating poverty
within a society involves three steps:
45
1. Space selection, which is described below

2. Identification, in which individuals with welfare levels below the
threshold are classified as poor and individuals with welfare levels
above the threshold are classified as nonpoor
3. Aggregation, our focus, which requires choosing an appropriate aggre-
gation method to measure the poverty level in a society.
In this book, we define the space for evaluating poverty as money metric
and single dimensional. The welfare indicator is either consumption expen-
diture or income:
• An individual’s consumption is the destruction of goods and services

through use by that individual. Consumption expenditure is the overall
consumption of goods and services valued at current prices, regardless
of whether an actual transaction has taken place.
• An individual’s income, in contrast, is the maximum possible expen-
diture the individual is able to spend on consumption of goods and
services, without depleting the assets held.
Whether it is income or consumption expenditure, welfare indica-

tors are constructed by aggregating various components. For example, an
individual’s consumption expenditure is constructed by aggregating the
commodities and services consumed by the individual using the prices paid.
Consumption expenditure as a welfare indicator is more commonly used for
assessing developing countries in Asia and Africa (Deaton and Zaidi 2002).
In contrast, using income as a welfare indicator is common when assessing
Latin American countries.
Although both income and consumption expenditure are used as wel-
fare indicators, consumption expenditure has certain advantages. Income
data, for example, may not lead to an accurate assessment of welfare when
incomes fluctuate significantly. Furthermore, in developing countries,
income data may be difficult to collect, and data accuracy is difficult to ver-
ify because most of the population may be employed in the informal sector.
To work around these problems, many developing countries collect
consumer expenditure survey data, which include detailed information
on goods and services consumed by individuals. Then they use the market
prices to compute the overall consumption expenditure. The surveys ask
about food consumption for several items over a specific reference period,
46
Chapter 2: Income Standards, Inequality, and Poverty
which may be a month or any longer period of time. If the reference period
is short (for example, one month), seasonality concerns may be overcome,
but a shorter reference period may also lead to more noise in the expenditure
data. Noise can be avoided by using a longer reference period, but difficulties
in recollection may bias expenditures downward.1
A person may consume many private and public goods from the long
list of commodities in a consumer expenditure survey. For a private good,
total expenditure is the amount of commodity consumed times that com-
modity’s price. Consumption expenditure for two individuals having the
same consumption patterns and requirements, therefore, should be twice the
consumption expenditure for either of the two.
This straightforward expenditure computation may not be possible when
the consumed commodities are, instead, public goods. Given that public
goods are nonrival and nonexcludable, the same amount of public goods
may be consumed by multiple individuals without additional cost. Multiple
individuals living together and sharing public goods enjoy economies of scale.
Examples of public goods include a radio, a water pump, bulk purchase dis-
counts of food items, and food preparation efficiencies (which may lower the
cost of fuel and time).
Although the goal is to construct a money-metric wealth indicator for
each person, fulfilling that goal may not be straightforward. Most of the
time, data for commodities and services consumed are collected at the
household level. A household typically consists of members with different
characteristics, such as age, sex, and employment status. Usually, an individ-
ual’s welfare indicator is calculated by dividing total household expenditures
by the number of people residing in that household. The result is called the
per capita expenditure.
Analyzing poverty on the basis of per capita expenditure, however,
ignores the fact that different individuals may have different needs.
The cost per person to reach a certain welfare level may be lower in
large households, because large households enjoy certain economies of
scale. For example, a child may not need the same share of income as
an adult member, or the food consumption expenditure may not be the
same across men and women within a household. The minimum income
needed to meet the subsistence needs of a household with four adults
may be much more than the subsistence income needed for a household
with two adults and two children. This intrahousehold allocation can be
adjusted using an equivalence scale tool.
47
There are various types of equivalence scales and economies of scale.

Also, there are different ways of determining these scales, such as evalu-
ating nutritional needs and behavioral needs. Differences in nutritional
needs are derived from various health studies. Data on behavioral needs are
obtained from econometric estimates that are based on observed commodity
allocations.
However, the observed allocation is suspect because what is observed
may not necessarily be what is actually needed. For example, if female chil-
dren are observed to consume less, does this mean that they need less, or are
they just discriminated against? There is no straightforward answer to this
question, unfortunately, because it is beyond the scope of most consumer
expenditure surveys.
Two adult equivalence (AE) scales are more commonly used than oth-
ers. The first is used by the Organisation for Economic Co-operation and
Development (OECD), which we denote by AEOECD. It is defined as
AEOECD = 1 + 0.7(NA − 1) + 0.5NC , (2.1)
where NA is the number of adults in the household, and NC is the number
of children in the household.
This scale actually serves as both an equivalence scale and an economy
of scale. Note that when there is only one adult member in the household,
AEOECD = 1. For a household with two adult members, AEOECD = 1.7
(AEOECD = 2 is incorrect because two adults sharing the same household
are assumed to enjoy economy of scale). For instance, if the actual total
income of a two-member household is Rs 17,000, then the per capita real
income of the household is equivalent to Rs 17,000/1.7 = Rs 10,000 and not
Rs 8,500, as it would be in the per capita case. This is an example of adjust-
ing for economy of scale. For a single parent household with two children,
however, the actual total income of Rs 17,000 is equivalent to a per capita
real income of Rs 8,500 because AEOECD = 1 + 2 × 0.5 = 2.
The second adult equivalent scale is used by the Living Standards
Measurement Study (LSMS), which we denote by AELSMS. It is defined as
AELSMS = (NA + ϱNC)ϑ, (2.2)
where NA is the number of adults in the household, and NC is the number
of children in the household.
In this scale, parameter ϱ measures the cost of a child compared to an
adult. Parameter ϑ captures the effect of economy of scale. Both parameters
48
are positive but not larger than one. When ϱ = 1, then the cost of a child
is equal to the cost of an adult. The lower the value of ϱ, the lower the cost
of each child compared to an adult. Similarly, when ϑ = 1, no economy of
scale is assumed. The lower the value of ϑ, the larger the economy of scale
is assumed to be.
For example, suppose there are five members in a household: three adults
and two children. If a child is assumed to be half as costly as an adult, then
ϱ = 0.5 and ϑ = 0.5. Then AELSMS = (3 + 0.5 × 2)0.5 = 2. Therefore, if the
actual total income of the household is Rs 20,000, then the real per capita
income of the household is equivalent to Rs 10,000. However, if no econ-
omy of scale is assumed and each child is considered as equally expensive as
an adult, then the household’s per capita income is only Rs 4,000.
In the subsequent analysis in this chapter, we assume that we are using
a dataset having all the information required for constructing a welfare
indicator either at the individual level or at the household level. The
dataset may cover the entire population or may just be a collection of
samples from the population. There are other important issues one should
take into account regarding a dataset (such as its survey design, sample
coverage, sample variability, and so on), which are not covered in this
chapter.2
To keep explanations and mathematical formulas simple, we make two
fundamental assumptions. First, we use income as the welfare indicator
and assume that information on income is available for every person in our
dataset. Second, we assume that every household contains only one adult
member. As a result of the second assumption, we do not need to make
any adjustment for the economy of scale and equivalent scale because each
member is an adult and lives in a single-member household. However, the
tools and techniques introduced in this chapter can be easily extended to
situations when the welfare indicator is consumption expenditure and more
than one person lives in a household.
Basic Concepts
Suppose our reference society X consists of N people, where the income of

person n is denoted by xn for all n = 1,2,…,N. Thus, the income distribution
data for society X has N incomes. For the sake of simplicity, we assume these
incomes are ordered so that x1 ≤ x2 ≤ … ≤ xN.
49
There are two different ways to represent an income distribution:
• The simplest income distribution is a vector of incomes. We denote

the society’s vector of incomes as X = (x1,x2,…,xN).
• The second way is to represent the income distribution in terms of
a cumulative distribution function (cdf) in which x is designated an
income distribution. We denote the average, or mean, of all elements
in x by x̄ = (x1 + … + xN)/N. For a large enough sample, the cdf may
be approximated by a density function.
Another, more intuitive, presentation of the cdf is the quantile function,

which is more suitable to our needs. Before moving into the discussion on
measurement, we will discuss these three concepts and examine their signifi-
cance in describing various aspects of an income distribution.
Density Function
An income distribution’s density function reports the percentage of the popu-

lation that falls within an income range. Suppose incomes in distribution x
range from $100 to $100,000, and we want to know what percentage of the
population earns income between $10,000 and $20,000. The answer can
be easily obtained by calculating the area underneath the density function
between $10,000 and $20,000.
Notice that the total area underneath the density function between $100
and $100,000 is 100 percent because incomes of the entire population fall
within this range. Thus, the density function is a frequency distribution that
is normalized by the total population in the distribution.
Figure 2.1 depicts the probability density function of income distribution
x. Recall that the minimum and maximum incomes in distribution x are x1
and xN, respectively. The horizontal axis reports the income and the verti-
cal axis reports the density. We denote the density function of distribution
x by fx, which is a bell-shaped curve in figure 2.1. The total area between x1
and xN underneath the density function fx is 100 percent. The share of the
population with incomes between b' and b'' is the shaded area.
Two interesting statistics may be found in figure 2.1:
• The median is the income in the distribution that divides the entire
population into two equal shares. In the figure, xM is the median of
50
Figure 2.1: Probability Density Function
Density
fx
x1 xMo xM b ′ b ″ xN
Income
distribution x. Hence, 50 percent of the area underneath fx lies to the

right of xM, and the remaining 50 percent lies to the left of xM.
• The mode is the income in the distribution that corresponds to the
largest density (locally). In figure 2.1, the distribution’s mode is
denoted by xMo.
Commonly, income distributions have one mode, but there can
be distributions with more than one mode. A density with two modes
is called bimodal and that with many modes is called multimodal.
When there is more than one mode, a society is understood to be
polarized in different groups according to their achievements. A
polarized society may produce social tensions among different groups,
which increases the chance of social unrest. These issues are discussed
in more detail in chapter 3.
In addition, a density function can be a useful tool for understanding the

skewness of an income distribution. Skewness is a measure of asymmetry in
the distribution of incomes. It arises when most incomes lie on any one side
of the mean of the distribution. If more observations are located to the left
of the distribution’s mean, then the distribution is positively skewed. If more
observations lie to the right of the mean, then the distribution is negatively
51
skewed. If there is an equal number of observations on both sides of the

mean, then there is no skewness, and the distribution is symmetric around
the mean. Income distributions are usually positively skewed.
Cumulative Distribution Function
A cdf, or cumulative distribution function, denotes the proportion of the

population whose income falls below a given level. A cdf may be easily
obtained from a density function and vice versa. For every income reported
on the horizontal axis of figure 2.1, a distribution function reports the area
to the left of the income underneath fx. Because the total area underneath fx
is 100 percent, the highest value that a distribution function can take is 100
percent. We denote the distribution function of x by Fx, and Fx(b) denotes
the percentage of the population whose income is no greater than b.
For example, if the number of people in society X having incomes less
than b is q, then Fx(b) = 100 × q/N. For any two incomes b' and b", Fx(b')
≤ Fx(b") when b' ≤ b" because having income less than b' must also imply
having income less than b". Therefore, a distribution function should not
decrease as income increases.
As seen in figure 2.2, the horizontal axis denotes income and the verti-
cal axis denotes the value of the cumulative distribution function. For xN,
which is the largest income in distribution x, the value of the distribution
function is Fx(xN) = 100 percent because no one in distribution x has an
income above xN.
Figure 2.2: Cumulative Distribution Function

Cumulative distribution
Fx(xN) = 100%
Fx(b ″) Fx
Fx(b ′)
Fx(bM) = 50%
Mean
bM b ′ b ″ xN
Income
52
At median bM, the distribution function’s value is Fx(bM) = 50 percent,

which implies that half the population has an income less than bM. In figure
2.1, the share of the population with income ranging between b' and b" is
represented by the shaded area, which, in figure 2.2, is denoted by the dif-
ference Fx(b") − Fx(b'). A distribution function provides another important
statistic: the mean of the distribution. In figure 2.2, the shaded area to the
left of Fx is the mean x̄ of distribution x.
Quantile Function
A quantile function is the inverse of a cdf. Recall that a distribution function

shows the percentage of the population whose income falls below a given
level of income. The quantile function, however, reports the level of income
below which incomes of a given percentage of the population fall.
We denote the quantile function of distribution x by Qx and by con-
struction Qx = Fx–1, where Fx–1 is the inverse of the cdf Fx. For example, the
level of income below which incomes of 25 percent of the population lie is
Qx(25). If 25 percent of Georgia’s population has income below GEL 2,000,
then QGEO (25) = GEL 2,000.
Figure 2.3 describes the quantile function corresponding to distribution
x. The horizontal axis denotes the population share, or the percentage of
Figure 2.3: Quantile Function
xN
Qx
Income
bM
x1 Mean
0 50 100
Population share (percent)
53
population. The vertical axis denotes the corresponding value of a quantile

function in terms of income. Of course, no one in the society can have any
income above Qx (100). Half of the population has an income less than the
median bM, so Qx (50) = bM. The shaded area underneath the quantile func-
tion is the mean x̄ of the distribution x.
Having introduced these basic concepts, we discuss income standards in
the next section.
Income Standards
An income standard gauges the size of a distribution by summarizing the

entire distribution in a single income level. Some income standards can be
viewed as stylized measures of a society’s overall level of well-being. Others
focus more narrowly on one part of the distribution or have no general wel-
fare interpretation. We begin this section by introducing common proper-
ties that an income standard should satisfy. We denote any income standard
by W and use subscripts to indicate specific measures or indices.
Desirable Properties
An income standard can satisfy several basic properties. We refer to the first
two properties—symmetry and population invariance—as invariance properties
because they describe changes in the distribution that leave the income
standard unaltered. The second pair of properties—weak monotonicity and
the weak transfer principle—are called dominance properties because they
require the income standard to rise (or not fall) when the income distribution
changes in a particular way. Finally, normalization and linear homogeneity are
calibration properties that ensure the income standard is measured by income.
The additional property of subgroup consistency is not a part of the basic prop-
erties, but it is desirable when evaluating income standards of subpopulations.
Symmetry requires that switching two people’s incomes leaves the
income standard evaluation unchanged. In other words, a person should
not be given priority on the basis of his or her identity when calculating a
society’s income standard. Thus, symmetry is also known as anonymity. In
technical terms, symmetry requires the income standard of distribution x to
be equal to the income standard of distribution x', if x' is obtained from x by
a permutation of incomes.
54
What is a permutation of income? An example will explain. Consider

the three-person income vector x = ($10k, $20k, $30k) so that the first,
second, and the third person receive incomes $10k, $20k, and $30k, respec-
tively. If the incomes of the first and second persons are switched, then the
new income vector becomes x' = ($20k, $10k, $30k). This new vector x' is
said to be obtained from vector x' by a permutation of incomes. The sym-
metry property thus can be stated as follows:
Symmetry: If distribution x' is obtained from distribution x by a per-

mutation of incomes, then W(x') = W(x).
The second property is population invariance. This property requires that

the income standard not depend on population size. That is, a replication of
an income vector results in the same income standard as the original sample
vector. Consider the income vector of society X to be x = ($10k, $20k, $30k).
Now suppose three more people join the society with the same income
distribution. The new income vector of society X is x' = ($10k, $10k, $20k,
$20k, $30k, $30k). Society X now has more overall income, but population
invariance requires that the income standard of society X remain unaltered.
What is the implication of population invariance? It allows us to
compare income standards across countries and across time with varying
population sizes. Furthermore, when combined with symmetry, population
invariance allows the income standard to depend only on information found
in a distribution function, which does not include the population size and
the identities of income receivers.
Population Invariance: If vector x' is obtained by replicating vector x

at least once, then W(x') = W(x).
The third property requires that if the income of any person in a society
increases, then the income standard should register an increase, or at least
should not fall. Implicitly, this property assumes that increasing someone’s
income is not harmful to the entire society.
There are two versions of this property. One is weak monotonicity, which
requires that the income standard not fall because of an increase in any-
one’s income. The other version is monotonicity, the stronger version, which
requires that the income standard register an increase if anyone’s income in
the society increases.
55
For vectors x and x', the notation x' > x implies that at least one element
in x' is strictly greater than that in x, and all other elements in x' are no less
than the corresponding elements in x. For example, if x' = ($20k, $10k, $30k)
and x = ($25k, $10k, $30k), then x' > x. However, if x' = ($20k, $10k, $30k)
and x = ($25k, $10k, $25k), then x' ⬎ x because the income of the third
person is lower in x than that in x'.
Weak Monotonicity: If distribution x' is obtained from distribution x

such that x' > x, then W(x') ≥ W(x).
Monotonicity: If distribution x' is obtained from distribution x such
that x' > x, then W(x') > W(x).
Some income standards are occasionally interpreted as social welfare mea-

sures. The fourth property, known as the transfer principle, is the key property that
enables this interpretation. A regressive transfer occurs when income is transferred
from a poorer person to a richer person. The transfer principle requires that a
regressive transfer between two people in a society should lower the income
standard. Conversely, a progressive transfer occurs when income is transferred
from a richer person to a poorer person. The transfer principle requires that a
progressive transfer between two people raise the income standard.
Here is a formal definition of these two kinds of transfers using vector x.
We have already assumed that incomes in x are ordered so that x1 ≤ x2 ≤ … ≤
xN. Let income d be transferred from person n to person m, where n < m and
0 < d < (xm − xn)/2. Denote the post-transfer income vector by x', where all
incomes except those for people n and m are the same as in x, but x'n = xn − d
and xm' = xm + d. Then x' is said to be obtained from x by a regressive transfer.
Now, let income d > 0 be transferred from person m to person n. Denote
the post-transfer income vector by x", where all incomes except those for
people n and m are the same as in x, but x"n = xn + d and xm " = xm − d such
that xm" > xn. Then x" is said to be obtained from x by a progressive transfer.
Consider the following example. Let the two income vectors of society
X at two different points in time be x = ($10k, $20k, $30k) and x' = ($15k,
$20k, $25k), where x' has been obtained from x by transferring $5k from the
third person to the first person. This is a progressive transfer.
Below is the formal statement of the transfer principle property. This
principle is also known as the Pigou-Dalton transfer principle after the
English economists Arthur Cecil Pigou and Hugh Dalton.3
56
Transfer Principle: If distribution x' is obtained from distribution x by

a regressive transfer, then W(x') < W(x). If distribution x" is obtained
from distribution x by a progressive transfer, then W(x") > W(x).
One justification of the transfer principle invokes a utilitarian form of

welfare function that takes welfare to be the average level of (indirect) util-
ity in society and assumes that all utility functions are identical and strictly
increasing (see Atkinson 1970). In this context, the intuitive assumption of
diminishing marginal utility yields the transfer principle. Diminishing marginal
utility requires that the loss to the poorer giver is greater than the gain to
the richer receiver because of a regressive transfer. Hence, overall welfare
falls, or, equivalently, the gain to the poorer receiver is greater than the loss
to the richer giver because of a progressive transfer—hence, welfare rises.
The fifth property is normalization. This property requires that if incomes
are the same across all people in a society, then the income standard should
be represented by that commonly held income. This property is intuitive.
For example, let the income vector of a three-person society be ($20k, $20k,
$20k). Then the income standard should be $20k.
Normalization: For the income distribution, x = (b, b, …, b), W(x) = b.
The sixth property is linear homogeneity. This property requires that if

an income distribution is obtained from another income distribution by
changing the incomes by some proportion, then the income standard should
also change by the same proportion. For example, if everyone’s income in
a society doubles, then the society’s income standard doubles. If everyone’s
income is halved, then the society’s income standard is halved.
Linear Homogeneity: If distribution x' is obtained from distribution x

such that x' = cx where c > 0, then W(x') = cW(x).
Subgroup consistency is the final property presented here. In some empiri-

cal applications, there is a natural concern for certain identifiable popula-
tion subgroups as well as for the overall population. We might be interested,
for instance, in the performances of various states or subregions of a country
to understand how the overall improvement in income standard is distrib-
uted across those regions.
57
When population subgroups are tracked alongside the overall population

value, there is a risk that the income standard could indicate contradictory
or confusing trends. For example, it may be possible that the income stan-
dards of some regions within a country improve while the income standards
of the rest of the country remain the same, but the income standard of the
country as a whole deteriorates. This type of result may cause confusion
because following the regional performances, one would expect the coun-
try’s overall performance to improve.
Thus, a natural consistency property for an income standard might be
that if subgroup population sizes are fixed but incomes are varying, when the
income standard rises in one subgroup and does not fall in the rest, the over-
all population income standard must rise. This property, known as subgroup
consistency, avoids inconsistencies arising from multilevel analyses of this sort.
As an example, suppose the income vector x with population size N is
divided into two subgroup vectors x' with population size N' and x" with
population size N" such that N' + N" = N. Let a new vector y be obtained
from x with the same population size N and its corresponding two subgroups
be y' with population size N' and y" with population size N". The subgroup
consistency property can be stated as follows:4
Subgroup Consistency: Given that the overall population size and

the subgroup population sizes remain unchanged, if W(y') > W(x')
and W(y") ≥ W(x"), then W(y) > W(x).
Having discussed the properties of the income standards, we now discuss

the commonly used income standards. We outline these income standards
and analyze their usefulness in terms of the properties they satisfy.
Commonly Used Income Standards
Four kinds of income standards are in common use: quantile incomes,

partial means, general means, and means based on the maximin approach.
(Among the maximin means, we discuss only the Sen mean in this book.)
We now describe each kind in greater detail.
Quantile Income
Quantile incomes provide information about a specific point on the distri-

bution. They can be directly calculated from a quantile function or a cdf.
58
The quantile income at the pth percentile is the income below which the
incomes of p percent of the population fall. For the income distribution x
with N people, the quantile income at the pth percentile is the income that
is larger than the incomes of the poorest pN/100 people.
We denote the quantile income at the pth percentile of distribu-
tion x by WQI (x; p). For example, if p = 50 percent, then the quantile
income at the pth percentile of distribution x is denoted by WQI (x; 50).
If WQI (x; 50) = $200, then it should be read as 50 percent of the population
in society X earns less than $200. Similarly, if WQI (x; 90) = $1,000, then
90 percent of its population earns less than $1,000.
Commonly reported quantile incomes used when gauging societies’
standard of living are the quantile incomes at the 10th percentile, 20th per-
centile, 50th percentile, 80th percentile, and 90th percentile. A close look
at the quantile income at the 50th percentile reveals that this is the income
below which half of the population of a distribution lies. Therefore, the
quantile at the pth percentile income is just the median of a distribution. For
a particular income distribution where each and every person earns equal
income, the quantile incomes at all percentiles are equal to each other,
ensuring that the quantile incomes satisfy the normalization property.
A quantile function is the most helpful tool for visualizing quantile
incomes. Figure 2.4 shows the quantile function for income distribution x.
Figure 2.4: Quantile Function and the Quantile Incomes
Qx(100) WQI(x;100)
Corresponding value of
a quantile function Qx
Qx(90) WQI(x;90)
Quantile income
Qx
Qx(75) WQI(x;75)
bM WQI(x;50)
Qx(25) WQI(x;25)
Qx(10) WQI(x;10)
0 10 25 50 75 90 100
Population share or percentiles
59
As in figure 2.3, the horizontal axis in figure 2.4 denotes the population share
in percentage, which lies between 0 and 100. The left-hand vertical axis
denotes the corresponding value of a quantile function Qx and the right-hand
vertical axis reports the quantile incomes.
By definition, the quantile income for a certain percentile is the value of
the quantile function at that percentile, so WQI (x; p) = Qx(p). In the figure,
WQI (x; 50) = bM is the median of distribution x. Likewise, WQI (x; 25) and
WQI (x; 75) are the first and the third quartiles of distribution x. The well-
known 10th and 90th percentiles of distribution x are WQI (x; 10) = Qx(10)
and WQI (x; 90) = Qx(90), respectively. Given that a cdf is an inverse of a
quantile function, quantile incomes can also be graphically portrayed and
calculated using a cdf.
What properties do quantile incomes satisfy? It is straightforward to verify
that any quantile income satisfies symmetry, normalization, population invari-
ance, linear homogeneity, and weak monotonicity. However, no quantile income
satisfies the other dominance properties: monotonicity, transfer principle,
and subgroup consistency. Quantile incomes do not satisfy monotonicity
because a person’s income may increase, but as long as it does not surpass a
certain quantile, that quantile income remains unaltered. Similarly, quantile
incomes do not satisfy the transfer principle because they do not change to a
transfer that takes place at a nonrelevant part of the distribution.
The income standards are not subgroup consistent because the quantile
incomes of the subregions may increase, but the overall quantile income may
fall. Consider the following example. Suppose the income vector of society
X is x = ($10k, $20k, $30k, $50k, $60k, $80k) and the income vector of two
subgroups is x' = ($10k, $20k, $30k) and x" = ($50k, $60k, $80k). The 67th
quantile of the three distributions is WQI (x'; 67) = $20k, WQI (x"; 67) = $60k,
and WQI (x; 67) = $50k. Now, suppose the subgroup income vectors over time
become y' = ($10k, $20k, $30k) and y" = ($45k, $65k, $80k). Apparently, the
quantile income at the 67th percentile of the first group does not change, but
that of the second does. In fact, WQI (x'; 67) = WQI (y'; 67) but WQI (y"; 67) >
WQI (x"; 67). What happens to the quantile income at the 67th percentile
of the overall distribution? It turns out that WQI (y; 67) = 45 < WQI (x; 67).
Partial Mean
The next set of commonly used means is the partial means. There are two
types of partial means: lower partial means and upper partial means. A lower
60
partial mean is obtained by finding the mean of the incomes below a specific
percentile cutoff. An upper partial mean is obtained by finding the mean of
incomes above a specific percentile cutoff. Lower partial means are more
commonly used than upper partial means.
The lower partial mean of the pth percentile is the average or mean
income of the bottom p percent of the population. The upper partial mean
of the pth percentile, in contrast, is the average or mean income of the
top (1 – p) percent of the population. We denote the lower partial mean
and upper partial mean of distribution x for percentile p by WLPM(x; p) and
WUPM(x; p), respectively. For example, if p = 50 percent, then the lower par-
tial mean of the pth percentile of distribution x is denoted by WLPM(x; 50).
If WLPM(x; 50) = $100 and WUPM(x; 50) = $10,000, then together they
should be read as the mean income of the bottom 50 percent of the population
is $100, and the mean income of the top 50 percent of the population is $10,000
(see example 2.1).
Example 2.1: Consider the income vector x = ($2k, $4k, $8k, $10k).
The lower partial mean of the 50th percentile of the distribution is
($2k + $4k)/2 = $3k, and that of the 75th percentile of the distribution
is ($2k + $4k + $8k)/3 = $4.7k. In contrast, the upper partial mean
of the 50th percentile of the distribution is ($8k + $10k)/2 = $9k and
that of the 75th percentile of the distribution is $10k.
The following is a graphical description of how partial means can

be calculated using quantile function Qx. The vertical axis of figure 2.5
denotes income, and the horizontal axis denotes population share. There
are two percentiles, p' and p", for describing the lower and upper partial
means. The lower partial mean of the p' percentile population is the
shaded area underneath the quantile function Qx to the left of p' divided
by p'. The lower partial mean is the average income of all people in society
X whose income is less than Qx(p'). Similarly, the upper partial mean of
the p" percentile population is the shaded area underneath the quantile
function Qx to the right of p" divided by (100 – p"). This upper partial
mean is the average income of all people in society X whose income is
larger than Qx(p").
Like the quantile incomes, any partial mean satisfies symmetry, normal-
ization, population invariance, linear homogeneity, and weak monotonicity, but
no partial mean satisfies monotonicity, transfer principle, and subgroup
61
Figure 2.5: Quantile Function and the Partial Means
xN
Qx(p″)
Income
Qx
Qx(p′)
0 p′ p″ 100
Population share
consistency. Like the quantile incomes, one can easily show using a simple
example that partial means do not satisfy subgroup consistency.
Quantile incomes and partial means are crude income standards because
they do not depend on the entire income distribution. Yet they are highly
informative and easy to understand. Especially when income data are miss-
ing for certain parts of the income distribution, these crude income stan-
dards are useful tools for understanding a society’s performance.
In contrast, when rich datasets are available, a study based on quan-
tile incomes and partial means may be limited because they do not reflect
changes in every part of the distribution. For example, if the income
of a person below the median increases—but not by enough to surpass
the median income—then the distribution median does not reflect any
change.
The following income standards are designed to consider the entire
distribution. These income standards will, in most cases, reflect a change in
any part of the distribution.
General Mean
General means are a family of normative income standards. Standards in this

family are normative because the formulation of each measure depends on
62
a parameter denoted by a, which can take any value between − ∞ and + ∞.

Unlike the quantile means and the partial means, general means take into
account the entire income distribution, but emphasize lower or higher incomes
depending on the value of a. Parameter a is familiar in the literature as the
order of general means.
For income distribution x, we denote the general mean of order a by
WGM(x; a). It is defined as
⎧⎛ x a + x a + …+ x a ⎞ 1a
⎪⎪⎜ ⎟⎠ if a ≠ 0.
1 2 N
WGM (x; a ) = ⎨⎝ N (2.3)

⎪ 1
⎪⎩(x1 × x 2 × …× x N ) N if a = 0
Although a may take any value between − ∞ and + ∞, four means in this
family are more well known than others: arithmetic mean, geometric mean,
harmonic mean, and Euclidean mean.
• For a = 1, WGM is known as the arithmetic mean (denoted by WA)

or the average x̄ of all elements in x and can be written as5
x1 + x 2 + L + x N
WA (x) = . (2.4)
N
• For a = 0, WGM becomes the geometric mean (denoted by WG) of all
elements in distribution x and can be expressed as
WG(x) = (x1 × x2 × ... × xN)1/N. (2.5)
If we take a natural logarithm on both sides of equation (2.5), we find
ln x1 + ln x 2 + L + ln x N
WL (x) = ln WG (x) = . (2.6)
N
WL(x) is the average of the logarithm of all incomes in distribution

x. The logarithm of incomes is frequently used for various analyses by
labor economists.
• For a = –1, WGM becomes the harmonic mean (WH) of distribution
x and can be expressed as
−1
⎛ x −1 + x 2−1 + L + x N−1 ⎞
WH (x) = ⎜ 1 ⎟⎠ . (2.7)
⎝ N
63
• Finally, another well-known mean is the Euclidean mean (WE),

obtained when a = 2. The Euclidean mean formula is
1
⎛ x12 + x 22 + L + x 2N ⎞ 2
WE (x) = ⎜ ⎟⎠ . (2.8)
⎝ N
Example 2.2 shows the results of calculating these means for a given
income vector.
• The arithmetic mean of x is ($2k + $4k + $8k + $10k)/4 = $6k.
• The geometric mean of x is ($2k × $4k × $8k × $10k)1/4
= $5.03k.
• The harmonic mean of x is [($2k−1 + $4k−1 + $8k−1 +
$10k−1)/4]−1 = $4.10k.
• The Euclidean mean of x is [($2k2 + $4k2 + $8k2 + $10k2)/4]1/2
= $6.78k.
Having been introduced to the family, one can now understand the
properties of general means and the way they depend on parameter a. All
means in this family satisfy symmetry, normalization, population invariance,
linear homogeneity, monotonicity, and subgroup consistency. Furthermore, for
a < 1, general means satisfy the transfer principle. Thus, the general means
satisfy all the dominance properties introduced earlier. One reason is that,
unlike the quantile means and the partial means, general means consider all
incomes in the distribution.
It is straightforward to show that general means satisfy symmetry, nor-
malization, population invariance, linear homogeneity, and monotonicity.
That general means satisfy subgroup consistency may be verified as follows:
if vector x is divided into subgroup vectors x' and x", then the general mean
of x can be expressed as
WGM(x; a) = WGM((WGM(x'; a), WGM(x"; a)); a). (2.9)
In other words, the general mean of x is the general mean of the general
means of x' and x". Then the monotonicity property ensures that subgroup
consistency is satisfied.
64
Another interesting property of WGM is its monotonic relationship with

parameter a, which requires that the value of WGM increase as a rises and
decrease as a falls. A lower a gives more emphasis to lower values within a
distribution and thus causes WGM to fall. Conversely, a higher a gives more
emphasis to higher values within a distribution, causing the value of WGM
to rise. Technically speaking, WGM(x; a) < WGM(x; a ') for any a < a '. We
refer to this property of general means as increasingness to a. It follows from
this property that WE(x) ≥ WA(x) ≥ WG(x) ≥ WH(x).
There is an exception, however, when the values of general means do
not change as a changes, and this happens when a distribution is degener-
ate. A society’s income distribution is degenerate if all people in that society
have equal incomes. For a degenerate income distribution, all general means
are equal; that is, WGM(x; a) = WGM(x; a ') for all a ≠ a '. Invariance of
general means to degenerate distribution is another way of ensuring that
they satisfy the normalization property.
Given that a ranges from − ∞ to + ∞, what is the range of WGM? Unlike
the value of a, however, WGM is not unbounded. Rather, it has a lower bound
and an upper bound. When a decreases and approaches − ∞, WGM(x; a) con-
verges to the minimum element in x. The society’s income standard in this
case is nothing, but the poorest person’s income is x1. In contrast, when a
increases and approaches + ∞, WGM(x; a) converges toward the maximum
element in x, and the society’s income standard equals the income of the
richest person, xN. Notice, however, that unlike the other general means,
these two extreme income standards—WGM(x; − ∞) and WGM(x; + ∞) —are
not sensitive to the entire distribution. That is, if any element in x other than
x1 and xN changes, these two income standards do not reflect that change.
Figure 2.6 describes the relationship between the family of generalized
means and parameter a. As already discussed, the general mean is the
arithmetic mean at a = 1, the geometric mean at a = 0, the harmonic
mean at a = −1, and the Euclidean mean at a = 2. Values of general means
increase with parameter a. They are bounded below by x1 = min{x} and are
bounded above by xN = max{x}.
One feature we should note carefully is that the general means are
undefined for a < 0 when there is at least one nonpositive element in an
income vector. For example, if an element of x is 0, then for a = −1, we
have (0)−1 = 1/0. Therefore, one requirement for any measure in this family
with a < 0 is that all elements in x be strictly positive.
65
Figure 2.6: Generalized Means and Parameter `
Generalized mean of order

xN
WE(x) WGM(x; α)
WA(x)
WG(x)
WH(x)
x1
–∞ –2 –1 0 1 2 ∞
Parameter
General Means as Welfare Measures

The transfer principle ensures that the general means may be interpreted as
social welfare measures. Actually, the general means for a < 1 are commonly
interpreted as measures of social welfare. This form of welfare function was
considered by Atkinson (1970), who then defined a helpful transforma-
tion of the function called the equally distributed equivalent income (ede).
The utility function that Atkinson assumed to obtain his particular ede was
1 1
U(x n ) = (x n ) a for a < 1 and a ≠ 0 and U(x ) = ln x for a = 0 for all n.
a n n
a
The ede represents the level of income x ede, which, if received by all people
in a society, yields the same welfare level as that of the original income dis-
tribution. Thus, like the general mean itself, the value of ede depends on the
parameter a, and for vector x, the ede of order a is EDE(x; a) = WGM(x; a).
Sen Mean
The usual mean can be reinterpreted as the expected value of a single income
drawn randomly from the population. Now, suppose that instead of a single
income, we were to draw two incomes randomly from the population (with
replacement). If we then evaluated the pair in terms of the lower of the two
incomes, this would lead to the Sen mean, which is defined as the expecta-
tion of the minimum of two randomly drawn incomes.6 These two random
incomes are drawn with replacement, which means that these two incomes
may belong to the same person in a society. If every income in distribution x
66
is compared with every other income in x with replacement, then there are
N2 possible comparisons. Thus, the Sen mean can be defined as
1 N N
WS (x) = ∑ ∑ min{x n′ x n′}.
N 2 n =1 n ′ =1
(2.10)
Because we are using the minimum of the two incomes, this number can be
no higher than the mean, and is generally lower. The Sen mean also empha-
sizes the lower incomes but in a way that differs from the general means with
α < 1, the lower partial means, or the quantile incomes below the median.
There is a straightforward way of calculating the Sen mean for an income
vector—by creating an N × N matrix that has a cell for every possible pair of
incomes and placing the lower value of the two incomes in the cell. Adding
all the entries and dividing by the number of entries (N2) to obtain their
mean provides the Sen mean. Consider example 2.3 to better understand
this way of calculating the Sen mean.
First, we construct the following matrix:
x $2k $4k $8k $10k

$2k $2k $2k $2k $2k
$4k $2k $4k $4k $4k
$8k $2k $4k $8k $8k
$10k $2k $4k $8k $10k
Each cell in this 4 × 4 matrix is the minimum of the top row and
the left column, both of which represent the ordered income vector x.
The Sen mean is the average of all elements in the matrix. Thus,
1
WS (x) = (7 × $2k + 5 × $4k + 3 × $8k + 1 × $10k) = $4.25k.
42
The Sen mean of x is lower than the arithmetic mean of x,
which is $6k.
There is another interesting way of understanding the Sen mean—the

weighted average of all elements of an income distribution—where the
weight on each element depends on the rank of the corresponding element.
Recall that we assumed x1 ≤ x2 ≤ … ≤ xN for distribution x so that the Nth
67
person has the highest income and the first person has the lowest income.
Thus, element xN receives the highest rank and element x1 receives the low-
est rank. The Sen mean attaches the highest weight to the lowest income,
the second-highest weight to the second-lowest income, and the lowest
weight to the highest income.
For distribution x, the Sen mean can be expressed as WS(x) = a1x1 + … +
aNxN, where aN = (2(N − n) + 1)/N2 for all n. Thus, the weight attached
to the highest income xN is aN = 1/N2; the weight attached to the second-
highest income xN–1 is aN–1 = 3/N2; and the weight attached to the lowest
income x1 is a1 = (2N − 1)/N2. The weight attached to the richest income
in the example above ($10k) is 1/16, whereas the weight attached to the
poorest income ($2k) is 7/16. Notice that the weights sum to one, that is,
1 N2
a1 + a 2 + ... + a N = (1 + 3 + 5 + ... + (2 N − 1) = = 1. (2.11)
N2 N2
Thus, the Sen mean can also be expressed as
1
WS (x) = ((2N − 1)x1 + (2N − 3)x 2 + L + 3x N −1 + x N ). (2.12)
N2
The Sen mean satisfies symmetry, normalization, population invariance, lin-

ear homogeneity, monotonicity, and the transfer principle. It does not, however,
satisfy subgroup consistency, which means it is possible that the Sen mean
of one region increases while the Sen mean for the other regions remains
the same and the overall Sen mean falls.
This failure to satisfy subgroup consistency can be shown using a simple
example. Suppose the income vector of society X is x = ($4k, $5k, $6k, $7k,
$14k, $16k) and the income vectors of two subregions are x' = ($4k, $5k,
$7k) and x" = ($6k, $14k, $16k). The Sen means of these three income
vectors are WS(x) = $6.22k, WS(x') = $4.67k, and WS(x") = $9.78k. Now,
suppose the income vector of society X changes to y = ($3.4k, $6.1k,
$6k, $6.5k, $14k, $16k) so that the income vector of the first subgroup
changes to y' = ($3.4k, $6.1k, $6.5k), whereas that of the other subgroup
remains unaltered such that y" = x". Note that the overall mean income
and the mean income of both groups remain unchanged. The Sen means
of the three income vectors become WS(y) = $6.24k, WS(y') = $4.64k, and
WS(y") = $9.78k. Clearly, the Sen mean of the first subgroup decreases
while that of the second subgroup remains the same; yet the overall Sen
68
mean goes up. This feature of the Sen mean is inherited by the inequality
and poverty measures that are based on the Sen mean—the famous Gini
coefficient and the Sen-Shorrocks-Thon index of poverty.
Finally, unlike Atkinson, Sen suggested going beyond the utilitarian
form. His key nonutilitarian example, the Sen mean, can be viewed as both
an ede and a general welfare function, because it satisfies the transfer prin-
ciple. If we denote the Sen ede as EDES(x), then EDES(x) = WS(x).
During our subsequent discussion in this chapter, we will see that these
five means (arithmetic, geometric, harmonic, Euclidean, and Sen) and their
various functional forms are often used in the measurement of welfare,
inequality, and poverty.
An income standard provides a point estimate of the evaluation of a certain

income distribution. We might ask one obvious question: Does the direc-
tion of comparison between distributions in a given point in time, or even
across time, using one income standard continue to hold for other income
standards? Let us clarify this concern with a few examples.
Consider two income vectors x = ($4k, $5k, $6k, $7k, $14k, $16k) and
y = ($3k, $5k, $6k, $9k, $14k, $16k). If we use arithmetic mean WA as an
income standard, then WA(x) = 8.7 and WA(y) = 8.8. Clearly, distribution
y has higher mean income than distribution x. What if we, instead, use the
Sen mean? We get WS(x) = 6.22 and WS(y) = 6.19. Thus, according to the
Sen mean, distribution x has higher welfare than distribution y.
How do the geometric mean and the Euclidean mean of these two
vectors compare? According to the geometric mean, distribution x has
higher welfare than distribution y because WG(x) = 7.57 and WG(y) = 7.52.
According to the Euclidean mean, distribution y has higher welfare than
distribution x because WE(x) = 9.81 and WE(y) = 10.02. What we see from
these comparisons is that different income standards rank two distributions
differently.
Are there situations when the various income standards agree with each
other? This question leads to a discussion of dominance and unanimity. If
there is a situation where we find a dominance relation holding between
two distributions, then there is no need to use different income standards to
evaluate that situation because all income standards would agree. If there is
no unanimous relation, then certain curves may help in understanding the
69
source of ambiguity. Thus, conducting a dominance analysis that is based on

these curves should be the first step in welfare comparison.
A second important motivation for dominance analysis might be focus,
or an identified concern with different parts of the distribution. Has the
rapid growth for the higher-income group been matched by growth of the
middle-income group or the lower-income group? We spend some time in
this subsection finding answers to these questions by plotting entire classes
of income standards using the various curves to be defined next. If one curve
always remains above another curve, then all income standards in that class
agree in ranking—for example, two income distributions. However, if the
curves cross, then situations may arise in which different income standards
in the same class disagree with each other.
A first such curve is the quantile function itself, which simultaneously
depicts incomes from lowest to highest. When all income quantiles are
the same, then one income distribution always lies above another income
distribution. When two distributions never cross, the situation is known as
first-order stochastic dominance (FSD). An income distribution x first order
stochastically dominates another distribution y, denoted by x FSD y, if and
only if (a) no portion of x’s quantile function lies below y’s quantile func-
tion and (b) at least some part of x’s quantile function lies above y’s quantile
function. Let us denote quantile function using the notations introduced
earlier. So x’s quantile function is denoted by Qx and that of y is denoted by
Qy. Then, the definition of FSD is as follows:
First-Order Stochastic Dominance: Distribution x first order stochasti-

cally dominates another distribution y if and only if Qx(p) ≥ Qy(p) for
all p in the range [0,100] and Qx(p) > Qy(p) for some p.
The concept of FSD may also be understood in terms of cumulative

distribution functions. Recall that a quantile function is just an inverse
of a cdf. Using the notations introduced earlier, we denote the cdf of x by
Fx and that of y by Fy. The formal definition of FSD in terms of cdfs is as
follows:
First-Order Stochastic Dominance: Distribution x first order

stochastically dominates another distribution y if and only if Fx(b) ≤
Fy(b) for all b in the range [0, ∞] and Fx(b) < Fy(b) for some b.
70
FSD ensures higher welfare according to every utilitarian welfare func-

tion with identical, increasing utility functions. The robustness implied by
an unambiguous comparison of quantile functions extends to all income stan-
dards and all symmetric welfare functions for which “more is better.” However,
if the resulting curves cross, the final judgment is contingent on which income
standard is selected. Even in this case, the quantile function can be helpful in
identifying the winning and losing portions of the distribution.
Figure 2.7 depicts the situation where x FSD y. Panel a shows the FSD
by quantile functions, and panel b shows the FSD by cdfs. In panel a, the
quantile function of x lies completely above that of y, which means that
every quantile income of distribution x is larger than the corresponding
quantile income of distribution y, so x FSD y. The same argument applies to
the cdfs in panel b, where the cdf of x lies to the right of y. Later, we will find
the concept of FSD that is based on cdfs useful, especially in poverty analysis.
The generalized Lorenz (GL) curve is a second curve that is useful for
dominance analysis. The generalized Lorenz curve graphs the area under the
quantile function up to each percent p of the population. Thus, any point
on a generalized Lorenz curve is the cumulative mean income held by the
bottom p percent of the population. We denote the generalized Lorenz func-
tion of distribution x by GLx, and that for the p percent of the population
by GLx(p). By construction, for income distribution x, GLx(100) = WA(x)
and GLx (0) = 0.
Figure 2.7: First-Order Stochastic Dominance Using Quantile Functions and Cumulative
Distribution Functions
a. Quantile function b. Cumulative distribution function

100
Fy
Income
Qx Fx
Qy
0 100 0
Population share Income
71
Figure 2.8 describes the construction of a generalized Lorenz curve from

a quantile function of a five-person income vector x = ($10k, $15k, $20k,
$25k, $30k). There are five percentiles: 20th, 40th, 60th, 80th, and 100th. In
panel a, we outline the quantile function of x, Qx. In panel b, we report the
generalized Lorenz curve of x, GLx. The mean of distribution x is WA(x) = 20.
A point on the generalized Lorenz curve denotes the area underneath the
quantile function for the corresponding percentile of the population. Up to
the 20th percentile of the population, the area under Qx is the area A.
In panel b, the corresponding value of GLx for the 20th percentile of the
population is denoted by point I. Thus, the value at point I is A/100 = 10 ×
20/100 = 2. Similarly, the value of GLx for the 40th percentile of the popula-
tion is denoted by point II, and the value at point II is (A+B)/100 = (10+15)
× 20/100 = 5. Repeating this approach, we find that the value of GLx for
the 100th percentile of the population is denoted by point V, and the
value at point V is (A + B + C + D + E)/100 = (10 + 15 + 20 + 25 + 30)
× 20/100 = 20. Note that the value at point V is the same as the mean of
distribution x, WA(x).
The generalized Lorenz curve is closely linked with lower partial means
(see Shorrocks 1983). Recall from our earlier discussion that the lower
partial mean for a certain percentile of population p is the area underneath
Figure 2.8: Quantile Function and Generalized Lorenz Curve
a. Quantile function b. Generalized Lorenz curve
30
25
Q
x
20 20 V
WA(x)
Income
Income
E
15 14 IV
D
C
10 III GLx
B 9
A II
5
I
2
0 20 40 60 80 100 0 20 40 60 80 100
Population share Population share
72
the quantile function divided by the percentile itself. Thus, the height of
the generalized Lorenz curve at any percentile of population p is the lower
partial mean times p itself, because the height of the generalized mean is the
area underneath the quantile function at corresponding percentile p, that is,
GLx(p) = pWLPM(x; p). If income distribution x has a large enough sample
size, the generalized Lorenz curve takes a form similar to the one described
in figure 2.9.
The horizontal axis in figure 2.9 shows the population share, and the ver-
tical axis denotes the height of the generalized Lorenz curve by income. The
generalized Lorenz curve for distribution x is denoted by GLx. The maximum
height of GLx is WA(x). The height of GLx for the 50th percentile of the
population is GLx(0.5).
If the total income in distribution x is distributed equally across all
people in the society and distribution y is obtained, then the generalized
Lorenz curve GLy becomes a straight line. The maximum height of GLy is
also WA(x), because redistribution of incomes does not change the mean
income. Notice that the height of GLy is higher than the height of GLx
for every percentile p. This implies that every partial mean of distribution
y is larger than the corresponding partial mean of distribution x. Thus, two
generalized Lorenz curves of this sort show a dominance relation between two
distributions in terms of partial means.
Figure 2.9: Generalized Lorenz Curve
GLx(100) WA(x)
GLy
Income
GLy(50) WA(x)/2
GLx
WA(x ′)
GLx(50)
GLx ′ (50) GLx ′
0 50 100
Population share
73
All partial means agree that distribution y has higher welfare than dis-
tribution x. Similarly, if there is another distribution x' whose generalized
Lorenz curve, GLx', lies completely below GLx (also shown in figure 2.9),
then all partial means agree that distribution x has higher welfare than dis-
tribution x'. The heights of the generalized Lorenz curves for distributions
y, x, and x' at the 50th percentile are GLy(50), GLx(50), and GLy'(50),
respectively. The generalized Lorenz curve represents second-order stochas-
tic dominance, which signals higher welfare according to every utilitarian
welfare function with identical and increasing utility function exhibiting
diminishing marginal utility. Example 2.4 provides a practical illustration of
generalized Lorenz calculations. The generalized Lorenz curve is also closely
related to the Sen mean. For distribution x, the Sen mean, WS(x), is twice
the area underneath GLx.
Example 2.4: Suppose per capita income in India is Rs 25,000. If only

3 percent of this mean income is received by the poorest 20 percent
of the population, then GLInd(20) = Rs 750.
Suppose incomes in India were redistributed, thereby keeping
the average income unaltered so that everyone in India has identical
income. Let us denote this income distribution by y. Then the
cumulative average income received by the poorest 20 percent of the
population is 20 percent and GLy(20) = Rs 5,000. Thus, GLy(20) –
GLInd(20) = Rs 5,000 – Rs 750 = Rs 4,250. The loss of welfare
because of unequal distribution of income for the poorest 20 percent
of the population is Rs 4,250. In relative terms, the loss of welfare is
4,250/5,000 = 85 percent.
However, note that the loss presented in terms of the height of the
generalized Lorenz curve is not the potential loss in the mean income
of the poorest 20 percent of the population. The mean income of the
poorest 20 percent of the population is GLInd(20)/0.2 = Rs 3,750.
Had income been equally distributed, the mean income of the
poorest 20 percent would have been Rs 25,000. In that scenario, the
potential loss of mean income is Rs 21,250. But in a relative sense,
the percentage loss in mean income is 25,205/25,000 = 85 percent,
which is the same as the percentage loss in terms of the height of the
generalized Lorenz curve. In fact, the percentage loss of welfare using
the height of the generalized Lorenz curve is always the same as the
percentage loss of mean income of that percentile.
74
Finally, a third curve depicts the general mean levels as parameter a varies.
We call this curve a general mean curve. This curve has already been outlined
in figure 2.6, where it is increasing in α; tends to the minimum income for
very low a ; rises through the harmonic, geometric, arithmetic, and Euclidean
means; and tends toward the maximum income as α becomes very large.
Why is this curve useful? At the beginning of this subsection, an example
showed that different generalized means may rank an income distribution
differently. So the general mean curve is useful for determining (a) whether
a given comparison of general means is robust to the choice of any income
standard from the entire class of general means, and, if not, (b) which of the
income standards is higher or lower.
General mean curves are also related to the quantile function and the
generalized Lorenz curve. A higher quantile function will always yield a
higher general mean curve, and a higher generalized Lorenz curve will raise
the general mean curve for a < 1, or the general means that favor the low
incomes. The general mean curve concept will be particularly relevant to
our later discussions of Atkinson’s inequality measure.
Growth Curves
Some analyses go beyond the ordinal question (Which distribution is

larger?) to consider the cardinal question: How much larger in percentage
terms is one distribution than another? This question is especially salient
when the two distributions are associated with the same population at two
points in time. Thus, the second question follows: At what percentage rate
did the income standard grow?
The most common and well-known way of understanding growth is by
the growth of per capita income or mean income. The arithmetic mean is
the income standard involved in this case. However, the defining proper-
ties of an income standard ensure that its rate of growth is a meaningful
number that can be compared with the growth rates of other income stan-
dards, either for robustness purposes or for an understanding of the quality
of growth.
As in our use of various curves in dominance analysis, we may also use
different growth curves to understand how robust the growth of an income
standard is and to understand whether the growth is of meaningful quality.
A growth curve depicts the rates of growth across an entire class of income
standards, in which the standards are ordered from lowest to highest.
75
In fact, each of the three dominance curves presented earlier suggests an

associated growth curve. First, the growth incidence curve assesses how the
quantile incomes are changing over time. Second, the generalized Lorenz
growth curve indicates how the lower partial means are changing over time.
Finally, the general mean growth curve plots the rate of growth of each general
mean over time against parameter a. In the remainder of this section, we
discuss the concepts of these different growth curves in greater detail.
Growth Incidence Curve
We start with the growth incidence curve. Consider two income distributions,
x and y, at two different periods of time, where x is the initial income distri-
bution. The quantile incomes of distribution x and distribution y at percen-
tile p are denoted by WQI(x; p) and WQI(y; p), respectively. The growth of
quantile income at percentile p is denoted by
WQI (y; p) − WQI (x; p)

g QI (x, y; p) = × 100%. (2.13)
WQI (x; p)
If every quantile registers an increase over time, then gQI(x, y; p) > for
all p. The curve’s height at p = 50 percent gives the median income’s growth
rate. Note that no part of this growth curve provides any information about
the growth of mean income. Varying p allows us to examine whether this
growth rate is robust to the choice of income standard, or whether the low-
income standards grew at a different rate than the rest.
Figure 2.10 depicts the growth curves of quantile incomes. The vertical
axis denotes the growth rate of quantile income and the horizontal axis denotes
the cumulative population share. Suppose there are two societies, X and X'.
The income distributions of society X at two different points in time are
x and y, while those of society X' are x' and y'. The dashed growth curve
gQI(x, y) denotes the quantile income growth rates of society X over time,
whereas the dotted growth curve gQI(x', y') denotes the quantile income
growth rates of society X' over time.
Suppose the growth rates of mean income across these two distributions
are the same and are denoted by –g > 0. Thus, the solid horizontal line at
–g denotes the growth rate if the growth rate had been the same for all per-
centiles or the cumulative population share.
76
Figure 2.10: Growth Incidence Curves
Growth rate of quantile income
A
B
g
gQI(x,y)
A′ B′ gQI(x ′,y ′)
0 20 40 60 80 100
Cumulative population share
What information do these two growth curves provide? Growth between

x and y is pro-poor in the sense that lower quantile incomes have positive
growth, whereas the upper quantile incomes have negative growth. Growth
between x' and y', in contrast, is not pro-poor because lower quantile
incomes have negative income growth, whereas upper quantile incomes
have positive growth. In society X, the growth rate of income for the 20th
percentile is much higher than that of the 40th percentile, as denoted by
point A and point B, respectively. Note that the growths are higher than the
mean growth rates. In society X', however, the income growth rate for the
20th percentile is almost the same as that of the 40th percentile, as denoted
by point A' and point B', respectively. We will discuss pro-poor growth in
greater detail in the poverty section of this chapter.
Generalized Lorenz Growth Curve
The next growth curve is the generalized Lorenz growth curve. Consider the
two income distributions, x and y, used previously. The lower partial means
of distribution x and distribution y at percentile p are denoted by WLPM
(x; p) and WLPM(y; p), respectively. The growth of partial means at percen-
tile p is denoted by
77
WLPM (y; p) − WLPM (x; p)

g LPM (x, y; p) = × 100%. (2.14)
WLPM (x; p)
If every quantile income registers an increase over time, then gLPM(x, y;

p) > 0 for all p. Given that GLx(p) = pWLPM(x; p), the growth of the lower
partial mean at a certain percentile is equal to the growth of the general-
ized Lorenz curve at that percentile. So the height of the generalized Lorenz
growth curve at p = 20 percent is the rate at which the mean income of the
lowest 20 percent of the population changed over time.
Unlike the growth incidence curve, this curve provides information
about the growth rate of mean income, which is the height of the curve at
p = 100 percent. Again, varying p allows us to examine whether this growth
rate is robust to the choice of income standard, or whether the low-income
standards grew at a different rate than that of the rest. If the growth rates of
the lower-income standards are found to be lower than the mean income,
then overall growth, indeed, has not been pro-poor. However, if all lesser
“lower partial means” grow at a faster rate than the higher “lower partial
means,” then growth is assumed to be pro-poor.
Figure 2.11 depicts the growth curves of lower partial mean incomes.
The vertical axis denotes the growth rate of lower partial mean income, and
the horizontal axis denotes the cumulative population share. Following the
same notations as the growth incidence curve, suppose that there are two
Figure 2.11: Growth Rate of Lower Partial Mean Income

Growth rate of partial mean income
C
D gLPM(x,y)
g
C′ D′
gLPM(x ′,y ′)
0 20 40 60 80 100
Cumulative population share
78
societies, X and X'. The income distributions of society X at two differ-

ent points in time are x and y, whereas those of society X' are x' and y'.
The dashed growth curve gLPM(x, y) denotes growth rates of lower partial
mean income of society X over time, whereas the dotted growth curve
gLPM(x', y') denotes growth rates of lower partial mean income of society
X' over time.
are the same and are denoted by g– > 0. Thus, the solid horizontal line at
g– denotes the growth rate if the growth rate had been the same for all per-
centiles or the cumulative population share.
x and y is pro-poor in the sense that mean incomes of the population’s bot-
tom percentiles have positive growth, whereas mean incomes of the popula-
tion’s upper percentiles have negative growth. Growth between x' and y', in
contrast, is not pro-poor because mean incomes of the population’s bottom
percentiles have negative income growth, whereas mean incomes of the
population’s upper percentiles have positive growth.
In society X, the growth rates of the mean income of the bottom 20th
percentile of the population and that of the bottom 40th percentile of the
population are denoted by point C and point D, respectively. In society X',
however, the growth rate of mean income of the bottom 20th percentile of
the population and that of the bottom 40th percentile of the population are
denoted by point C' and point D', respectively. Note that growth of mean
income is the growth at the 100th percentile income where the two growth
curves meet because they have been assumed to have the same growth rate
of mean income.
General Mean Growth Curve
The final of the three growth curves is the general mean growth curve.
Considering the income distributions x and y discussed previously, we
denote the general mean of order a of distribution x and distribution y by
WGM(x; a) and WGM(y; a), respectively. The growth of general mean of
order a is denoted by
WGM (y; a) − WGM (x; a)

g GM (x, y; a) = × 100%. (2.15)
WGM (x; a)
79
When every general mean registers an increase over time, gGM(x,y; a) > 0.
When a = 1, the curve’s height is the usual mean income growth rate. This
rate is equal to the growth of the generalized Lorenz growth curve at p = 100
percent. At a = 0 the curve shows the growth rate for the geometric mean,
and so forth. As we will see later, each of these growth curves can help
in understanding the link between growth and change in inequality
over time.
Figure 2.12 shows the growth curves of general mean incomes. The verti-
cal axis denotes the growth rate of general mean income, and the horizontal
axis denotes the values of parameter a. Following the same notations as the
previous two growth incidence curves, suppose that there are two societies,
X and X'. Income distributions of society X at two different points in time
are x and y, whereas those of society X' are x' and y'. The dashed growth
curve gGM(x, y) denotes the growth rates of general mean income of soci-
ety X over time, whereas the dotted growth curve gGM(x', y') denotes the
growth rates of general mean income of society X' over time.
–
are the same and are denoted by g > 0. Thus, the solid horizontal line at
–
g denotes the growth rate if the growth rate had been the same for all a.
x and y is pro-poor in the sense that general means for lower a, which focus
more on the lower end of the distribution, have positive growth, whereas
Figure 2.12: General Mean Growth Curves

Growth rate of general mean income
gG ( ′)
M x,y) ′,y
(x
M
gG
g
–∞ –2 –1 0 1 2 ∞
Parameter
80
general means for larger a have negative growth. Growth between x' and y', in
contrast, is not pro-poor because the general means for lower a have negative
income growth whereas the general means for larger a have positive growth.
The mean income growth rates are the heights of the two growth curves at
a = 1, which are equal by assumption for this example. Heights at a = 0 and
a = –1 are growth rates of the geometric and harmonic means, respectively.
Inequality Measures
The second aspect of a distribution is spread, which is evaluated using a

numerical inequality measure, assigning each distribution a number that
indicates its inequality level. There are two ways of understanding and inter-
preting an income inequality measure. One way is through the properties it
satisfies. The other way is by using a fundamental link between inequality
measures and income standards. We begin with the first approach by out-
lining the desirable properties an inequality measure should satisfy. In this
section, any inequality measure is denoted by the notation I. Specific indices
are denoted by using corresponding subscripts.
An inequality measure should satisfy four basic properties: symmetry, popula-

tion invariance, scale invariance, and the transfer principle. Like income stan-
dards, these properties may be classified into categories. Invariance properties
leave the inequality measures invariant to certain changes in the dataset, and
they include symmetry, population invariance, scale invariance, and normaliza-
tion. The normalization property calibrates the measure’s value when there is
no inequality. Dominance properties cause inequality measures to change in
a particular direction. Properties in this category include the transfer principle
and transfer sensitivity. Other properties, such as subgroup consistency and
additive decomposability, are compositional properties relating subgroups and
overall inequality levels. Most of these properties are similar in interpreta-
tion to the corresponding properties of income standards.
The first property, symmetry, requires that switching the income levels of
two people leaves the evaluation of a society’s inequality unchanged. In other
words, a person should not be given priority on the basis of his or her identity
when evaluating a society’s inequality. In more technical terms, it requires
81
the inequality measure of distribution x to be equal to the inequality measure

of another distribution x' if x' is obtained from x by a permutation of incomes.
For example, recall the three-person income vector x = ($10k, $20k,
$30k) so that the first, second, and the third persons receive incomes $10k,
$20k, and $30k, respectively. If the incomes of the first and second persons
are switched, then the new income vector becomes x' = ($20k, $10k, $30k).
This new vector x' is said to be obtained from vector x by a permutation of
incomes.
Symmetry: If distribution x' is obtained from distribution x by

permutation of incomes, then I(x') = I(x).
The second property, population invariance, requires that the level of

inequality within a society is invariant to population size, in the sense that
a replication of an income vector results in the same inequality level as the
original sample vector. What is the implication of this property? Consider
the income vector of society X, x = ($10k, $20k, $30k). Now, suppose three
more people join the society with the same income distribution so that the
new income vector of society X is x' = ($10k, $10k, $20k, $20k, $30k, $30k).
The population invariance property requires that the inequality level in society
X remain unaltered. This property allows us to compare the inequality level
across countries and across time with varying population sizes. Furthermore,
population invariance allows the inequality measure to depend on a distribu-
tion function, which normalizes the population size to one.
Population invariance: If a vector x' is obtained by replicating vector

x at least once, then I(x') = I(x).
The third property, scale invariance, requires that if an income distribu-

tion is obtained from another distribution by scaling all incomes by the same
factor, then the inequality level should remain unchanged. For example,
if everyone’s income in a society is doubled or halved, then the level of
inequality of the society does not change. The scale invariance property
ensures that the inequality being measured is a purely relative concept and
is independent of the distribution’s size.
Scale invariance is analogous to the linear homogeneity property for
income standards, which ensures that the relative status of every person
82
remains unchanged when compared to the income standard, even after all
incomes are scaled up or down by the same factor. This similarity supports
the idea that the relative inequality level remains unchanged.7
Scale Invariance: If distribution x' is obtained from distribution x'

such that x' = cx, where c > 0, then I(x') = I(x).
The fourth property, normalization, requires that if incomes are the same
across all people in a society, then no inequality exists within the society
and the inequality measure should be zero. Normalization is a natural
property. For example, if the income vector of a three-person society is
($20k, $20k, $20k), then the inequality measure should be zero. Even if
everyone’s income increases 10-fold and the new income vector is ($200k,
$200k, $200k), the inequality measure should still be zero.
Normalization: For the income distribution x = (b, b ,..., b), I(x) = 0.
The fifth property is the transfer principle, which requires that a regressive
transfer between two people in a society should increase inequality and a
progressive transfer between two people should reduce inequality. Regressive
and progressive transfers were defined earlier for income standards.
Transfer Principle: If distribution x' is obtained from distribution x

by a regressive transfer, then I(x') > I(x). If distribution x" is obtained
from distribution x by a progressive transfer, then I(x") < I(x).
In inequality measurement, there is also a weaker version of the transfer

principle, which requires that a regressive transfer between two people in a
society not decrease inequality and that a progressive transfer between two
people not increase inequality. Thus, the weaker principle allows the pos-
sibility that the level of inequality may remain unaltered because of progres-
sive or regressive transfers.
Weak Transfer Principle: If distribution x' is obtained from

distribution x by a regressive transfer, then I(x') ≥ I(x). If distribution
x" is obtained from distribution x by a progressive transfer, then
I(x") ≤ I(x).
83
The transfer principle requires an inequality measure to decrease if the

transfer is progressive. However, it does not specify the amount by which
inequality should fall, and it is not concerned with the part of the distribu-
tion where the transfer is taking place. The same amount may be transferred
between two poor people or between two rich people. Should the transfer
have the same effect on the inequality measure no matter where it takes
place? Consider the four-person income vector x = ($100, $200, $10,000,
$20,000). First, suppose $20 is transferred from the second person to the
first person. The post-transfer income vector is x' = ($120, $180, $10,000,
$20,000). Thus, transferring 10 percent of the second person’s income has
increased the first person’s income by 20 percent.
Now, suppose instead that the same $20 transfer takes place between the
third and the fourth person. The post-transfer income vector is x" = ($100,
$200, $10,020, $19,980), where transferring 0.1 percent of the fourth person’s
income has increased the third person’s income by 0.2 percent. This transfer
makes hardly any difference in the large incomes of the two richer people.
It may seem that a transfer of the same amount between two poor people
and two rich people should not have the same effect on the overall inequal-
ity. However, the sixth property, transfer sensitivity, requires an inequality
measure be more sensitive to transfers at the lower end of the distribution. In
other words, this property requires that the inequality measure change more
if a transfer takes place between two poor people than if the same amount of
transfer takes place between two rich people the same distance apart.
Suppose the initial income distribution is x = (x1, x2, x3, x4), where x1 < x2
< x3 < x4, x2 – x1 = x4 – x3 > 0. Note that the distance between x1 and x2 is the
same as the distance between x3 and x4. Suppose distribution x' is obtained
from distribution x by a progressive transfer of amount d < (x2 – x1)/2 between
x2 and x1, that is, x' = (x1 + d, x2 – d, x3, x4), and distribution x" is obtained
from distribution x by a progressive transfer of the same amount d between
x3 and x4, that is, x" = (x1, x2, x3 + d, x4 – d ). Thus, the same amount of
progressive transfer has been made between two poorer people and two richer
people, who are equally distant from each other. Both x' and x" are more
equal than x according to the transfer principle, but can we compare x' and
x"? The answer is yes. In fact, any transfer sensitive inequality measure should
judge distribution x' as more equal than distribution x". Shorrocks and Foster
(1987) have reinterpreted the transfer sensitivity property in terms of favor-
able composite transfer (FACT). When a distribution is obtained from another
84
distribution by a progressive transfer at the lower end of a distribution and

simultaneously by a regressive transfer at the upper end of the same distribu-
tion, such that the variance remains unchanged, then the latter distribution
is stated to be obtained from the former distribution by FACT. Thus, the
transfer sensitivity property may be stated as follows:
Transfer Sensitivity: If distribution x' is obtained from distribution x

by FACT, then I(x') < I(x').
When one distribution is obtained from another distribution by FACT,

then the corresponding Lorenz curves intersect each other. In this case, the
transfer principle cannot rank two distributions. However, if a Lorenz curve
crosses the Lorenz curve of another distribution once from above, and the
coefficient of variation (standard deviation divided by the mean) of the former
distribution is no higher than that of the latter distribution, then all transfer
sensitive measures agree that the former distribution has less inequality.8
The seventh property is subgroup consistency, which is conceptually the
same as the corresponding property for income standards. This property
requires that if the sizes and means of a subgroup population are fixed, then
overall inequality must rise when the inequality level rises in one subgroup
and does not fall in the rest of the subgroups.
For example, suppose that income vector x with population size N is
divided into two subgroup vectors: x' with population size N' and x" with
population size N" such that N' + N" = N. Let a new vector, y, be obtained
from x with the same population size N, and let its two subgroups be denoted
by y' with population size N' and y" with population size N". The subgroup
consistency property can be stated as follows:
Subgroup Consistency: Given that subgroup population sizes and

subgroup means remain unchanged, if I(y') > I(x') and I(y") ≥ I(x"),
then I(y) > I(x).
There is a closely related property that is often useful for understanding

how much of the overall inequality can be attributed to inequality within
subgroups and how much can be attributed to inequality across subgroups,
given a collection of population subgroups. For example, the population of
a country may be divided across various subgroups, such as across rural and
85
urban areas, states, provinces, and other geographic regions; across ethnic
and religious groups; across genders; or across age groups. One may want to
evaluate the source of inequality, such as whether overall income inequality
is due to unequal income distribution within sex or unequal income distri-
bution across sex.
The eighth property is additive decomposability, which requires overall
inequality to be expressed as a sum of within-group inequality and between-
group inequality. Within-group inequality is a weighted sum of subgroup
inequalities. Between-group inequality is the inequality level obtained when
every person within each subgroup receives the subgroup’s mean income.
Kanbur (2006) discussed the policy significance of this type of inequality
decomposition. It is often found that the contribution of the between-group
term is much lower than the within-group term, and, thus, policy priority
is directed toward ameliorating within-group rather than between-group
inequality. These types of policy conclusions should be carefully drawn,
because the lower between-group term may receive much larger social
weight than its within-group counterpart. Also, the between-group term’s
share of overall inequality may increase as the number of groups increases.
How to incorporate these issues into inequality measurement requires fur-
ther research, and solving these issues is beyond the scope of this book.
However, if the policy interest is in understanding how the between-group
inequality as a share of total inequality has changed over time for a fixed
number of groups, then the decomposability property is very useful.
To formally outline the additive decomposability property, we will use
two groups to simplify the interpretation, but the definition can be extended
to any number of groups. Suppose the income vector x with population size
N is divided into two subgroup vectors: x' with population size N' and x"
with population size N" such that N' + N" = N. Let us denote the means of
these three vectors by x̄, x̄', and x̄". The additive decomposability property
can be stated as follows:
Additive Decomposability: If income distribution x is divided into

two subgroup distributions x' and x", then I(x) = W'I(x') + W"I(x") +
I(x̄',x̄"), where W' and W" are weights.
The between-group contribution is I(x̄', x̄")/I(x) and the within-group

contribution is [W' I(x') + W" I(x")]/I(x), as seen in example 2.5.
86
Example 2.5: Consider the five-person income vector x = ($10k, $15k,

$20k, $25k, $30k), which is divided into two subgroups x' = ($10k,
$30k) and x" = ($15k, $20k, $25k). The mean of x' is x̄' = $20k, and
the mean of x" is also x̄" = $20k. Let an additively decomposable
inequality index I be used to estimate the inequality level. The total
within-group inequality is W'I(x') + W"I(x"). However, there is no
between-group inequality in this case, because the mean incomes
of both groups are equal. So the between-group contribution I(x̄',
x̄")/I(x) is 0.
Inequality and Income Standards
There is a second way of understanding inequality measures: through

income standards. This, in fact, relies on an intuitive link between inequal-
ity measures and pairs of income standards: a and b. Let a be the smaller
income standard, and let b be the larger income standard. It is natural
to measure inequality in terms of the relative distance between a and b,
such as I = (b − a)/b, or some other increasing function of the ratio b/a.
Indeed, scale invariance and the weak transfer principle essentially require
this form for the measure. We will find in our subsequent discussions that
virtually all inequality measures in common use are based on twin income
standards.
Commonly Used Inequality Measures
Commonly used inequality measures are mostly related to the five kinds
of income standards we discussed earlier. The inequality measures that we
discuss in this section are quantile ratios, partial mean ratios, Gini coefficient,
Atkinson’s class of inequality measures, and generalized entropy measures.
Quantile Ratio
A quantile ratio compares incomes of higher and lower quantile incomes.

Inequality across quantile incomes provides a useful way to understand
income dispersion across the distribution. Because no quantile ratio considers
the entire distribution, this measure is a crude way of presenting inequality.
87
For income distribution x, let the quantile income at the pth percentile
be denoted by WQI(x; p), and let the quantile income at the p'th percentile
be denoted by WQI(x; p'), such that p > p'. A quantile ratio is commonly
reported as a ratio of the larger quantile income to the smaller quantile
income. However, this view leads the values of inequality measures to range
from one to ∞. This range is not comparable to other inequality measures,
which commonly range from zero to one. In this book, we formulate the
quantile ratio in such a way that it ranges from zero to one. The p/p' quantile
ratio is represented by the following formula:
WQI (x; p) − WQI (x; p ′) WQI (x; p ′)

IQR (x; p / p ′) = = 1− . (2.16)
WQI (x; p) WQI (x; p)
In this case, the quantile income at the pth percentile WQI(x; p) is the
higher income standard, and the quantile income at the p'th percentile
WQI(x; p') is the lower income standard.
The higher the quantile ratio, the higher the level of inequality across
two percentiles of the population in the society. A quantile ratio is zero
when both the upper and the lower quantile incomes are equal. A quantile
ratio reaches its maximum value of one when the lower quantile income
WQI(x; p') is zero. This means that no one in the lower percentile earns
any income and that the upper quantile income is positive. Note that if all
people in the society have equal incomes, then any quantile ratio is zero.
However, a quantile ratio of zero does not necessarily mean that incomes are
equally distributed across everyone in the society.
The quantile ratios used most often include the 90/10 ratio, 80/20 ratio,
50/10 ratio, and 90/50 ratio. The 90/10 ratio, for example, captures the dis-
tance between the quantile income at the 90th percentile and the quantile
income at the pth percentile as a proportion of the quantile income at the
10th percentile. How should the number IQR(x; 90/10) = 0.9 be interpreted?
There are, in fact, several ways to interpret the number:
• The number may be directly read as the gap between the lowest
income of the richest 10 percent and the highest income of the poorest
10 percent of the population, being 90 percent of the lowest income of
the richest 10 percent of the population.
• The number may be seen as the highest income of the poorest
10 percent of the population, being 10 percent (1 − 0.9 = 0.1) of the
lowest income of the richest 10 percent of the population.
88
• The number can be interpreted as the lowest income of the richest

10 percent of the population, being 10 times (1/(1 – 0.9)) larger
than the highest income of the poorest 10 percent of the population.
Similarly, IQR(x; 90/50) = 0.75 implies that the lowest income of the
richest 10 percent of the population is 1/(1 − 0.75) = 4 times larger
than the highest income of the poorest 50 percent of the population.
Quantile ratios may be classified into three categories: upper end quantile
ratio, lower end quantile ratio, and mixed quantile ratio. The first two categories
capture inequality within any one side of the median, and the third category
captures inequality in one side of the median versus that of the other side of
the median. For example, IQR(x; 90/50) is an upper end quantile ratio, and
IQR(x; 50/10) is a lower end quantile ratio, whereas IQR(x; 90/10) is a mixed
quantile ratio.
What properties does a quantile ratio satisfy? A quantile ratio, as defined
earlier, satisfies symmetry, normalization, population invariance, and scale
invariance. Thus, a quantile ratio satisfies all four invariance properties.
What about the dominance properties? It turns out that a quantile ratio
satisfies none of the dominance properties.
The following example shows that a quantile ratio does not satisfy the weak
transfer principle. Suppose the highest income of the poorest 10 percent of
the population is $100 and the lowest income of the richest 10 percent of the
population is $2,000. Then IQR(x; 90/10) = ($2,000 − $100)/$2,000 = 0.95.
Now, suppose that a regressive transfer takes place between the poorest
person in the society and the richest person among the poorest 10 percent
of the population such that the highest income in that group increases to
$120. Then the post-transfer quantile ratio is IQR(x; 90/10) = ($2,000 −
$120)/$2,000 = 0.94.
Therefore, the quantile ratio shows a decrease in inequality even when
a regressive transfer has taken place. If a quantile ratio does not satisfy the
weak transfer principle, then it cannot satisfy its stronger version—the
transfer principle, or transfer sensitivity. The quantile ratios are not addi-
tively decomposable and also do not satisfy subgroup consistency.
Partial Mean Ratio
A partial mean ratio is an inequality measure comparing an upper partial

mean and a lower partial mean. Like quantile ratios, no partial mean ratio
89
considers the entire income distribution; thus, it is also a crude way of

understanding inequality.
For income distribution x, let the upper partial mean for percentile p
be denoted by WUPM(x; p) and the lower partial mean for percentile p' be
denoted by WLPM(x; p'). A partial mean ratio is also commonly reported
as a ratio of both partial means ranging from one to ∞. However, as with
the quantile ratio, we formulate the partial mean ratio in such a way that
it ranges from zero to one. The p/p' partial mean ratio is represented by the
following formula:
WUPM (x; p) − WLPM (x; p′ ) W (x; p′ )
IPMR (x; p / p′ ) = = 1 − LPM . (2.17)
WUPM (x; p) WUPM (x; p)
The higher the partial mean ratio, the higher the level of inequality
across two percentiles of a society’s population. A partial mean ratio is
zero when both upper and lower partial mean incomes are equal. A quan-
tile ratio reaches its maximum value of one when the lower partial mean
income WLPM(x; p') is zero and the upper partial mean income is positive.
Note that if all people in the society have equal incomes, then any partial
mean ratio is zero. However, a partial mean ratio of zero does not necessarily
imply that incomes are equally distributed across all people in the society.
The most well-known partial mean ratio was devised by Simon Kuznets
and is known as the Kuznets ratio. It is based on two income standards: the
mean of the poorest 20 percent of the population and the mean of the rich-
est 40 percent of the population. Using our formulation, the Kuznets ratio
equivalent inequality measure of distribution x is denoted by IPMR(x; 20/40).
How should the number IPMR(x; 20/40) = 0.8 be interpreted? Again, there
are several ways to interpret this measure:
• The difference in mean income between the richest 20 percent of

the population and the poorest 40 percent of the population is
80 percent of the mean income of the richest 20 percent of the
population.
• The mean income of the poorest 40 percent of the population is
(1 − 0.8) = 0.2 or 20 percent or one-fifth of the mean income of the
richest 20 percent of the population.
• The mean income of the richest 20 percent of the population is
1/(1 − 0.8) = 5 times larger than the mean income of the poorest
40 percent of the population.
90
What properties does a partial mean ratio satisfy? A partial mean ratio,
as defined in equation (2.17), satisfies symmetry, normalization, population
invariance, and scale invariance. Thus, a partial mean ratio satisfies all four
invariance properties. What about the dominance properties? A quantile
ratio satisfies the weak transfer principle but does not satisfy the transfer
principle, transfer sensitivity, and subgroup consistency. It does not satisfy
the transfer principle because some regressive and progressive transfers may
leave the inequality measure unchanged, since a partial mean ratio does not
consider the entire income distribution.
Atkinson’s Class of Inequality Measures
Atkinson’s class of inequality measures, developed by Sir Anthony Atkinson,

is based on general means (see Atkinson 1970). All inequality measures in
this family are constructed by comparing the arithmetic mean and another
income standard from the family of general means. Recall that each mea-
sure’s formulation in the general means family depends on a parameter
denoted by a, which can take any value between − ∞ and + ∞.
In the Atkinson family of inequality measures, a is called the inequality
aversion parameter. The lower the value of a, the higher a society’s aver-
sion toward inequality. In other words, the more averse a society is toward
inequality across the population, the more emphasis it gives to lower
incomes in the distribution by choosing a lower value of a. The Atkinson
class of inequality measures for a < 1 may be expressed as
WA (x) − WGM (x; a) W (x; a)

IA (x; a) = = 1 − GM . (2.18)
WA (x) WA (x)
The Atkinson index of order a is the difference between the arithmetic

mean and the general mean of order a divided by the arithmetic mean. Any
Atkinson index lies between zero and one, and inequality increases as the
index moves from zero to one. The minimum level of inequality, zero, is
obtained when the total income is equally distributed across everyone in the
society. Unlike the quantile ratios and the partial mean ratios, if IA(x; a) = 0
for any a < 1, then, by implication, the total income in the society is equally
distributed. This is because any inequality measure in this family is con-
structed by considering the entire distribution.
91
We already know from our discussion of income standards that the value
of general means falls as a falls and vice versa. As a decreases, the distance
between WA(x) and WGM(x; a) increases, implying that IA increases as a
falls for a particular income distribution. Among the entire class of mea-
sures, three are used more frequently: a = 0, a = –1, and a = –2. For a = 0,
the general mean takes the form of the geometric mean. The corresponding
Atkinson’s inequality measure for distribution x is expressed as
WA (x) − WG (x) W (x)

IA (x; 0) = = 1− G . (2.19)
WA (x) WA (x)
For a = –1, the general mean is known as the harmonic mean. The cor-
responding Atkinson’s inequality measure for distribution x is expressed as
WA (x) − WH (x) W (x)

IA (x; −1) = = 1− H . (2.20)
WA (x) WA (x)
For a = –2, the general mean has no such name, and we will call it
simply WGM(X; –2). The corresponding Atkinson’s inequality measure for
distribution x is expressed as
WA (x) − WGM (x; −2) W (x; −2)

IA (x; −2) = = 1 − GM . (2.21)
WA (x) WA (x)
Following the relationship between the Atkinson’s class of inequality

measures and parameter a, we can state that IA(x; −2) < IA(x; −1) < IA(x; 0)
unless all incomes in distribution x are equal (see example 2.6).
Example 2.6: Consider the income vector x = ($2k, $4k, $8k, $10k)
used previously in the general means example. The arithmetic mean
is WA(x) = $6k, the geometric mean is WG(x) = $5.03k, the harmonic
mean is WH(x) = $4.10k, and WGM(x; –2) = $3.44k.
Thus,
IA(x; 0) = ($6k − $5.03k)/$6k = 0.162.
IA(x; −1) = ($6k − $4.10k)/$6k = 0.317.
IA(x; −2) = ($6k − $3.44k)/$6k = 0.427.
What is the interpretation of the number IA(x; 0) = 0.162? First, note

that IA(x; 0) is based on two income standards: the arithmetic mean of x
92
and the geometric mean of x. The arithmetic mean represents the level
of welfare obtained when the overall income is distributed equally across
everyone in the society. This is an ideal situation when there is no inequality
in the society.
The geometric mean, in contrast, is the equally distributed equivalent
(ede) income, which, if received by everyone in the society, would yield
the same welfare level as in x for the degree of inequality aversion a = 0. So
IA(x; 0) = 0.162 implies that the loss of welfare because of inequality in dis-
tribution x is 16.2 percent of what the welfare level would be if the overall
income had been equally distributed.
Suppose the society becomes more averse to inequality and a is reduced
to −1. In this case, the equally distributed equivalent income is the har-
monic mean of x. The loss of total welfare because of unequal distribution
increases from 16.2 percent to 31.7 percent. Likewise, the percentage loss
of welfare would increase to 42.7 percent if the society became even more
averse to inequality and a fell to −2.
What properties does any index in this family satisfy? Any measure in
this family satisfies all four invariance properties: symmetry, population invari-
ance, scale invariance, and normalization. In addition, unlike the quantile
ratios and the partial mean ratios, measures in this class satisfy the transfer
principle, transfer sensitivity, and subgroup consistency.
If distribution x' is obtained from distribution x by at least one regres-
sive transfer, then the level of inequality in x' is strictly higher than that
in x. Furthermore, if transfers take place between poor people, then the
inequality measure changes more than if the same amounts of transfers take
place among rich people. Finally, because these measures satisfy subgroup
consistency, they do not lead to any inconsistent results while decomposing
across subgroups. If inequality in certain subgroups increases while inequal-
ity in the others does not fall, then overall inequality increases. However,
measures in this class are not additively decomposable.
Gini Coefficient
The Gini coefficient, developed by Italian statistician Corrado Gini (1912),

is the most commonly used inequality measure. It measures the average dif-
ference between pairs of incomes in a distribution, relative to the distribu-
tion’s mean. The most common formulation of the Gini coefficient for the
distribution x is
93
N N
1
IGini (x) = ∑ ∑ x n − x n′ .
2N 2 × WA (x) n =1 n’=1
(2.22)
Note that equation (2.22) may be broken into two components: WA(x)
(the mean of the distribution) and (∑N n=1∑ n'=1|xn – xn'|)/2N (the average
N 2
difference between pairs of incomes). The second component is divided

by its number of elements, 2N2. There are 2N2 elements because each ele-
ment in x is compared with another element in x including itself twice.
This original Gini coefficient formula can be simplified further. The second
component of the Gini coefficient can be written as
1 N N 1 N N
∑ ∑ n n′ A
2 N 2 n =1 n ′ =1
x − x = W (x) − ∑ ∑ min {x n , x n′}= WA (x) − Ws (x), (2.23)
N 2 n = 1 n ′= 1
where WS(x) is the Sen mean of distribution x. Therefore, the Gini coefficient
may be simply formulated by using the arithmetic mean and the Sen mean.
Like any measure in Atkinson’s class, the Gini coefficient can be expressed as
WA (x) − WS (x) W (x)

IGini (x) = = 1− S . (2.24)
WA (x) WA (x)
Thus, the Gini coefficient is the difference between the arithmetic

mean and the Sen mean divided by the arithmetic mean. The coef-
ficient lies between zero and one, and inequality increases as the index
moves from zero to one. The minimum inequality level, zero, is obtained
when income is equally distributed across everyone in the society.
Like Atkinson’s measures, if IGini(x) = 0, then, by implication, income in
the society is equally distributed. Again, this is because any inequality
measure in this family is constructed by considering the entire distribution
(see example 2.7).
What is the interpretation of IGini(x) = 0.292? First, IGini(x) is based on
two income standards: the arithmetic mean of x and the Sen mean of x.
The arithmetic mean represents the level of welfare obtained when the
overall income is distributed equally across all people in the society. This is
an ideal situation when there is no inequality in the society. The Sen mean,
in contrast, is an ede income, which, if received by everyone in the society,
would yield the same welfare level as in x. So IGini(x) = 0.292 implies that
the loss of welfare because of inequality in distribution x is 29.2 percent of
94
that we used previously. First, we calculate the Gini coefficient using
the formulation in equation (2.22). It can be easily verified that
WA(x) = $6k. The second component is
1
( 2 − 2 + 2 − 4 + 2 − 8 + 2 − 10 + 4 − 2 + 4 − 4 + 4 − 8 + 4 − 10
2 × 42
+ 8 − 2 + 8 − 4 + 8 − 8 + 8 − 10 + 10 − 2 + 10 − 4 + 10 − 8 + 10 − 10 )
1
= (0 + 2 + 4 + 6 + 8 + 2 + 0 + 4 + 6 + 6 + 4 + 0 + 2 + 8 + 6 + 2 + 0)
32
56
= = 1.75.
32
Thus, IGini(x) = 1.75/6 = 0.292.

Next, we calculate the Gini coefficient using equation (2.24). The
Sen mean of distribution x is WS(x) = $4.25k. Thus, IGini(x) = ($6k −
$4.25k)/$6k = 1.75/6 = 0.292.
the welfare level if overall income had been equally distributed. We will see
later that the Gini coefficient has an interesting relationship with the well-
known Lorenz curve.
The Gini coefficient satisfies all invariance properties: symmetry, population
invariance, scale invariance, and normalization. In addition, it satisfies the transfer
principle. If distribution x' is obtained from distribution x by at least one regres-
sive transfer, then the level of inequality in x' is strictly higher than that in x.
However, the Gini coefficient is neither transfer sensitive nor subgroup con-
sistent. It is not transfer sensitive because the Gini coefficient changes by the
same amount whether transfers take place between poor people or between
rich people. That the Gini coefficient is not subgroup consistent means that if
the inequality in some subgroups increases while inequality in other subgroups
does not fall, then the overall inequality may register a decrease.
The following is an example showing that the Gini coefficient is neither
transfer sensitive nor subgroup consistent. Consider the vector x = ($4k,
$5k, $6k, $7k, $14k, $16k). If a progressive transfer of $0.5k takes place
between the first person and the second person, then x' = ($4.5k, $4.5k, $6k,
$7k, $14k, $16k). If a progressive transfer of the same amount takes place
95
between the two richer people, then x" = ($4k, $5k, $6.5k, $6.5k, $14k,
$16k). As a result, IGini(x') = IGini(x") = 0.279. Thus, the Gini coefficient
cannot distinguish between these two transfers.
The next example shows that the Gini coefficient is not subgroup consis-
tent. We use the same example that we used to show that the Sen mean does
not satisfy subgroup consistency. The original income vector x = ($4k, $5k,
$6k, $7k, $14k, $16k) becomes, over time, y = ($3.4k, $6.1k, $6k, $6.5k,
$14k, $16k). The income vector of the first subgroup x' = ($4k, $5k, $7k)
becomes y' = ($3.4k, $6.1k, $6.5k), whereas the income vector of the sec-
ond subgroup remains unaltered. The Sen mean of the first group falls from
WS(x') = $4.67k to WS(y') = $4.64k, whereas the mean income remains
unchanged at WA(x') = WA(y') = $5.33k. So the inequality of the first
group increases from IGini(x') = 0.125 to IGini(y') = 0.129. What happens
to the overall inequality? It turns out that the overall Sen mean increases
from WS(x) = $6.22k to WS(y) = $6.24k, whereas the overall mean income
remains unchanged at WA(x) = WA(y) = $8.67k. The overall inequality
decreases from IGini(x) = 0.282 to IGini(y) = 0.280.
However, unlike the Atkinson class of measures, the Gini coefficient is
additively decomposable, but with an added residual term. If distribution x is
divided into population subgroups x' with population size N' and x" with
population size N", then the decomposition formula of the Gini coefficient is
IGini(x) = w'IGini(x')+w" IGini(x") + IGini(x–', x– ") – residual, (2.25)
where the weights are w' = (N'/N)2(x̄'/x̄) and w" = (N"/N)2(x̄"/x̄). Note,
however, that the weights may not sum to one. The residual term is not zero
if and only if the groups’ income ranges overlap. If we consider the example
above, where the income vector x = ($4k, $5k, $6k, $7k, $14k, $16k) is
divided into two subgroup vectors: x' = ($4k, $5k, $7k) and x" = ($6k, $14k,
$16k). These vectors overlap as $7k > $6k. Thus, the residual term will
not vanish. However, if the two subgroups were x' = ($4k, $5k, $6k) and
x" = ($7k, $14k, $16k), then the residual term would be zero.9
Generalized Entropy Measures
The final inequality measures we consider are in the class of generalized

entropy measures. Two well-known Theil measures are also in this class. The
common formula for the generalized entropy measures of order a for any
distribution x is
96
⎧ 1 ⎡1 N ⎛ xn ⎞
a
⎤
⎪ ⎢ ∑ n =1 ⎜ ⎟ − 1⎥ if a ≠ 0,1
⎪ a(a − 1) ⎢⎣ N ⎝ x⎠ ⎦⎥
⎪
⎪1 N x ⎛x ⎞
IGE (x; a) = ⎨ ∑ n =1 n ln ⎜ n ⎟ if a = 1. (2.26)
⎪N x ⎝ x⎠
⎪1 ⎛ x⎞
⎪ ∑ n =1 ln ⎜ ⎟
N
if a = 0
⎪⎩ N ⎝ xn ⎠
At first glance, the formula above looks complicated. However, measures

in this class are closely related to general means. Every index in this class,
except one, can be expressed as a function of the arithmetic mean and the
general mean of order a. For a ≠ 0, 1, the class of generalized entropy mea-
sures can be written as
1 ⎛ ⎢⎣WGM (x; a ')⎥⎦a − ⎢⎣WA (x)⎥⎦a ⎞
IGE (x; a ) = ⎜ a ⎟, (2.27)
a (a − 1) ⎝ [ W (x)] ⎠
A
where we replace the term x̄ by WA(x) (the arithmetic mean), and where
WGM(x; a) denotes the general mean of order a. Thus, a generalized
entropy measure for any a ≠ 0,1 may be easily calculated once we know the
arithmetic mean and the general mean of order a.
For a = 1, the generalized entropy is Theil’s first measure of inequality
and can be written as
1 N xn ⎛ xn ⎞
IT1 (x) = ∑ ln ⎜
N n =1 WA (x) ⎝ WA (x) ⎟⎠
. (2.28)
This is the only measure in this class that cannot be expressed as a function
of general means and does not have a natural twin-standards representation.
For a = 0, the generalized entropy index is Theil’s second measure of
inequality, which is also known as the mean logarithmic deviation and can be
expressed as a function of the arithmetic mean, WA(x), and the geometric
mean, WG(x), as follows:
WA (x)
IT 2 (x) = ln WA (x) − ln WG (x) = ln . (2.29)
WG (x)
Besides the two Theil measures, the other commonly used measure in
the entropy class is the index for a = 2, which is closely related to the coef-
ficient of variation (CV). The CV is the ratio of the standard deviation and
97
mean. For a = 2, the general entropy measure is half the CV squared and
can be expressed as
2 2
1 ⎢⎣WE (x)⎢⎣ − ⎢⎣WA (x)⎢⎣ 1 Var(x) CV 2
IGE (x; 2) = 2
= 2
= , (2.30)
2 2 2
⎣⎢ WA (x)⎣⎢ ⎣⎢ WA (x)⎢⎣
where Var(x) is the variance of the distribution x, which is the square
of its standard deviation. In equation (2.29), WE(x) is the Euclidean
1
mean, as in equation (2.8) and [WE (x)]2 = ∑ n =1 x 2n . Clearly,
N
1 N
⎢⎣WE (x)⎢⎣ − ⎢⎣WA (x)⎢⎣ = ∑ n =1 x n − x is the variance of x (see example 2.8).
2 2 N 2 2
that we used in the general means example. The arithmetic mean
is WA(x) = $6k, the geometric mean is WG(x) = $5.03k, and the
Euclidean mean is WE(x) = $6.78k.
We now calculate the two Theil inequality measures and the
squared coefficient of variation:
IGE(x; 2) = ([WE(x)]2 − [WA(x)]2)/(2[WA(x)2] = (6.782 − 62)/(2 × 62)
= 0.279.
IT2(x) = ln[WA(x)/WG(x)] = ln [$6k/$5.03k] = 0.176.
The calculation of Theil’s first measure is not as straightforward
as that of the previous two measures. However, it can be calculated
using the following steps. First, create a new vector from vector x by
dividing every element by the mean of x as (2/6, 4/6, 8/6, 10/6). Then
1 ⎡ 2 ⎛ 2 ⎞ 4 ⎛ 4 ⎞ 8 ⎛ 8 ⎞ 10 ⎛ 10 ⎞ ⎤
IT1 (x) = ln ⎜ ⎟ + ln ⎜ ⎟ + ln ⎜ ⎟ + ln ⎜ ⎟ = 0.15.
4 ⎢⎣ 6 ⎝ 6 ⎠ 6 ⎝ 6 ⎠ 6 ⎝ 6 ⎠ 6 ⎝ 6 ⎠ ⎥⎦ .
Having introduced the measures in the generalized entropy class, now

we try to understand their behavior. First, what is the range of any measure
in this class? The lower bound of any measure in this class is zero, which is
obtained when incomes in a society are equally distributed across all people.
However, unlike the Atkinson’s measures and the Gini coefficient, general-
ized entropy measures may not necessarily be bounded above by one.
Next, how do the measures in this class relate to the parameter? There
are, in fact, three distinct ranges: a lower range a < 1, an upper range a > 1,
98
and a limiting case where a = 1. For the lower range, a < 1, measures in this
class are monotonic transformations of the Atkinson’s class of measures and
can be written as
⎧ ⎢⎣1 − IA (x ;a − 1)⎥⎦a
⎪ if a ≠ 0, a < 1
⎪ a (a − 1)
IGE (x; a ) = ⎨ , (2.31)
⎪ ln 1
if a = 0
⎪⎩ 1 − IA (x; 0)
where IA(x; a) is the Atkinson’s inequality measure for parameter a.

For the range a < 1, the entropy measures behave the same way as the
Atkinson’s measures. Over the range a > 1, the general mean places
greater weight on higher incomes and yields a representative income that
is typically higher than the mean income. An example is the squared coef-
ficient of variation.
All measures in the generalized entropy class satisfy the invariance
properties: symmetry, normalization, population invariance, and scale invari-
ance. Furthermore, they all satisfy the transfer principle and subgroup con-
sistency. However, transfer sensitivity is satisfied only by the measures in
this class with a < 2. Measure IGE(x; 2) is, in fact, transfer neutral like
the Gini coefficient. It turns out that the generalized entropy measures
are the only inequality measures that satisfy the usual form of additive
decomposability (see Shorrocks 1980). If distribution x is divided into
two population subgroups, x' with population size N' and x" with popula-
tion size N", then the decomposition formula of the generalized entropy
measure for a ≠ 0,1 is
IGE(x; a) = w'IGE(x'; a) + w"IGE(x"; a) + IGE(x̄', x̄"; a), (2.32)
where the weights are w' = (N'/N)(x̄'/x̄)a and w" = (N"/N)(x̄"/x̄)a. For
example, when a = 2, the weights are w' = (N'/N)(x̄'/x̄)2 and w" = (N"/N)
(x̄"/x̄)2.
Note that the weights may not always sum to one. However, for the two
Theil measures, the weights do sum to one. The first Theil measure can be
decomposed as
IT1(x) = w'IT1(x') + w"IT1(x") + IT1(x̄', x̄"), (2.33)
where the weights are w' = x̄'/x̄ and w" = x̄"/x̄. Although it is difficult to
get an intuitive interpretation of the first Theil measure, the additive
99
decomposability property makes the first Theil measure useful in under-

standing within-group and between-group inequalities. The second Theil
measure can be decomposed as
IT2(x) = w'IT2(x') + w"IT2(x") + IT2(x̄', x̄"), (2.34)
where the weights are w' = (N'/N) and w" = (N"/N).
Inequality and Welfare
The Gini coefficient and the inequality measures in Atkinson’s family share
a social welfare interpretation. As we have already discussed, they can be
expressed as I = (x̄ − a)/x̄, where x̄ is the mean income of the distribution
x and a is an income standard that can be viewed as a welfare function
(satisfying the weak transfer principle). Note that the distribution in which
everyone has the mean income has the highest level of welfare among all
distributions with the same total income, and the distribution’s measured
welfare level is just the mean itself. This finding results from the normaliza-
tion property of income standards.
Thus, the mean WA(x) = x̄ is the maximum value that the welfare func-
tion can take over all income distributions of the same total income. When
incomes are all equal, a = WA(x) and inequality is zero. When the actual
welfare level a falls below the maximum welfare level WA(x), the percentage
welfare loss I = (WA(x) − a)/WA(x) is used as a measure of inequality. This
is the welfare interpretation of both the Gini coefficient and the Atkinson’s
class of measures.
The simple structure of these measures allows us to express the welfare
function in terms of the mean income and the inequality measure. A quick
rearrangement leads to a = WA(x)(1 – I), which can be reinterpreted as
saying that the welfare function a can be viewed as an inequality-adjusted
mean. If there is no inequality in the distribution, then (1 – I) = 1 and
a = WA(x). If the inequality level is I > 0, then the welfare level is obtained
by discounting the mean income by (1 – I) < 0.
For example, if we take I to be the Gini coefficient, IGini(x), then the Sen
mean (or Sen welfare function) can be obtained by multiplying the mean by
[1 – IGini(x)], that is, WS(x) = WA(x)[1 – IGini(x)]. Similarly, if we take I to
be the Atkinson’s measure with parameter a = 0, IA(x; 0), then the welfare
function is the geometric mean, and the geometric mean can be obtained by
multiplying the mean by [1 – IA(x; 0)], that is, WG(x) = WA(x)[1 – IA(x; 0)].
100
An inequality measure estimates, with a single number, the inequality level

in a society. A question may naturally arise: Do all inequality measures
compare two income distributions in the same way? In other words, if an
inequality measure evaluates income distribution x to be more equal than
distribution y, would another inequality measure evaluate distributions x
and y in the same way? The answer depends on the two inequality measures
we use for evaluation. Not all inequality measures evaluate various distribu-
tions in the same manner.
We can clarify this concern with an example. Consider the two income
vectors x = ($4k, $5k, $6k, $7k, $14k, $16k) and y = ($3.4k, $6.1k, $6k,
$6.5k, $14k, $16k). These two vectors have the same mean. The Gini coef-
ficient indicates that the inequality level in x is 0.282, which is higher than
the inequality in y (0.280). However, the Atkinson’s measure that is based
on the geometric mean shows that the inequality level in x is 0.127, which is
lower than the level of inequality in y (0.132). Therefore, different inequal-
ity measures may disagree in different situations.
Is there any condition in which different inequality measures agree with
each other? The answer is yes. Inequality measures that satisfy the four
basic properties—symmetry, population invariance, scale invariance, and
the weak transfer principle—agree with each other when Lorenz dominance
holds between two distributions. To understand Lorenz dominance, we need
to understand the Lorenz curve.
The Lorenz curve of an income distribution shows the proportion of total
income held by the poorest p percent of the population.10 We denote the
Lorenz curve of distribution x by Lx. Then Lx(p) is the share of total income
held by the poorest p percent of the population. Indeed, Lx(100) = 100
percent and Lx(0) = 0 percent. Suppose the total income of Nigeria is N25
trillion and only N1 trillion is received by the poorest 20 percent of the
population. Then LNig(20) = 4 percent. Suppose that income in Nigeria
is redistributed, keeping the total income unaltered, so that everyone has
identical income. Let us denote the equal income distribution by y. Then
the percentage of total income enjoyed by the poorest 20 percent of the
population is 20 percent, and Ly(20) = 20 percent.
In figure 2.13, the horizontal axis denotes the cumulative share of the
population (p), and the vertical axis shows the share of total income.
Note that the lowest and the highest values for both axes are 0 and 100,
101
Figure 2.13: Lorenz Curve
Lx(100) 100
Ly Lx ′
Income share
Lx
Ly(20) A 20
Lx ′(20) B 14
Lx(20) C 4
0
0 20 100
Population share
respectively. For income distribution x, Lx represents its Lorenz curve,

denoted by the dotted curve. Following the example of Nigeria, Lx(20) = 4
percent, which is the height of the curve Lx at point C.
If distribution y is obtained from distribution x by distributing income
equally across the population, then the Lorenz curve becomes a 45-degree
straight line, Ly (the solid line in figure 2.13). In this case, the share of the
population’s bottom 20 percent in distribution y is Ly(20) = 20 percent. This
is obtained at point A on Lorenz curve Ly.
Now, suppose the income distribution in Nigeria improves over time
and the new income distribution is denoted by x'. The Lorenz curve for x' is
denoted by the dashed curve Lx' in figure 2.13. The share of the bottom 20
percent in the total income increases from 4 percent to 14 percent. This is
shown at point B on the Lorenz curve Lx'.
Notice that every portion of Lorenz curve Lx' lies above that of Lorenz
curve Lx. This is what we mean by Lorenz dominance: the income share
of every cumulative population share in x' is higher than that in x. Thus,
distribution x' Lorenz dominates distribution x'. Similarly, distribution x
Lorenz dominates both distributions x and x'.
Any inequality measure satisfying the four basic properties—symmetry,
population invariance, scale invariance, and the weak transfer principle—
would evaluate distribution y as more equal than distributions x and x' and
distribution x' as more equal than distribution x. Thus, before comparing
distributions using different inequality measures, the distributions’ Lorenz
102
curves should be compared. If one distribution’s Lorenz curve dominates

that of another distribution, then all inequality measures satisfying these
four basic properties would consider the former distribution to be more equal
than the latter.
Well-known inequality measures satisfying these four basic properties are
the Gini coefficient, measures in the Atkinson’s family, measures in the gen-
eralized entropy family, and partial mean ratios. What happens when two
Lorenz curves cross? In this situation, Lorenz dominance does not hold, and
the inequality level needs to be judged using inequality measures when dif-
ferent inequality measures may agree or disagree with each other. However,
even in this case, the Lorenz curve can be helpful in identifying the winning
and losing portions of the distribution.
The Lorenz curve also has interesting relationships with income stan-
dards and inequality measures. First, consider its relationship with the
generalized Lorenz curve. A Lorenz curve may be obtained from a general-
ized Lorenz curve by dividing the latter by the mean. Thus, for distribu-
tion x, Lx(p) = GLx(p)/WA(x). The construction of a Lorenz curve can be
easily understood by following the construction of the generalized Lorenz
curve in figure 2.8. Next, recall that the height of the generalized Lorenz
curve at a certain percentile of population p is the lower partial mean
times p itself, that is, GLx(p) = p × WLPM(x; p). Therefore, the height
of the Lorenz curve at a certain percentile of population p is the ratio of
the lower partial mean to the overall mean times p itself, that is, Lx(p) = p ×
[WLPM(x; p)/WA(x)]. Note that the ratio of the lower partial mean to the
overall mean itself may be used to construct a partial mean ratio, denoted
by IPMR(x; 100/p).
Finally, an interesting relationship exists between the Lorenz curve and
the Gini coefficient. The Gini coefficient of distribution x is twice the area
between the Lorenz curves Lx and Ly in figure 2.13. Similarly, the Gini coeffi-
cient for distribution x" is twice the area between the Lorenz curves Lx' and Ly.
Inequality and Growth
The twin-standard view of inequality offers fresh insights into the relation-
ship between growth and inequality. Almost all inequality measures are
constructed in terms of a larger income standard b and a smaller income
standard a, and these income standards are expressed as 1 – a/b or b/a – 1.
Suppose income standard a changes to a' over time with growth rate g– a,
103
that is, a' = (1 + g– a)a, and income standard b changes to b' over time with
growth rate g– b, that is, b' = (1 + g– b)b. The inequality measure then changes
from I = 1 – a/b to I' = 1 – a'/b'. To have a fall in inequality, we require I' <
I or 1 – a'/b' < 1 – a/b, which occurs when g– a > g– b. Therefore, for a reduction
in inequality, the smaller income standard a needs to grow faster than the
larger income standard b.
Consider the example of the Gini coefficient, which is constructed from
two income standards. The larger income standard is the arithmetic mean
WA, and the smaller income standard is the Sen mean WS. Let us denote
the growth rate of the mean income by g– and the growth rate of the Sen
mean by g–S. The Gini coefficient will register a fall in inequality when the
growth rate of the Sen mean is larger than the growth rate of the arithmetic
mean, that is, g–S > g–. Similarly, inequality over time, in terms of the Gini
coefficient, increases when g–S < g–.
What about the Atkinson’s measures and the generalized entropy mea-
sures? Measures in these classes, including Theil’s second measure, are based
on the arithmetic mean and on any income standard from the class of gen-
eral means. For a < 1, the arithmetic mean is the larger income standard,
and the other general mean–based income standard is the smaller income
standard. In this case, if the growth rate of the smaller income standard of
order a is denoted by g–GM(a), then inequality decreases when g–GM(a) > g–.
If inequality is evaluated by Theil’s second index, then inequality falls when
the growth of geometric mean g–GM(0) is larger than that of the arithmetic
mean, that is, g–GM(0) > g–. For a > 1 in the generalized entropy measure,
the arithmetic mean is the smaller income standard, and the other general
mean–based income standard is the larger one. Inequality falls, according
to these indices, when the growth rate of the arithmetic mean g– is higher.
Is there any way to tell if all inequality measures in the Atkinson family
and the generalized entropy family have fallen? Yes, it is possible to do so
just by looking at the general mean growth curve, as described in figure 2.12.
A generalized mean growth curve is the loci of the growth rates of all
income standards in the class of general means. Comparing distributions x
and y for the general mean growth curve gGM(x,y) in figure 2.12 shows that
all inequality measures in Atkinson’s class and the generalized entropy class
agree that the inequality has fallen because the growth rates of the lower
income standards are higher than ḡ. The growth rates of the larger income
standards are lower than ḡ. However, for the general mean growth curve
gGM(x',y') in the same figure, all inequality measures in Atkinson’s class and
104
the generalized entropy class agree that the inequality has risen because the
growth rates of the lower income standards are lower than ḡ, whereas the
growth rates of the larger income standards are higher than ḡ.
In a similar manner, the growth incidence curve may be used to under-
stand the change in inequality using quantile ratios. If the growth rate of
the upper quantile income is larger than the growth rate of a lower quantile
income, then inequality has risen over time. In contrast, if the growth rate
of a lower quantile income is larger than the growth rate of the higher quan-
tile income, then inequality has fallen. For example, consider the growth
incidence curve gQI(x,y) in figure 2.10. If inequality is measured by the
90/10 measure IQR(x; 90/10), then inequality has fallen. Furthermore, for
growth incidence curve gQI(x',y'), the level of inequality has increased for
the same inequality measure.
Poverty Measures
The third aspect of a distribution is base, which is evaluated using a numeri-

cal poverty measure, assigning each distribution a number reflecting its
level of deprivation. In this section, before proceeding further, we introduce
additional notations that are more specific to poverty measures than income
standards and inequality measures. The income distribution of society X
with N people can be summarized by the vector x = (x1,x2, …, xN), where
xn is the income of person n. We also assume that the income distribution is
ordered, that is, x1 ≤ x2 ≤ … xN.
Any poverty measure is constructed in two steps. The first step is iden-
tification, where each person is identified as poor or nonpoor by using a
threshold called the poverty line, denoted by z. More specifically, a person
is identified as poor if his or her income falls below the poverty line z and
nonpoor if his or her income is greater than or equal to z. We denote the
number of poor in our reference society X by q. So the number of nonpoor
is N − q. Because elements in income distribution x are ordered, people 1,…,
q are poor and people q + 1, …, N are nonpoor.
Suppose society X consists of four people with the income vector x = ($1k,
$2k, $50k, $70k). If the poverty line is set at $10k, this means that a person
must have $10k to meet the minimum necessities to lead a healthy life. This
requirement would identify the first two people as poor with earnings $1k
and $2k, whereas the third person and the fourth person are identified as
105
nonpoor. In this example, society X has two poor people and two nonpoor
people. We summarize the incomes of the poor in vector x by the vector xq.
Poverty analysis is concerned only with the poor or the distribution’s
base, which should be the group targeted for public assistance. It naturally
ignores the incomes of nonpoor people in a society. In this way, the identifi-
cation step allows us to construct a censored distribution or censored vector
of incomes for society X, which we denote by x* = (x*1,x*2, …,x*N) such that
x*n = xn if income xn is less than the poverty line z and xn* = z if income xn is
greater than or equal to the poverty line z.
For the four-person income vector x = ($1k, $2k, $50k, $70k) in the
previous example, the censored vector is denoted by x* = ($1k, $2k, $10k,
$10k). Notice that incomes of the two nonpoor people are replaced by
the poverty line, and portions of their income above the poverty line are
ignored. A policy maker’s objective should be to include poor people at or
above the poverty line. Including all poor people at or above the poverty
line results in a nonpoverty censored distribution of income. We denote the
nonpoverty censored distribution of society X corresponding to poverty line
z by x– z* such that x– z* = (z,z,…,z).
The second step for constructing a poverty measure is aggregation. In this
step, incomes of individuals who are identified as poor using the poverty
line in the identification stage are aggregated to obtain a poverty measure.
Therefore, a poverty measure depends on both the incomes of the poor and
the criterion that is used for identifying the poor—that is, the poverty line.
In fact, it turns out that any poverty measure is obtained by aggregating ele-
ments in the censored distribution x∗.
In this section, we denote a poverty measure by P, where specific indi-
ces are denoted using corresponding subscripts. We denote the poverty
measure of distribution x for poverty line z by P(x; z). Alternatively, it may
be denoted by P(x∗). There are two different ways to understand a poverty
measure: one is based on the properties it satisfies and the other is through
its link with income standards. First, we discuss the properties that a poverty
measure should satisfy.
A useful poverty measure should satisfy some desirable properties. Like

income standards and inequality measures, poverty measure properties can
fall into two categories:
106
• Invariance properties leave poverty measures invariant to certain

changes in the dataset. Properties in the invariance category are sym-
metry, normalization, population invariance, scale invariance, and focus.
• Dominance properties cause a poverty measure to change in a particu-
lar direction. Properties in the dominance category are monotonicity,
transfer principle, transfer sensitivity, and subgroup consistency.
Six of these properties—symmetry, population invariance, scale invariance,

focus, monotonicity, and transfer principle—are called basic properties. Many
of these properties are analogous to the corresponding properties of income
standards and inequality measures.11
The first invariance property, symmetry, requires that switching the
income levels of two people while the poverty line remains the same leaves
poverty unchanged. In other words, a person should not be given priority on
the basis of his or her identity when evaluating the level of poverty within
a society. Formally, it requires that the poverty measure of distribution x be
equal to the poverty measure of another distribution x', if x' is obtained from
x by a permutation of incomes without changing the poverty line.
For example, recall the four-person income vector ($1k, $2k, $50k,
$70k). If the poverty line is z = $10k, then the first two people are poor and
the last two people are nonpoor. Now, if the income of the first and the
fourth individuals are switched, the new income vector becomes x' = ($70k,
$2k, $50k, $1k). This new vector x' is said to be obtained from vector x by
a permutation of incomes.
Symmetry: If distribution x' is obtained from distribution x by

permutation of incomes and the poverty line z remains the same,
then P(x'; z) = P(x; z).
The second invariance property, normalization, requires that the poverty

measure be zero if no one’s income in the society is less than the poverty
line. This is a natural property. For example, if the income vector of the
four-person society is ($1k, $2k, $50k, $70k), but the poverty line in this
case is $1k, then any poverty measure should be 0, reflecting that there are
no poor in the society.
Normalization: For any income distribution x and poverty line z, if

min{x} ≥ z, then P(x; z) = 0.
107
The third invariance property, population invariance, requires that pov-

erty be invariant to the population size, in the sense that a replication of
an income vector results in the same level of poverty as the original sample
vector if the poverty line does not change. The implication of this property
is as follows. Consider the income vector of society X, x = ($1k, $2k, $50k,
$70k). Suppose four more people with the same income distribution join the
society so that the new income vector is x' = ($1k, $1k, $2k, $2k, $50k, $50k,
$70k, $70k). The population invariance property requires that the poverty
level in society x remains unaltered, at least if the poverty line does not
change. This allows us to compare the extent of poverty across countries and
across time with varying population sizes. Furthermore, this property allows
any poverty measure to depend on a distribution function, which normalizes
the population size to one.
Population Invariance: If vector x' is obtained by replicating

vector x at least once and the poverty line remains unaltered, then
P(x'; z) = P(x; z).
The fourth invariance property, scale invariance, requires that if an

income distribution is obtained from another income distribution by
scaling all incomes and the poverty line by the same factor, then the pov-
erty level should remain unchanged. For example, if everyone’s income
and the poverty line in a society are tripled or halved, then the level of
deprivation of the society does not change. The scale invariance prop-
erty ensures that the measure is independent of the unit of measurement
for income. Consider the following example, where the income of each
person in vector x = ($1k, $2k, $50k, $70k) increases by three times and
becomes x' = ($3k, $6k, $150k, $210k) over time. If the poverty line also
increases from, say, $6k to $18k, then the level of deprivation should not
change over time.12
Scale Invariance: If distribution x' is obtained from distribution x

such that x' = cx and z' = cz where c > 0, then P(x'; cz) = P(x; z).
The fifth and final axiom in the invariance properties is focus, which
requires that if the income of a nonpoor person in a society changes but
does not fall below the poverty line, then the level of poverty should not
108
change. This property ensures that the measure focuses on the poor incomes
in evaluating poverty. In fact, focus ensures that the income distribution
is censored at the poverty line before evaluating a society’s poverty. For
example, suppose the initial income vector is x = ($1k, $2k, $50k, $70k)
and the poverty line income is $6k. Thus, the third person and the fourth
person are nonpoor. If the income of either the third or the fourth person
increases, but the poverty line remains unaltered at $6k, then the society’s
poverty level does not change.
Focus: If distribution x' is obtained from distribution x by increasing

the income of a nonpoor person while the poverty line remains the
same at z, then P(x'; z) = P(x; z).
The next group of properties are dominance properties. The first of these
properties requires that if the income of a poor person in a society increases,
then the poverty level should register a fall, or at least it should not increase.
There are two versions of this property. One is weak monotonicity, which
requires that poverty should not increase because of an increase in a poor
person’s income. The other is monotonicity, the stronger version, which
requires that poverty should fall if a poor person’s income in the society
increases.
These two properties are the same as the two corresponding properties
of income standards, except the ones introduced here are solely concerned
with incomes of the poor. For example, suppose the initial income vector
is x = ($1k, $2k, $50k, $70k) and the poverty line income is $6k so that
the first two people are identified as poor. If a new vector x' is obtained by
increasing the income of either the first or the second person, while the
poverty line remains unchanged, then according to the weak monotonicity
property, poverty should not be higher in x', and, according to the monoto-
nicity property, poverty should be lower in x'.
Weak Monotonicity: If distribution x' is obtained from distribution x

by increasing the income of a poor person while keeping the poverty
line unchanged at z, then P(x'; z) ≤ P(x; z).
Monotonicity: If distribution x' is obtained from distribution x by
increasing the income of a poor person while keeping the poverty
line unchanged at z, then P(x'; z) < (x; z).
109
The second dominance property is the transfer principle, which requires

that a regressive transfer between two poor people in a society increase pov-
erty and a progressive transfer between two poor people reduce poverty.13
(For definitions of regressive and progressive transfers, refer to the section
discussing the transfer principle for income standards.) Suppose the initial
income vector is x = ($1k, $2k, $50k, $70k) and the poverty line income
is $6k, so the first two people are poor. If a new vector x' is obtained by a
progressive transfer between the first and the second person such that x'=
($1.5k, $1.5k, $50k, $70k) and the poverty line is still fixed at $6k, then pov-
erty in x' should be lower. Note that the transfer principle property allows
the number of poor to change as a result of a regressive transfer because the
richer poor may become nonpoor because of a regressive transfer.14
Transfer Principle: If distribution x' is obtained from distribution x

by a regressive transfer between two poor people while the poverty
line is fixed at z, then P(x'; z) > P(x; z). If distribution x" is obtained
from another distribution x by a progressive transfer between two
poor people while the poverty line is fixed at z, then P(x"; z) < P(x; z).
As in inequality measurement, we also define a weaker version of trans-

fer principle in poverty measurement. It requires that a regressive transfer
between two people in a society not decrease poverty and a progressive
transfer between two people not increase poverty. Thus, the weaker prin-
ciple allows the possibility that the poverty level may remain unchanged
because of a progressive or a regressive transfer.
Weak Transfer Principle: If distribution x' is obtained from

distribution x by a regressive transfer between two poor people while
the poverty line is fixed at z, then P(x'; z) ≥ P(x; z). If distribution
x" is obtained from another distribution x by a progressive transfer
between two poor people while the poverty line is fixed at z, then
P(x"; z) ≤ P(x; z).
The transfer principle requires a poverty measure to decrease if the trans-

fer is progressive. However, it is not concerned with which part of the dis-
tribution the transfer is taking place. A same amount of transfer may take
place between two extremely poor people, who are further away from the
110
poverty line, or between two moderately poor people, who are much closer
to the poverty line.
Should the effect of transfer, no matter where it takes place, have
the same effect on the poverty level? We elaborate this situation with an
example. Consider the five-person income vector x = ($80, $100, $800,
$50, 000, $70,000). Let the poverty line be set at $1,050. Then the first four
people are identified as poor because their incomes are below the poverty
line. First, suppose $10 is transferred from the second person to the first per-
son. Then the post-transfer income vector is x' = ($90, $90, $800, $1,000,
$50,000, $70,000). Transferring 10 percent of the second person’s income
has increased the first person’s income by 12.5 percent.
Suppose, instead, that the same $10 transfer takes place between the
third and the fourth persons, who are also poor. The post-transfer income
vector is x'' = ($80, $100, $810, $990, $50,000, $70,000), where transfer-
ring 1 percent of the fourth person’s income increases the third person’s
income by 1.25 percent. This transfer makes hardly any difference in the
large pool of income of the two richer poor people. Therefore, one might
feel that a transfer of the same amount between two extreme poor and
two richer poor should not have the same effect on the society’s overall
poverty.
The third dominance property, transfer sensitivity, requires a poverty
measure to be more sensitive to a transfer between poor people at the lower
end of the income distribution of the poor. In other words, this property
requires that a poverty measure should change more when a transfer takes
place between two extremely poor people than between two richer poor
people. In terms of the example above, the level of deprivation should be
lower in x' than in x''.
Suppose the initial income distribution is x and distribution x" is obtained
from distribution x by a progressive (or regressive) transfer between two
extremely poor people. Suppose further that distribution x" is obtained from dis-
tribution x by a progressive (or regressive) transfer of the same amount between
two richer poor people. The following is the transfer sensitivity property:
Transfer Sensitivity: A poverty measure that satisfies transfer

sensitivity places greater emphasis on progressive (or regressive)
transfers at the lower end of the distribution of the poor than at the
upper end of the distribution of the poor; so P(x'; z) < (>) P(x"; z).
111
The final dominance property is subgroup consistency, which is concep-

tually the same as the corresponding property for income standards and
inequality measures. This property requires that if subgroup population sizes
are fixed, then overall inequality must rise when poverty rises in one sub-
group and does not fall in the rest of the subgroups. For example, suppose
that income vector x with population size N is divided into two subgroup
vectors: x' with population size N' and x" with population size N" such
that N' + N" = N. Let a new vector, y, be obtained from x with the same
population size N, and let its two corresponding subgroups be denoted by
y' with population size N' and y" with population size N". The subgroup
consistency property can be stated as follows:
Subgroup Consistency: Given that subgroup population sizes remain

unchanged, if P(y';z) > P(x';z) and P(y";z) ≥ P(x";z), then P(y;z) > P(x;z).
There is a property closely related to subgroup consistency that is often

useful for understanding how much of the overall poverty is attributed to
the poverty of a particular group, given a collection of population subgroups.
For example, a country’s population may be divided into subgroups such as
rural and urban areas, states, provinces, and other geographic regions; ethnic
and religious groups; genders; or age groups. Often, one may want to evalu-
ate a particular group’s contribution. The additive decomposability property
requires that overall poverty is expressed as a population-weighted average
of subgroup poverty levels. This property is similar in spirit to the corre-
sponding properties of income standards and inequality measures. However,
it is more analogous to that of income standards in the sense that there are
no within-group and between-group terms as we see for a decomposable
inequality measure.
To formally outline the property, we will use two groups to simplify
the interpretation, but the definition can be extended to any number of
groups. Suppose income vector x with population size N is divided into two
subgroup vectors: x' with population size N' and x" with population size N"
such that N' + N" = N. The additive decomposability property can be stated
as follows (see example 2.9):
Additive Decomposability: If income distribution x is divided into two

′ ′
subgroup distributions x' and x", then P(x ′; z) = N P(x ′; z) + N P(x′′; z).
N N
112
Example 2.9: Consider the six-person income vector x = ($80, $100,

$800, $1,000, $50,000, $70,000), which is divided into two subgroups
x' = ($80, $100, $50,000) and x" = ($800, $1,000, $70,000). Suppose
the poverty line is z = $1,100, which is the same across both subgroups.
Note that N' = 3, N" = 3, and N = 6, and, thus, N'/N = N"/N = 3/6
= 0.5. Then any additively decomposable poverty index can be
expressed as P(x;$1,100) = 0.5P(x';$1,100) + 0.5P(x";$1,100).
Poverty and Income Standards

The second way of understanding poverty measures is through the income
standards discussed earlier. Like inequality measures, most poverty measures
are based on a comparison between two income standards: a higher income
standard b and a lower income standard a. However, there is a crucial dif-
ference between inequality measures and poverty measures. In inequality
measures, the higher and lower income standards are two different income
standards applied to the same income vector. In poverty measures, the
higher and lower income standards are the same income standards applied
to two different income vectors: one is the censored distribution and the other
is the nonpoverty censored distribution. Recall that a censored distribution is
obtained from an original income distribution by replacing the income of the
nonpoor by the poverty line. The nonpoverty censored distribution is that
income distribution where all incomes are equal to the poverty line income.
It turns out that the higher income standard for poverty measures is the
poverty line itself. Why is that so? This can be understood by the normaliza-
tion property of income standards, which requires that if all incomes are equal
in an income distribution, then an income standard of the distribution should
be equal to that commonly held income. Because in a nonpoverty censored
income distribution all incomes are equal to the poverty line, any income
standard of the nonpoverty censored distribution should be equal to the pov-
erty line itself, that is, b = z. Many well-known poverty measures take the form
P = (z − a)/z or the form P = a/z or a monotonic transformation of either form.
Commonly Used Poverty Measures
In this section, we introduce various poverty measures that are in com-

mon use. We classify them into two categories. The first category lists basic
113
poverty measures, and the second category lists advanced poverty measures.
There are two basic poverty measures in common use: headcount ratio and
poverty gap measure.
Headcount Ratio
The headcount ratio (PH) is a crude measure of poverty that simply counts
the number of people whose incomes are below the poverty line z and
divides that number by the total number of people in the society. In society
X with population size N, if there are q poor people, then the headcount
ratio is simply q/N. It is obvious that the headcount ratio lies between zero
and one. If all people are poor in a society, then the headcount ratio is one.
When there are no poor, it is zero.
The headcount ratio can also be understood using income standards
applied to the nonpoverty censored distribution and a doubly censored dis-
tribution. What is a doubly censored distribution, and how do we obtain
it? A doubly censored distribution x** is obtained from an original income
distribution x by replacing nonpoor incomes with the poverty line income z
and by replacing the poor incomes with zero. Therefore, income distribution
x is censored upward at poverty line z for nonpoor and again censored at zero
for the poor. The term doubly censored comes from the fact that distribution
x*z* is obtained by censoring distribution x twice.
The arithmetic mean is the income standard used to understand head-
count ratio. The arithmetic mean of the nonpoverty censored distribution is
poverty line z, and the arithmetic mean of the doubly censored distribution
is called the dichotomous mean. If there are q poor people, or N − q nonpoor
people, in society X, then the dichotomous mean of the society is
N− q N− q
WA (x ** ) = q × 0 + z= z. (2.35)
N N
The headcount ratio of distribution x is a normalized shortfall of the

dichotomous mean from the mean of the nonpoverty censored distribution
(see example 2.10). Thus, the headcount ratio can be expressed as
N−q
WA (xz* ) − WA (x** ) z− z
N q
PH (x; z) = *
z
= = . (2.36)
WD (xz ) z N
114
Example 2.10: How is the headcount ratio calculated by different

methods? Consider the four-person income vector x = ($800, $1,000,
$50,000, $70,000). If the poverty line is set at z = $1,100, then two of
the four people are poor. Thus, the headcount ratio is PH(x;z) = 2/4 = 0.5
or 50 percent.
How can the headcount ratio be calculated using the concept of
doubly censored distribution?
The doubly censored vector of x is x*z* = (0, 0, $1,100, $1,100)
and the nonpoverty censored distribution is x̄*z = ($1,100, $1,100,
$1,100, $1,100).
Then WA(x̄*z) = 4 × $1,100/4 = $1,100 and WA(x*z*) = 2 ×
$1,100/4 = $550.
Hence, PH(x;z) = ($1,100 − $550)/$1,100 = 0.5.
The headcount ratio is the most well-known and most widely used
poverty measure because its interpretation is highly intuitive and simple.
However, the effectiveness of the headcount ratio depends on which prop-
erties the headcount ratio satisfies. It satisfies all invariance properties:
symmetry, normalization, population invariance, scale invariance, and focus.
However, it does not satisfy any dominance property except subgroup consis-
tency. The headcount ratio is not sensitive to changes in the income level
of the poor as long as incomes do not cross the poverty line. This is why
the headcount ratio does not satisfy the other dominance properties and
monotonicity, which require poverty measures to change as the incomes of
the poor change. The headcount ratio satisfies subgroup consistency because
the headcount ratio is additively decomposable, as shown by example 2.11.
Poverty Gap Measure
The second basic poverty measure is the poverty gap measure. Like headcount
ratio, it is also widely used. The poverty gap measure (PG) is the average
normalized shortfall with respect to the poverty line across the poor. In
society X, the normalized income shortfall of a person, say, n, is calculated as
(z − x*n)/z, which means that the normalized income shortfall of a nonpoor
person is zero. The average normalized income shortfall is the average of all
normalized income shortfalls within a society. We denote the normalized gap
vector of x by g* = ((z − x*1)/z,…,(z − x*N)/z). Then the poverty gap measure is
115
Example 2.11: Consider the six-person income vector x = ($80, $100,

$800, $1,000, $50,000, $70,000), which is divided into two subgroups
x' = ($80, $100, $800) and x" = ($1,000, $50,000, $70,000).
Suppose the poverty line, z = $1,100, is the same across both
subgroups.
Note that N' = 3, N" = 3, and N = 6; thus, N'/N = N"/N = 3/6 = 0.5
is the population share of each group.
The headcount ratio of x is PH(x;z) = 4/6 = 2/3; the headcount
ratio of x' is PH(x';z) = 3/3 = 1; and the headcount ratio of x" is
PH(x";z) = 1/3.
Thus, the overall headcount ratio may be obtained from the sub-
group headcount ratios. The population-weighted average headcount
ratio of the subgroups is 0.5P(x';z) + 0.5P(x";z) = 0.5 × 1 + 0.5 × 1/3
= 2/3.
1 N z − x *n
PG (x; z) = WA (g * ) = ∑
N n =1 z
. (2.37)
The poverty gap measure may also be understood and interpreted by

using two income standards. The higher income standard is the poverty line
z itself, obtained by taking an arithmetic mean of the nonpoverty censored
distribution x̄*z . The lower income standard is obtained by applying the
arithmetic mean to the censored income distribution x*. Thus, the poverty
gap measure can be expressed as
WA (xz* ) − WA (x * ) z − WA (x * ) 1 N z − x *n
PG (x; z) = = = ∑ . (2.38)
WA (xz* ) z N n =1 z
There is a third way to interpret the poverty gap measure, which is as a

product of the headcount ratio and the average normalized income shortfall
among the poor. The average normalized income shortfall among the poor
1 q
is PIG (x; z) = ∑ n =1(z − x n )/z. The poverty gap measure can be expressed as
q
N−q q 1 q z − xn
PG (x; z) = ×0+ × ∑ = PH × PIG (x; z). (2.39)
N N q n =1 z
The poverty gap measure lies between zero and one. Zero is obtained
when there are no poor in the society. A value of one is obtained when
116
everyone in the society is poor and has zero income. When everyone in
a society is poor, then the poverty gap measure is the average normalized
income shortfall among the poor, PIG, because the headcount ratio is one in
this situation, that is, PH = 1 (see example 2.12).
Example 2.12: How is the poverty gap measure calculated by different

$50,000, $70,000). The poverty line is set at z = $1,100. The cen-
sored income vector is x* = ($800, $1,000, $1,100, $1,100).
• Use the method in equation (2.37) to calculate the pov-
erty gap measure. The poverty gap vector is g* = (300/1100,
100/1100,0,0). Then the poverty gap measure is PG(x;z) =
WA(g*) = 0.09.
• The method in equation (2.38) uses two income standards. The
mean of the censored distribution is WA(x*) = 1,000. The non-
poverty censored distribution is x̄*z = ($1,100, $1,100, $1,100,
$1,100). Thus, the mean of the nonpoverty censored distribu-
tion is WA(x*) = 1,100. Hence, the poverty gap measure is
PG(x;z) = (1,100 − 1,000) / 1,100 = 0.09.
• The method in equation (2.39) uses the headcount ratio and
the income gap ratio to calculate the poverty gap measure. We
already know that the headcount ratio of x is 0.5. The income
gap ratio of x may be obtained by taking the mean of the first
two elements of Gx and so PIG(x;z) = 2/11. Thus, the poverty gap
measure is PG(x;z) = 0.5 × 2/11 = 0.09.
What properties does the poverty gap measure satisfy? It satisfies all
invariance properties: symmetry, normalization, population invariance, scale
invariance, and focus. Among dominance properties, it satisfies only mono-
tonicity and subgroup consistency and does not satisfy the transfer principle
and transfer sensitivity. Although it does not satisfy the transfer principle, it
satisfies the weak transfer principle, which means that the poverty gap mea-
sure does not increase (or decrease) because of a regressive (or progressive)
transfer but also does not fall (or increase). The poverty gap measure satis-
fies the monotonicity property, meaning that if the income of a poor person
increases, then (unlike the headcount ratio) the poverty gap increases. The
poverty gap measure satisfies subgroup consistency because, like the head-
count ratio, it is additively decomposable.
117
There is a long list of advanced poverty measures. These measures may

not necessarily be as intuitive and as easy to understand as the two basic
measures, but they are capable of moderating the limitations of the two basic
measures. The advanced measures discussed in this book include the Watts
index, the Sen-Shorrocks-Thon index, the squared gap measure, the Foster-
Greer-Thorbecke indices, the mean gap measure, and the Clark-Hemming-
Ulph-Chakravarty indices.
Watts Index
The Watts index was proposed by Watts (1968), and it is the average dif-
ference between the logarithm of the poverty line and the logarithm of
incomes. For income distribution x with population size N and poverty line
z, the Watts index can be written as
1 N
PW (x; z) = ∑ (ln z − ln x *n).
N n =1
(2.40)
The lowest value the Watts index can take is zero, which is obtained
when no one is poor in the society. However, unlike the headcount ratio
and the poverty gap measure, the Watts index has no maximum value.
Like the two basic measures, the Watts index can also be expressed as a
difference between two income standards. The income standard used for the
headcount ratio and the poverty gap measure is the arithmetic mean, where-
as the income standard for the Watts index is the geometric mean. The
higher income standard is obtained by applying the geometric mean to the
nonpoverty censored distribution x̄*z. Because the geometric mean satisfies
normalization, the higher income standard is equal to the common ele-
ment in x*, which is the poverty line z itself. The lower income standard is
obtained by applying the geometric mean to the censored income distribu-
tion x*. The Watts index is the logarithm of the ratio of the higher and the
lower income standards.
The other way of interpreting the measure is by calculating the differ-
ence of their logarithms (see example 2.13). The formulation of the Watts
index in terms of income standards is
⎡ W (x * ) ⎤ ⎡ z ⎤
PW (x; z) = ln ⎢ G z ⎥ = ln ⎢ ⎥ = ln z − ln ⎡⎣WG (x * )⎤⎦ . (2.41)
⎣ WG (x* ) ⎦ ⎣ WG (x * ) ⎦
118
Example 2.13: How is the Watts index calculated by different meth-

ods? Consider the four-person income vector x = ($800, $1,000,
$50,000, $70,000), with the poverty line set at z = $1,100. The cen-
sored vector is x* = ($800, $1,000, $1,100, $1,100). The logarithm of
the poverty line is Inz = In1,000 = 7.
• Use the method in equation (2.40) to calculate the Watts index.
The logarithmic differences between the poverty line and the
censored incomes are (7 − In800, 7 − In1,000,0,0) = (0.3, 0.1, 0, 0),
the mean of which is 0.103. Thus, PW(x;z) = 0.1.
• Calculate the Watts index using the income standards. The
geometric mean of x* is WG(x*) = 991.9 and In[WG(x*)] = 6.9.
Therefore, by equation 2.41, PW(x;z) = 7 − 6.9 = 0.1. Thus, both
calculation and understanding of the Watts index are much easier
in terms of income standards.
The Watts index satisfies all invariance properties: symmetry, normaliza-

tion, population invariance, scale invariance, and focus, as well as all dominance
properties: monotonicity, transfer principle, transfer sensitivity, and subgroup
consistency. It satisfies the transfer principle because poverty falls when
income is transferred from a richer poor person to a poorer poor person. It
satisfies transfer sensitivity because it is more sensitive to a transfer at the
lower end of the distribution than at the upper end of the income distribu-
tion of the poor. It satisfies the subgroup consistency property because, like
the two basic measures, it is additively decomposable.
Sen-Shorrocks-Thon Index
The Sen-Shorrocks-Thon (SST) poverty index was originally formulated in

terms of a basic poverty measure and an inequality measure. The poverty
gap measure is the basic poverty measure used for constructing the SST, and
the Gini coefficient is the inequality measure. Thus, the SST index can be
expressed as
PSST(x;z) = PG(x;z) + [1− PG(x;z)]IGini(x∗). (2.42)
Note that the Gini coefficient is applied to the censored income distri-
bution x*.15 This measure is sensitive to inequality among the poor, which
is evident from its formulation in equation (2.42). If there is no inequality
119
among the poor, then PSST(x;z) reaches its minimum. As inequality increases,
the values of PSST(x;z) increase because 1 − PG(x;z) > 0, which results from
the fact that PG(x;z) lies between zero and one. The Gini coefficient lies
between zero and one. When there are no poor in a society, the SST index is
zero. The maximum value of one is obtained when everyone in the society is
poor and has zero income.
The SST index has an interesting relationship with the average normal-
ized income shortfall among the poor, PIG. When everyone is poor in a
society, but has equal income, then the SST index is equal to the average
normalized income shortfall among the poor, that is, PSST(x;z) = PIG(x;z).
This is because in this situation IGini(x*) is zero and PH = 1. When the
inequality level among the poor increases while the average normalized
income shortfall remains the same, the SST index becomes larger than the
average normalized income shortfall.
The SST index can also be interpreted by an income standard. The
income standard in this case would be the Sen mean. The SST index is the
normalized difference between the Sen mean of the nonpoverty censored
distribution and the Sen mean of the censored distribution. The Sen mean
satisfies the normalization property of income standards. Thus, the Sen
mean of the nonpoverty censored distribution is the poverty line itself, that
is, WS(x̄*) = z. The Sen mean of the censored distribution x* is denoted by
WS(x*). The SST index16 can be presented as
WS (xz* ) − WS (x* ) z − WS (x * )
PSST (x; z) = = . (2.43)
WS (xz* ) z
Given a censored distribution, once the Sen mean is calculated using the
procedure discussed in the income standard section, the SST index can eas-
ily be obtained by applying equation (2.43). How do equations (2.42) and
(2.43) give the same result? That question can easily be answered as
z − WS (x * ) z − WA (x * ) WA (x * )
= + IGini (x * ) = PG + (1 − PG )IGini (x * ). (2.44)
z z z
In the previous section, when discussing dominance and ambiguity
results for income standards, we mentioned that the Sen mean is related to
the generalized Lorenz curve. The SST index is based on the Sen mean and
thus is naturally related to the generalized Lorenz curve, which has been
graphically depicted in Zheng (2000). Example 2.14 shows how to calculate
the SST index.
120
Example 2.14: How is the Sen-Shorrocks-Thon index calculated

by different methods? Consider the four-person income vector
x = ($800, $1,000, $50,000, $70,000); the poverty line is set at
z = $1,100. The censored vector is x* = ($800, $1,000, $1,100, $1,100).
• Calculate the SST index using equation (2.42). The poverty gap
measure, as we already know, is 0.09. The Gini coefficient of x* is
0.062. Then PSST(x;z) = 0.09 + (1 − 0.09) × 0.062 = 0.15.
• Calculate the SST index using equation (2.43). The Sen
mean of x* is 937.5. Thus, the SST index is PSST(x;z) =
(1,100 − 937.5)/1,100 = 0.15.
What properties does the SST index satisfy? It satisfies all invariance
properties: symmetry, normalization, population invariance, scale invariance,
and focus. However, it does not satisfy all dominance properties because it
is based on the poverty gap measure and the Gini coefficient. It inherits
the monotonicity property from the poverty gap measure, and it inherits
the transfer principle from the Gini coefficient. However, neither the Gini
coefficient nor the poverty gap ratio satisfies transfer sensitivity; conse-
quently, the SST index does not satisfy transfer sensitivity. Furthermore,
the Gini coefficient is neither subgroup consistent nor additively decom-
posable in the usual way. This shortcoming is also inherited by the SST
index.
Despite these shortcomings, the SST index is useful because it can be
broken down into the poverty gap measure and the Gini coefficient. In fact,
the poverty gap measure can be further broken down into the headcount
ratio (PH) and the average income gap of the poor (PIG).
Squared Gap Measure
The next poverty measure in the advanced measures category is the squared
gap measure. This measure is calculated by averaging the square of the nor-
malized income shortfalls and is denoted by
2
1 N ⎛ z − x *n ⎞
PSG (x; z) = ∑
N n =1 ⎜⎝ z ⎟⎠
. (2.45)
One way of interpreting the squared gap measure is as the weighted aver-
age of normalized income shortfalls, where each normalized income shortfall
121
is weighted by itself. This method of weighting puts greater emphasis on

larger shortfalls during aggregation. Thus, a transfer of income from a richer
poor person to a poorer poor person should reduce poverty. Like the SST
index, the squared gap measure can also be expressed as a function of the
headcount ratio (PH), the average normalized income shortfall (PIG), and the
generalized entropy measure for a = 2 of the incomes of the poor (denoted
by the vector xq), such that
PSG(x;z) = PH[PIG
2 + 2(1 − P )2I (xq;2)].
IG GE (2.46)
The squared gap measure lies between zero and one (see example 2.15).
A zero value is obtained when there are no poor people in the society
because the headcount ratio is zero. The maximum value of one is reached
when everyone in the society is poor and has zero income.
Example 2.15: How is the squared gap measure calculated by different

$50,000, $70,000). The poverty line is set at z = $1,100. The cen-
sored vector is x* = ($800, $1,000, $1,100, $1,100).
• Use the method in equation (2.45) to calculate the squared gap
measure. The squared gap vector is sg* = ([300/1100]2, [100/1100]2,
0, 0). Then the squared gap measure is PSG(x;z) = WA(sg*) = 0.02.
• The method in equation (2.46) uses the headcount ratio, average
normalized income shortfall, and generalized entropy measure
to calculate the squared gap measure. We already know that the
headcount ratio of x* is 0.5 and that the poverty gap measure is
0.18. The inequality measure IGE(x q ; 2) among the poor is 0.006.
Then the squared gap measure is PSG(x;z) = 0.5[0.182 + 2 × (1 −
0.18)2 × 0.006] = 0.02.
What properties does the squared gap measure satisfy? It satisfies all
invariance properties: symmetry, normalization, population invariance, scale
invariance, and focus. However, among the dominance properties, it satisfies
monotonicity, the transfer principle, and subgroup consistency, but it does not
satisfy transfer sensitivity because the headcount ratio, the income gap ratio,
and the generalized entropy of order 2 do not satisfy this property. Hence,
like the basic poverty measures and the SST index, the squared income gap
122
measure is transfer neutral. However, unlike the SST index, it satisfies sub-
group consistency because it is additively decomposable.
Foster-Greer-Thorbecke (FGT) Family of Indices
This family of measures was proposed by Foster, Greer, and Thorbecke

(1984). The FGT family of measures has the following formulation:
a
N ⎛ z − xj ⎞
*
1
PFGT = (x; z,a ) = ∑ n =1 ⎜ ⎟ , (2.47)
N ⎝ z ⎠
where a ≥ 0. The parameter a can be interpreted as the inequality aver-

sion parameter among the poor, which is conceptually the same as that for
Atkinson’s class of inequality measures. As a increases, a society’s aversion
toward inequality among the poor increases.
Notice that there is a minor difference between parameter a in this
case and parameter a in Atkinson’s class of inequality measures, where a
lower value of a leads to greater aversion toward inequality. This differ-
ence exists because inequality is measured in the income space and poverty
is measured in the normalized gap space, where large gaps imply worse
situations.
Measures in the FGT family take the form of various well-known poverty
measures introduced earlier for different values of a. For example, for a = 0,
the formulation in equation (2.45) becomes the headcount ratio because
(z − x*n/z)0 =1 when xn < z and because (z − x*n/z)0 = 0 when xn ≥ z. Thus,
PFGT(x;z,0) = q/N = PH(x;z). For a = 1, the formula becomes the poverty
gap measure, which is the average of all normalized income shortfalls. For
a = 2, the formula is the squared gap measure, which is the average of the
square of all normalized income shortfalls.
As a increases and becomes very large, PFGT approaches a Rawlsian
measure17 placing more emphasis on the largest normalized income gap of
the poorest person. However, note that the value of PFGT for any distri-
bution decreases as a increases, and, for a very large a, the overall value
of PFGT may be infinitesimally small. This occurrence can be verified by
expressing the FGT formulation in equation (2.47) in general mean form
using equation (2.3) as follows:
PFGT(x;z,a) = [WGM(g∗; a)]a for a > 0. (2.48)
123
Recall that the general mean of a distribution converges toward the

maximum or largest element in a vector or distribution. The largest element
in the gap vector g* belongs to the poorest person in the society.
We have already discussed the properties that the headcount ratio, the
poverty gap measure, and the squared gap measure satisfy. Thus, we know
what properties the FGT family of indices satisfies when a = 0, 1, and 2. The
additional property that the measures in this family satisfy is transfer sensi-
tivity when a > 2, which implies that if a similar amount of transfer takes
place between two poorer poor people and two richer poor people, then this
measure is able to distinguish between these two situations.
An aspect that is not so intuitive in this family of measures is interpreta-
tion of the inequality aversion parameter. A larger value of a implies greater
aversion to inequality among the poor. However, when there is no inequal-
ity in the society, should the poverty measure alter because of a change in α?
For example, suppose that in a society of 100 people, everyone is poor and all
people have an equal income of $500. If the poverty line is z = $1,000, then
the normalized income gap of each person is one-half in this society. Given
that there is no inequality in the society, it should not matter how averse the
society is to inequality because there is no inequality.
However, the FGT family of measures may not remain the same for all α.
For the simple example considered above, PFGT(x;z,1) = PG(x;z) = 1/2 and
PFGT(x;z,2) = PSG(x;z) = 1/4. However, this problem can be easily solved
by calculating a monotonic transformation of the original FGT family of
measures as
P'FGT(x;z,a) = [PFGT(x;z,a)]1/a = WGM(g*; a) for a > 0. (2.49)
Note that this formula is not valid for the headcount ratio when a = 0.
For the example above, P'FGT(x;z,a) = 1/2 for all a > 0 because the general
mean satisfies the normalization property of income standards.
Mean Gap Measure
The mean gap measure of poverty can be obtained by taking the Euclidean
mean (WE) of the normalized income shortfalls. This is a monotonic trans-
formation of the squared gap measure. More specifically, the mean gap mea-
sure is the square root of the squared gap measure. The mean gap measure
can be expressed as
124
1
1 ⎛ 1 N ⎛ z − x* ⎞ 2 ⎞ 2
PMG (x; z) = WE (g ) = P = ⎜ ∑ ⎜
* 2
SG
n
⎟⎠ ⎟ . (2.50)
N
⎝ n =1 ⎝ z ⎠
There is another interpretation of the mean gap measure: P'FGT(x;z,2).

Because the mean poverty gap is a monotonic transformation of the squared
gap measure, it satisfies all the properties that are satisfied by the squared gap
measure except the additive decomposability. One advantage of the mean
gap measure compared with the squared gap measure is that the values of
the mean gap measure are commensurate with the values of the poverty
gap measure as discussed using equation (2.49). Values of the squared gap
measure tend to be much smaller than the poverty gap measure, and these
numbers are not comparable to each other.
Unlike the squared gap measure, values of the mean gap measure tend
to be higher than those of the poverty gap measure, because it uses the
Euclidean mean instead of the arithmetic mean. For example, for the four-
person income vector x = ($800, $1,000, $50,000, $70,000) and poverty line
z = $1,100, the poverty gap measure is 0.09, whereas the mean gap measure
is (0.02)1/2 = 0.14. However, had the income of the poor been equally dis-
tributed, the income vector would have been x' = ($800, $1,000, $50,000,
$70,000), and the poverty gap measure would remain the same as that of x
(that is, 0.09), but the mean gap measure would be 0.13.
Like the squared gap measure, the mean gap measure also lies between
zero and one. Moreover, this measure has an interesting relationship with
the average normalized income shortfall. When everyone in a society is
poor, but there is no inequality, then the squared gap measure is equal to
the average normalized income shortfall among the poor because CV = 0
and PH = 1. Thus,
PMG = PSG = PH ⎡⎣P12G + z (1 − P1G )2 IGE (xa; z)⎤⎦ = P12G = P1G . (2.51)
Clark-Hemming-Ulph-Chakravarty (CHUC) Family of Indices
The final measure in our discussion of poverty measures is the Clark-

Hemming-Ulph-Chakravarty (CHUC) family of indices (see Clark,
Hemming, and Ulph 1981; Chakravarty 1983). This family is an extension
of the Watts index. The CHUC index is based on the generalized mean
125
and is the normalized shortfall of the generalized mean of the observed cen-
sored income distribution x* from the generalized mean of the nonpoverty
censored income distribution x̄*. Again, the generalized mean satisfies the
normalization property of income standards; thus, the generalized mean of
the nonpoverty censored income distribution is the poverty line itself. The
CHUC index for a ≤ 1 can be expressed as
WGM (xz*;a ) − WGM (x *;a ) z − WGM (x *;a )

PCHUC (x; z) = = . (2.52)
WGM (xz*;a ) z
The CHUC index lies between zero and one. The minimum value of
zero is obtained when there are no poor people in a society. However, the
maximum value of the CHUC index cannot be larger than one. When
everyone in a society is poor, having equal income, this measure is equal
to the average normalized income shortfall. It satisfies all invariance and
dominance properties. However, not all measures in this class are addi-
tively decomposable. For a = 1, the CHUC index is the poverty gap mea-
sure, and for a = 0, the CHUC index is a monotonic transformation of the
Watts index.
Advantages and Disadvantages of Each Measure
We have shown that the two basic measures—the headcount ratio and the
poverty gap measure—do not satisfy transfer-related properties and so are
not sensitive to inequality across the poor. Besides not being sensitive to
inequality, the headcount ratio does not satisfy monotonicity, which, if it is
used as a target for public policy, may cause inefficiency in public spending.
All of the subsequent advanced poverty measures, in contrast, are sensitive
to inequality across the poor. The SST index and the mean gap measure are
both equal to the poverty gap measure when everyone in a society is poor
and no inequality exists among them. These two measures become larger
than the poverty gap measure when the income gap remains the same, but
inequality among the poor increases.
Each advanced measure, however, has its own pros and cons. Let us
begin with the SST measure. We know from our previous discussion that
this measure is not subgroup consistent, which means that it may lead to
inconsistent outcomes when group-level analysis is of interest. This measure
is also not transfer sensitive, which means that if a similar amount of transfer
126
takes place between two poorer poor people and two richer poor people,
then this measure cannot distinguish between the two situations.
What, then, are the SST index’s advantages? The first is that it can
be neatly broken down into the headcount ratio, the average normalized
income shortfall among the poor, and the well-known Gini coefficient. If
one is not interested in group-level analysis, then this measure can be bro-
ken down into these three components to understand the source of change
in poverty. In fact, the Gini coefficient can be broken down further into a
within-group and a between-group component using the Gini decomposi-
tion formula introduced earlier. The within-group component assesses
inequality among the poor, and the between-group component measures
inequality between the average income of the poor and the poverty line.
This decomposition reveals whether the change in the measure’s inequal-
ity component is caused by the change in inequality among the poor or due
to a change in the average income of the poor compared to the poverty line.
Note that there is no within-group inequality among the nonpoor because they
all have the same income equal to the poverty line. Furthermore, there is no
residual term, which is commonly seen in the Gini decomposition, because
there is no income overlap between the poor and the nonpoor.
Second, consider the squared gap measure. This measure has many posi-
tive features, such as it is additively decomposable and subgroup consistent.
Furthermore, like the SST index, it can be broken down into the head-
count ratio, the average normalized income shortfall among the poor, and
the generalized entropy measure order of 2 among the poor to understand
the poverty composition. However, like the SST index, this measure is not
transfer sensitive, which means that if a similar amount of transfer takes
place between two poorer poor people and two richer poor people, then this
measure cannot distinguish between these two situations.
Also, the generalized entropy measure order of 2 may be a bit unintuitive
in the sense that it may range from zero to infinity, unlike the Gini coefficient
that ranges from zero to one. The same pros and cons apply to the mean gap
measure, which is just a monotonic transformation of the squared gap measure.
Third, consider the Watts index. This measure appears to be a perfect
measure of poverty in the sense that it satisfies all the properties that we dis-
cussed earlier: it is additively decomposable, is transfer sensitive, and satisfies
the transfer principle and all other properties. However, this measure has two
shortcomings. One is that it is not applicable when there are zero incomes
because the logarithm of zero is undefined. The second shortcoming is that
127
it does not have an intuitive interpretation like the two basic measures, the
SST index and the squared gap measure and its monotonic transformation
(the mean gap measure). Also, like these other measures, it does not have an
upper bound of one. Finally, the CHUC class of indices is a generalization of
the Watts index. Like the Watts index, its members satisfy all the properties
discussed earlier and also lie between zero and one. However, measures in
this class are not defined for zero incomes when α ≤ 0.
Policy Relevance of Poverty Measures
Besides gauging the level of deprivation in a society, a poverty measure can

have crucial policy relevance. In fact, different measures may have different
policy implications. We discuss three policy implications below with cer-
tain examples. First is the influence of poverty measures as targeting tools.
Second is the relevance of poverty measures in guiding public policies. Third
is the use of the additive decomposability property for geographic targeting.
How Do Different Poverty Measures Influence the Targeting Exercise?
Besides gauging the level of deprivation in a society, a poverty measure is a

useful tool that can influence a policy maker’s targeting exercise. An impor-
tant question that is often asked is the following: if a policy maker has allot-
ted a certain amount of the budget that he or she can spend on the welfare
program for the poor, how should that budget be allocated among the poor?
For instance, consider the following six-person society with income vector
x = ($80, $100, $800, $1,000, $50,000, $70,000). The poverty line is set at
$1,100 so that four people are poor and two people are nonpoor.
It is evident that the society’s policy maker requires at least $2,420 so
that he or she can drive all four poor people out of poverty. Suppose that
the policy maker can allot only $1,000 toward the welfare program for the
poor. Then how should that budget of $1,000 be allocated among the poor?
The answer depends on which poverty measure is used to assess the society’s
deprivation. Different poverty measures provide different answers for this
targeting exercise.
We begin this analysis when the society’s poverty is assessed by the
headcount ratio. The easiest way for a policy maker to reduce the headcount
ratio is to bring as many poor people as possible up to the poverty line.
Therefore, the first $100 of the allotted budget would be spent on the richest
128
poor person (with an income of $1,000). The next $300 would be spent on
the second-richest poor person (with an income of $800).
After bringing these two poor people out of poverty, the policy maker
still has $600 in his or her budget that remains unused. How and whom
should this amount assist? Given that the headcount ratio does not satisfy
the monotonicity property, because even if this entire amount is transferred
to either of the two remaining poor people, the poorest people still remain
under the poverty line and do not add to the headcount ratio. The policy
maker in this situation would have no incentive to spend the remaining
budget. This lack of incentive creates inefficiency in public spending.
Although poverty is reduced by 50 percent, the poverty status of the two
severely deprived people remains unchanged.
What if the society’s poverty is assessed by the poverty gap measure? Recall
that, unlike the headcount ratio, the poverty gap measure satisfies monotonicity;
but, like the headcount ratio, it does not satisfy the transfer principle or transfer
sensitivity. Thus, it is not sensitive to inequality among the poor. What implica-
tion does it have on the targeting exercise? In this case, the policy maker will
be inclined to spend his or her entire budget because the poverty gap measure
satisfies monotonicity. An increase in a poor person’s income, even when he or
she is not driven out of poverty, reduces the poverty gap measure. Therefore,
unlike the headcount ratio, inefficiency in public spending does not arise.
Then how should the budget of $1,000 be allocated among the poor? The
straightforward way is to spend the budget on any of the four poor people as
long as they do not surpass the poverty line income. Given that the poverty
gap measure is not sensitive to inequality among the poor, it does not matter
who among the poor receives the assistance. For example, in one case, out
of the budget of $1,000, the richest poor person, with an income of $1,000,
may receive $100; the second-richest poor person may receive $300; and the
third-richest poor person may receive the rest, or, in another case, the poor-
est person, with an income of $80, may receive the entire amount. In both
cases, the improvements in the poverty gap measure are the same. Thus, the
poverty gap measure is insensitive to whoever receives the assistance. The
poorest section of a society may perpetually remain poor in spite of showing
decent progress in terms of the poverty gap measure.
How would this policy exercise be affected when the society’s poverty is
gauged by a distribution-sensitive poverty measure? A distribution-sensitive
measure requires that assistance should go to the poorest of the poor first.
Thus, out of the $1,000 budget allotted for the poor, the first $20 should go
129
to the poorest person whose income is $80 so that the incomes of the two
poorest poor people are made equal. Then the rest of the budget should be
equally divided between the two poorest people so that, after allocating
the entire budget, the income distribution becomes x' = ($590, $590, $800,
$1,000, $50,000, $70,000).
What if, instead of $1,000, there was $1,600 allotted to the welfare of
the poor? Then the first $20 would be transferred to the poorest person.
Next, out of $1,580, $1,400 would be divided equally between the two poor-
est people so that the incomes of all three of the poorest people would be
equalized at $800. Finally, the rest of the budget of $180 is equally divided
among the three poorest poor so that the post-allocation income vector
is x" = ($860, $860, $860, $1,000, $50,000, $70,000). All distribution-
sensitive poverty measures support this type of targeting. However, not all
measures reflect similar amounts of decrease in poverty, which depends on
how these measures weight different people.
Can Poverty Measures Influence Public Policy?
Like the targeting exercise, can a poverty measure influence public policy?
Consider an example of a developing country where the major staple food is
rice. As with other agricultural producers, rice producers are poor and their
incomes are scattered around the country’s poverty line income. Some rice
producers earn enough income to live just above the poverty line, but many
rice producers are unfortunate enough to live below the poverty line.
There are other poor people in the country, such as those whose major
occupation is agricultural labor, plantation labor, or other unskilled jobs.
These poor people are the poorest in the country, and their major source
of energy and nutrition is the staple food, rice. Rice is, in fact, a necessary
commodity in that country, and the government controls its price.
Being benevolent, the government wants to see a reduction in poverty
by adjusting the price of rice. Which of the following two policy options
would reduce poverty?
• Option 1: Reduce the price of rice.

• Option 2: Increase the price of rice.
Suppose poverty in the country is assessed by the headcount ratio. If

the government decides to choose option 1 and reduce the price, then rice
130
producers would be adversely affected because their income would fall, and
rice consumers would benefit because their real incomes would increase.
Given that most rice consumers are poorer than rice producers, one does
not know whether more or fewer people would become poor. Thus, the
impact on the headcount ratio is uncertain.
However, if the price of rice increases, then producers gain, but the
poorer consumers lose because their real incomes fall. Given that the
already poor consumers become poorer, this is not taken into account by
the headcount ratio because it does not satisfy monotonicity. Therefore,
the number of poor people would most likely fall, thereby leading to a fall
in the country’s headcount ratio. Thus, the potential assessment of poverty
using the headcount ratio would incline the government to choose option
2 and increase the price because poverty, according to the headcount ratio,
would fall.
Note, however, that the decrease in the headcount ratio has ignored the
change in inequality among the poor. The marginally poor producers would
become better off because of the price increase, but the severely poor people
would be worse off for the same reason. This occurrence is very similar to the
idea of regressive transfer. The higher price paid by the poorer consumers is
obtained by the lesser poor producers as profit.
Any inequality-sensitive poverty measure, such as the squared gap,
the Watts index, or the SST index, would be sensitive to such inequality
among the poor. Suppose the poverty level in that country is now assessed
with one such measure that is sensitive to inequality among the poor. If
the government now chooses option 1 and reduces the price of rice, then
the poorer consumers benefit at the cost of a reduction in the producers’
income. The result is uncertain. If some producers become poorer than
some consumers, then the poverty measure may increase. But if the pro-
ducers remain less poor than the consumers, then the poverty measure
may fall.
However, if option 2 is chosen and the rice price rises, then inequality
among the poor increases and, most certainly, the poverty measure would
increase. Hence, the potential assessment of poverty using any inequality-
sensitive poverty measure would incline the government to not raise the
price because poverty, according to any inequality-sensitive measure, would
increase. The conclusion is that different poverty measures would incline the
government to choose different policies.
131
Additive Decomposability and Geographic Targeting
A poverty measure of a population subgroup reflects the level of depriva-

tion for that subgroup. A higher value of a population subgroup’s poverty
measure reflects a higher level of deprivation. The poverty measures we
have discussed in this chapter satisfy population replication invariance to
be able to compare the poverty levels of different population sizes, so these
measures are invariant to population size. However, a population subgroup
with a higher level of poverty does not necessarily imply that the subgroup
has a larger contribution to overall poverty.
A subgroup’s contribution to overall poverty also depends on the popula-
tion distribution across subgroups. Therefore, targeting a region or a group
based on only a poverty measure may not be completely accurate. We also
need to take the population distribution into account. If P is an additively
decomposable poverty measure and the income distribution x with total popu-
lation size N is divided into M subgroups—x1 with population size N1, x2 with
population size N2, …, xM with population size NM—then the contribution
of group m to total poverty is NmP(xm;z)/NP(x;z), where z is the poverty line.
Consider the situation when poverty is assessed by the headcount ratio.
A population subgroup’s headcount ratio denotes the population percentage
identified as poor. Interpreting a population subgroup’s contribution to over-
all poverty in terms of the headcount ratio is intuitive. If the total number
of poor is q, and qm is the number of poor in subgroup m, then the overall
headcount ratio is q/N and that of subgroup m is qm/Nm for all m = 1,…, M.
Then subgroup m’s share of overall poverty is Nm[qm/Nm]/N[q/N] = qm/q.
Thus, the contribution of the subgroup’s poverty to overall poverty in terms
of the headcount ratio is just the share of overall poor in that subgroup.
For example, consider table 3.9 in chapter 3, which shows the distribu-
tion of the poor across Georgian subnational regions for years 2003 and
2006. Suppose that, in 2003, the headcount ratio of the subnational region
Kvemo Kartli is 44.4 percent, which is more than twice the headcount ratio
of 20.9 percent in Tbilisi. However, the share of total poor living in Tbilisi
is, in fact, slightly larger than that living in Kvemo Kartli, because the popu-
lation size of Tbilisi is more than twice that of Kvemo Kartli. In 2006, the
headcount ratio of Kvemo Kartli decreased to 35.1 percent, which is still
10 percent higher than the headcount ratio of Tbilisi, but the share of the
poor living in Tbilisi increased to 20.4 percent alongside only 12.2 percent
in Kvemo Kartli. Therefore, the Georgian government needs to understand
132
that, despite having a lower headcount ratio, a massive number of poor

people reside in Tbilisi.
The share of subgroup poverty in overall poverty also has an intuitive inter-
pretation that can be relevant for geographic targeting. Using the same nota-
tions as in the previous paragraph, we can express the poverty gap measure as
q
[∑i = 1(z − xi)]/Nz,
q
where [∑i = 1(z − xi)] is the total sum of financial assistance required to bring
all poor people just to the poverty line to eradicate poverty. If the distribu-
tion x is divided into M subgroups as earlier, then the poverty gap measure
of subgroup m is
q
[∑ i m= 1(z − xi)]/Nmz,
q
where [∑ i m= 1(z − xi)] is the total amount of financial assistance required to
eradicate poverty in subgroup m. The contribution of subgroup m’s poverty
gap measure to the overall poverty gap ratio is
q q q q
[Nm∑ i m= 1(z − xi)]/Nmz]/N[∑i = 1(z − xi)]/Nz = ∑ i m= 1(z − xi)/∑i = 1(z − xi). (2.53)
Therefore, a subgroup’s contribution is nothing but the share of total

financial assistance that should be received by that subgroup to eradicate pov-
erty. Thus, the contribution in terms of the poverty gap measure may be used
to understand the requirement for fund allocation across geographic regions.
The subgroup contribution of other additively decomposable poverty
measures that are sensitive to inequality, such as the squared gap or the
Watts index, may not have such an intuitive implication for targeting.
However, their additively decomposable property enables us to understand
the subgroup’s contribution to overall poverty and monitor the targeting
exercise. Although for these examples we have considered only the popula-
tion subgroups in terms of subnational regions, the population may well be
grouped alternatively by gender, occupation, or household head character-
istics, as depicted in chapter 3.
Poverty, Inequality, and Welfare
Poverty measures that satisfy the transfer principle are called distribution-
sensitive poverty measures. The distribution-sensitive poverty measures
133
introduced earlier were the Watts index, the SST index, the FGT family of
measures for α > 1, and the CHUC family of indices. Each of these distribu-
tion-sensitive poverty measures is built on a specific income or gap standard
that is closely linked to an inequality measure. For example, the Watts index
is closely linked with Theil’s second measure of inequality, the SST index
is closely linked with the Gini coefficient, the FGT family of indices for
α > 1 is linked with the generalized entropy measures, and the CHUC fam-
ily of indices is linked with Atkinson’s family of measures.
For the Watts index, SST index, and CHUC family of indices, the
inequality measure is applied to the censored distribution x*, with greater
censored inequality being reflected in a higher level of poverty for a given
poverty gap level. The FGT indices for α > 1, however, use generalized
entropy measures applied to the gap distribution g*, with greater gap inequal-
ity leading to a higher level of poverty for a given poverty gap level.
Recall from our earlier discussion in the income standard section that
certain income standards can be viewed as welfare functions, and this link
provides yet another lens for interpreting poverty measures. The Sen mean
used in the SST index and the general means for α ≤ 1 that are behind the
CHUC indices can be interpreted as welfare functions. In each poverty
measure, the welfare function is applied to the censored distribution to
obtain the censored income standard, which is now seen to be a censored
welfare function that takes into account poor incomes and only part of non-
poor incomes up to the poverty line. For these measures, poverty and cen-
sored welfare are inversely related—every increase in poverty can be seen as
a decrease in censored welfare.
A poverty measure assesses the level of poverty within a society by a single

number for a given poverty line. Two obvious questions arise: (a) Does a
single poverty measure evaluate two distributions in the same way for all
poverty lines? and (b) Do all poverty measures evaluate two income distri-
butions in the same way? More specifically, according to the first question, if
one distribution has more poverty than another distribution for a particular
poverty line, is there any certainty that the former distribution would have
more poverty than the latter for any other poverty line?
Consider the following example with two four-person income distribu-
tions x = ($800, $900, $5,000, $70,000) and x' = ($200, $1,200, $1,600,
134
$70,000). Let poverty be measured by the headcount ratio. If the poverty

line is $1,000, then distribution x has more poverty than distribution x'.
What happens if the policy maker decides that the correct poverty line
should be $800? Then distribution x has no poor people, but distribution
x" has one poor person. Similarly, if the poverty line is $2,000, then, again,
distribution x has less poverty than distribution x". Hence, the choice of
poverty line affects the poverty comparison.
According to the second question, if one poverty measure determines
income distribution x to have more poverty than distribution x', would
other poverty measures compare these two distributions in the same way?
This situation is analogous to our discussion of dominance and ambiguity
for inequality and income standards. The answer is not too optimistic and
depends on the poverty measure used—not all poverty measures evaluate
different distributions in the same manner.
Consider the same two four-person income vectors used above: x = ($800,
$900, $5,000, $70,000) and x' = ($200, $1,200, $1,600, $70,000). Let the
poverty line be z = $1,000. We have already seen that the headcount ratio
reflects more poverty in distribution x than in distribution x'. How does the
poverty gap measure PG compare these two distributions? It turns out that
PG(x; z) = 0.08 < PG(x'; z) = 0.18. Distribution x has less poverty than distribu-
tion x'. Thus, these two basic measures disagree with each other.
Is there any way we can devise situations where we have unanimous
results? To start, we try to answer the first question using a concept intro-
duced at the beginning of this chapter: the cumulative distribution function, or
cdf.18 Recall that the cdf of distribution x denotes the proportion of people in
the distribution whose income falls below a given income level. In the pov-
erty analysis context, if that income level is the poverty line z, then the
value of the cdf at z is nothing but the headcount ratio at poverty line z (see
figure 2.14 below).
Poverty Incidence Curve
The horizontal axis of figure 2.14 denotes income, and the vertical axis
denotes the values of a cumulative distribution function. If the poverty line
is set at z, then the headcount ratio is PH(x; z), which is the percentage of
people in distribution x who have incomes less than z. Similarly, PH(x; z')
and PH(x; z") are the headcount ratios of distribution x corresponding to
poverty lines z' and z'', respectively.
135
Figure 2.14: Poverty Incidence Curve and Headcount Ratio
Cumulative distribution (%)

100
Fx ′
PH(x ′; z ″) Fx
PH(x ′; z ′)
PH(x; z ″)
PH(x; z ′)
PH(x ′; z)
PH(x; z)
z z′ z″ xN
Income
Suppose there is another distribution x'. One can see in figure 2.14 that
the headcount ratios corresponding to poverty lines z, z', and z" lie above
the respective headcount ratios for distribution x. Is there any other poverty
line that reflects a higher headcount ratio in x than in x'? The answer is no.
The cdf of x lies to the right of the cdf of x', which means that the headcount
ratio for x' for no poverty line can be lower than the headcount ratio for x.
When a cdf lies to the right of another cdf, first-order stochastic dominance
(introduced earlier) occurs. When such dominance relation holds between
two cdfs, not only do the headcount ratios agree for all poverty lines, but the
poverty gap measure, the squared gap measure, the mean gap measure, the
Watts index, and the CHUC indices also agree for all poverty lines.
This approach also answers the second question, which asks when all
poverty measures agree. Therefore, if the first-order stochastic dominance
holds, then there is no need to compare any two distributions by any poverty
measure introduced earlier with respect to varying the poverty line. The
choice of poverty measure and the choice of poverty line simply do not mat-
ter when the first-order dominance condition holds. The cdf in the context
of poverty measurement is also known as the poverty incidence curve.
Poverty Deficit Curve
What if two poverty incidence curves cross? Then a unanimous relationship

in terms of the headcount ratio does not hold. However, there are two other
136
poverty-value curves that lead to a unanimous relationship in terms of the

poverty gap measure and the squared gap measure. These two curves are
known as the poverty deficit curve and the poverty severity curve.
When the poverty deficit curve of one distribution lies above the poverty
deficit curve of another distribution, then the former distribution has higher
poverty—in terms of the poverty gap measure for all poverty lines—than
the latter distribution. Similarly, if the poverty severity curve of a distribu-
tion lies above the poverty severity curve of another distribution, then the
former distribution has higher poverty in terms of the squared gap measure
for all poverty lines. We now elaborate these two concepts.
Figure 2.15 outlines the poverty deficit curve concept. We use the pov-
erty incidence curve (panel a) to construct the poverty deficit curve (panel
b). The poverty incidence curve of distribution x is denoted by Fx. The
height of a poverty deficit curve at a poverty line is the area underneath
the poverty incidence curve to the left of the poverty line. In figure 2.15,
the height of the poverty incidence curve at poverty line z is denoted by
height B, which is the shaded area below the poverty incidence curve Fx
to the left of z. For instance, for the poverty line z, if q people are identi-
fied as poor, then Fx(z) = q/N percent, which is the percentage of the poor
population.
What does the area underneath the incidence curve denoted by B
mean? To understand, first note that the lightly shaded area denoted by
A is the average income of the q poor people times the share of the poor.
Figure 2.15: Poverty Deficit Curve and the Poverty Gap Measure
a. Poverty incidence curve b. Poverty deficit curve

C
Deficit
Fx
Dx
Fx(z)
Dx ′
A B
B
z xN z xN
Income Income
137
This can be easily verified from the quantile function as described earlier
in figure 2.5.
Recall that an income distribution’s cdf is just the inverse of the relevant
distribution’s quantile function. Thus, A is WA(xA)(xq)q/N = (x1 + … + xq )/N.
Another interpretation of area A is that it is the per capita income of an aver-
age poor person in the society. The combined area A + B denotes the society’s
per capita income, which, if held by each poor person, means that the poor will
not be poor anymore.
This per capita income is qz/N. Thus, area B, which is also the height of
the poverty deficit curve Dx at poverty line z, is the difference between the
area A + B and the area A, or the average income shortfall or the deficit,
that is, [z − WA(xq)]q/N. This deficit is the minimum per capita income of
the society, which, if transferred to the poor, will lift the poor out of poverty.
Area B is also zPG(x; z). The maximum height of the poverty deficit curve
is denoted by C, which is xN − WA(x).
Example 2.16: Suppose in a country of 100 million (m) people with a

per capita income of $20,000, 30 million people are poor. The aver-
age income of these poor people is $400. So the per capita income
held by an average poor person is ($1,000 − $400) × 30m ÷ 180m.
If the poverty line is $1,000, then the deficit is ($1,000 − $400) ×
30m ÷ 100m = $180.
Thus, $180 per capita, which is only 0.9 percent of the per capita
income of the country, is the minimum amount required to bring all
30 million poor people out of poverty.
Note that the larger height of the poverty deficit curve Dx compared
to the poverty deficit curve Dx' at z reflects a larger poverty gap measure
in distribution x than in distribution x' at poverty line z. It is evident from
figure 2.15 that the poverty deficit curve Dx lies above the poverty deficit
curve Dx' for all poverty lines. Hence, distribution x has higher poverty than
distribution x' for all poverty lines in terms of the poverty gap measure.
This type of unanimity result, however, fails to hold when two poverty
deficit curves cross each other. We should then check the poverty severity
curve of these two distributions. If the poverty severity curve of a distribu-
tion lies above the poverty severity curve of another distribution, then the
former distribution has higher poverty than the latter in terms of the squared
gap measure or the mean gap measure for all poverty lines.
138
Poverty Severity Curve
Panel a of figure 2.16 displays the poverty deficit curve that we will use to
show how a poverty severity curve is constructed. As explained earlier, the
height B of a poverty deficit curve is proportional to the poverty gap measure
and is the poverty gap measure times the poverty line. As shown in panel b,
the height of the poverty severity curve Sx at poverty line z is D, which is the
area underneath the poverty deficit curve Dx. Area D is proportional to the
squared gap measure. Therefore, the larger the height of the poverty sever-
ity curve Sx than the poverty severity curve Sx at z, the larger the squared
gap measure in distribution x than in distribution x' at poverty line z. It
turns out that the poverty severity curve Sx lies above the poverty severity
curve Sx' for all poverty lines. Hence, distribution x has higher poverty than
distribution x' for all poverty lines.
Note that the dominance by the poverty deficit curve is equivalent to
the second-order stochastic dominance, and the dominance by the poverty
severity curve is equivalent to the third-order stochastic dominance.19
When there is dominance in terms of poverty incidence curves, all pov-
erty measures satisfying the invariance properties and monotonicity agree
with each other when ordering distributions according to the level of pov-
erty for any poverty line. Such dominance relationships do not always hold.
When two poverty incidence curves cross, one distribution has higher or
lower poverty only for a part of the entire range of incomes. In fact, different
poverty measures may order two distributions differently.
Figure 2.16: Poverty Severity Curve and the Squared Gap Measure
a. Poverty deficit curve b. Poverty severity curve
E
C
Severity
Deficit
Dx Sx
Sx ′
B D
D
z xN z xN
Income Income
139
One way of examining the robustness of poverty comparisons is by cal-

culating the vector of poverty levels of different measures for a fixed pov-
erty line. For instance, the headcount ratio, the poverty gap measure, the
squared gap measure, the Watts index, and the SST index can be depicted
in a five-dimensional vector. If there are two distributions x and x', then the
five-dimensional vector of x for poverty line z is
(PH(x; z), PG(x; z), PSG(x ;z), PW(x ;z), PSST(x ;z)),
and the five-dimensional vector of x' for poverty line z is
(PH(x'; z), PG(x'; z), PSG(x'; z), PW(x'; z), PSST(x'; z)).
Vector dominance between these two vectors would then be interpreted
as a variable measure poverty ordering that ranks distributions when all five
measures unanimously agree. If each element in the vector x is greater than
each corresponding element in the vector x', then distribution x has unani-
mously more poverty than distribution x' for poverty line z.
Sensitivity Analysis with Respect to the Poverty Line
The dominance analysis discussed earlier helps us understand whether one dis-
tribution has more or less poverty than another distribution. It is not concerned
about the level of poverty, which is often of particular policy interest. The num-
ber of poor people in a country or the fact that many poor people have been
moved out of poverty over a particular time period are always matters of great
concern. These data, of course, depend on the particular poverty line chosen.
As discussed in the introductory chapter, there are three different types
of poverty lines:
• An absolute poverty line may be adjusted with the rate of inflation over
time, but it is not adjusted with income growth over time.
• A relative poverty line is not fixed over time, and it changes with income
growth. For example, if a poverty line is set at 50 percent of the median
income, then the poverty line changes as the median income changes
over time. Or the poverty line may be set at 50 percent of mean
income. In this case, the growth rate of the poverty line over time is
the same as the growth rate of per capita income over time.
• A hybrid poverty line is created by taking a weighted average of an
absolute poverty line and a relative poverty line.
140
No matter how a poverty line is chosen, one can argue that it is arbitrary.
It is possible to propose a feasible alternative, which may change the
perspective of poverty significantly. Thus, one must examine the sensitiv-
ity of poverty with respect to the poverty line. One way of conducting the
sensitivity analysis is to change the poverty by certain percentages, then
estimate how much the poverty level has changed.
For example, suppose the headcount ratio of society x is 25 percent for
poverty line z = $10,000. Let this figure increase to 30 percent when the
poverty line is increased to $10,200. This means that a 2 percent increase in
the poverty line increases the headcount ratio by 5 percent. The lower the
change in the poverty estimate because of change in the poverty line, the
more reliable the point estimate based on a particular poverty line. If there is
too much variation, then the poverty estimate may not be considered reliable.
Growth and Poverty
When a country is rapidly growing, one must evaluate the quality of the
growth. By growth, we generally mean a country or society’s growth in mean
income, and, by merely looking at the growth, there is no way of knowing
who has benefited from this growth. This growth may result from a rise in
incomes of the richer part of the distribution or from a rise in incomes of the
poorer part of the distribution.
There are various ways of understanding if the growth is pro-poor or
anti-poor. First, we may be interested in knowing directly if poverty has
increased or decreased because of the growth. Second, we may want to
know if the growth has relatively benefited or hurt the population with lower
incomes. In this case, it is not enough just to understand if poverty has
increased or decreased; it is also important to understand whether the situ-
ation of the poor has changed in comparison to others in the distribution.
Third, we may be interested in knowing if the growth has lowered poverty
more than a counterfactual-balanced growth path would. In this case, one
may be interested in knowing how much of the change in poverty is due to
growth and how much is due to the redistribution.
Consider some examples to clarify these various ways of understand-
ing pro-poor growth. Suppose the society consists of four people and the
income vector is x = ($80, $100, $200, $260). The society’s mean income
is $160. First, if the poverty line income is $120, then two people are
poor. Suppose that, over time, incomes of these four people change to
141
x' = ($100, $125, $160, $575). The society’s mean income has grown by
50 percent to $240. If the poverty line remains unchanged at $120, then
the headcount ratio goes down. In fact, poverty goes down for any poverty
measure that satisfies the monotonicity property. Thus, if one is merely
interested in knowing if poverty has decreased because of growth, then the
growth has been pro-poor for a fixed poverty line. If, instead of $120, the
poverty line is set at $180, then the change in poverty may not appear to
be pro-poor by all measures. For example, despite growth of 50 percent, the
headcount ratio deteriorates. Thus, in terms of the headcount ratio, the
growth in the distribution appears to be anti-poor.20
Given that a fixed poverty line is difficult to defend, we must understand
the change in poverty for a variable poverty line. The approach is analogous
to the dominance analysis. If one poverty curve (incidence, deficit, or sever-
ity) dominates another poverty curve, then poverty has improved unambigu-
ously in the dominant distribution because of growth. Besides merely knowing
the direction of change in poverty, we may be interested in the magnitude of
the reduction in poverty relative to the growth in mean—the growth elastic-
ity of poverty. The growth elasticity of poverty is defined as the percentage
change in poverty resulting from a 1 percent change in the mean income.
If the elasticity is greater than one, then the percentage change in poverty
has been larger than the percentage change in mean income, or the growth
of mean income. For an application of the growth elasticity of poverty using
the headcount ratio, see Bourguignon (2003). To understand the change in
the growth or elasticity of poverty for a variable poverty line, various poverty
growth curves can be constructed (similar to the various growth curves dis-
cussed in the income standard section).
A second way of understanding a change in poverty as pro-poor is by look-
ing at the gain of the poor relative to the gain in the mean. Reconsider the two
income vectors in the previous example. The growth rate of the mean was
50 percent. Have the incomes of individuals at the bottom of the distribution
improved enough to catch up with the growth in mean? The answer is no. The
growth of the poorest person’s income was 25 percent. The income growth of
the two poorest people also totaled 25.0 percent, and the growth of the three
poorest people totaled 1.3 percent. Then how was the 50 percent growth
achieved? It was achieved because the richest person’s income grew by about
121 percent. Thus, this second way understands the relationship between
poverty and growth from an inequality perspective and may be referred to as
an inequality-based approach, as discussed in chapter 1.
142
The tools we used to understand the relationship between growth and

inequality can also be used here. Comparing the growth rates of two income
standards may provide some insight. If a lower income standard grows faster
than the mean, then incomes of the poorer section of the distribution must
have grown faster than the mean. In contrast, if an upper income standard
grows faster than the mean, then incomes of the richer section of the dis-
tribution must have grown faster than the mean. For example, one may
compare the growth rate of the Sen mean (emphasizing lower incomes) vis-
à-vis the growth of the average. Indeed, the growth in the Sen mean is only
24 percent compared to 50 percent growth in mean income.
One can also use other income standards, such as the general means,
for this exercise. For example, Foster and Székely (2008) computed the
growth in general means for different a to show that although the growth
rate mean incomes in Mexico and Costa Rica were the same, the growth
of general means was starkly different. In Mexico, the growth in mean
income was mostly driven by the increase in the income of the richer
section of the population. In Costa Rica, the growth in mean was driven
by the increase in the income of the poorer section. The same amount of
growth may have improved the situation of the poor in Costa Rica, but it
may have deteriorated the situation of the Mexican poor.
One may also be interested in understanding the composition of change
in poverty because of growth and because of change in inequality.21 As discussed
in chapter 1, pro-poor growth may be understood as a difference between
the growth rate of an original distribution and a counterfactual distribution
that has the same mean and relative distribution as the original distribution.
Then the overall change in poverty can be split into a change because of
growth and a change because of redistribution.
Consider the following simple example using the vectors above:
x = ($80, $100, $200, $260) and x' = ($100, $125, $160, $575). The mean
of x is $160, whereas the mean of x' is $240. We now rescale each element
of vector x' in such a way that it has the same mean as x, and we denote the
transformed vector by x". Thus, x" = (66.7, 83.3, 106.7, 383.3).
Let us simply measure poverty by the headcount ratio (this exercise can
be performed using any poverty measure). For the poverty line of $120, the
headcount ratio in x is 2/4, which decreases to 1/3 in x'.
How was this reduction obtained? Distribution x" is obtained from x by
redistribution while keeping the mean unchanged. The headcount ratio for
x", as a result, increases from two-fourths to three-fourths. Thus, poverty has
143
increased because of redistribution. However, distribution x' may be seen as

being obtained from distribution x" by merely increasing everyone’s income
by the same proportion with balanced growth. As a result, the headcount
ratio falls from three-fourths to one-fourth. Hence, the improvement in
poverty in this case has resulted from growth rather than redistribution.22
Exercises
1. Consider the following table that enables you to construct a cumula-

tive distribution function (cdf) from income data.
Number of
Category Income
people pi F(xi) (pi ë xi)
(i) ($ xi)
(ni)
1 12,000 10
2 13,000 15
3 14,000 40
4 15,000 20
5 16,000 15
There are five income categories (Xi) in the economy. Each category
contains a certain number of people (ni).
a. What is the total number of people (n) in the economy?
b. Let pi denote the proportion of people in each category. Fill in the
column corresponding to pi for each i. The probability mass function is
defined as a function that gives the probability of a discrete variable
taking the same value. Now draw the probability mass function.
Hint: Draw a diagram with x on the horizontal axis and p on the
vertical axis.
c. Let F(xi) denote the proportion of people who have an income no
higher than xi. Fill in the column corresponding to F(xi) for each i.
Now draw the cdf.
Hint: Draw a diagram with x on the horizontal axis and F(x) on
the vertical axis.
d. What is the relationship between pi and F(xi)?
e. Calculate the proportion of people having an income less than
$14,100. What is the proportion of people having an income more
than $14,900?
f. What is the average income for the economy?
g. Fill in the last column, and find the sum of all cells in that column.
What does the sum give you?
144
h. Use the cdf to calculate the area to the left of the cdf bounded by
x = 0 and F(x) = 1. What do you get?
i. Calculate the median, the 95th percentile, and the 20th percentile
using the cdf that you drew in 1c.
2. The Gini coefficient is probably the most commonly used index of
relative inequality. What are some of the advantages and disadvan-
tages of this measure?
3. The variance of logarithm (VL) is an inequality measure that is com-
puted as
1 N
VL (x) = ∑[ln x n − WL (x)]2,
N n =1
where WL(x) is the mean of the logarithm of elements in x as defined

in the chapter.
a. Verify that the variance of logarithms satisfies scale invariance. What
property of the variance of logarithms ensures scale invariance?
b. Graph the Lorenz curves for the two distributions x = (1,1,1,1,41)
and y = (1,1,1,21,21). Can the curves be ranked?
c. Find the variance of logarithms of the two distributions. What is
wrong here?
d. Find the mean log deviation (the second Theil measure) of the
two distributions. What is correct here?
4. Construct an inequality measure that violates replication invariance.
5. Are the following statements true, false, or uncertain? In each case,
support your answer with a brief but precise explanation.
a. The Kuznets ratios satisfy the Pigou-Dalton transfer principle.
b. Distribution y = (1,2,3,2,41) is more unequal than distribution
x = (1,8,4,1,36) in terms of the Lorenz criterion.
c. The four basic properties of inequality measurement are enough to
compare any two income distributions in terms of relative inequality.
d. If everyone’s income increases by a constant dollar amount,
inequality must fall.
6. Consider the distribution x = (1,3,6).
a. Draw the Lorenz curve, and calculate the area between the
45-degree line and the curve.
b. Calculate the Gini coefficient for x. What is the relationship
between the Gini coefficient and the calculated area?
145
7. Consider the distribution x = (3,6,9,12,24,36).

a. Divide the distribution into the following two subgroups: x1 = (3,6,9)
and x2 = (12,24,36). Calculate the Gini coefficient for x, x1, and x2.
Using the traditional additive decomposability formula, check if
the Gini coefficient is decomposable in this situation.
b. Divide distribution x into the following two subgroups:
x3 = (3,24,36) and x4 = (6,9,12). Again, using the traditional
additive decomposability formula, check if the Gini coefficient is
decomposable in this situation.
c. What is the difference between these two circumstances? Explain.
d. What is the residual for the Gini coefficient in these two
circumstances?
8. For the two distributions x = (2,100; 700; 1,100; 200) and y = (3,410;
620; 2,170; 6,510), do the following:
a. Calculate the WGM(.; a) and use it to calculate the Atkinson
measure IA(.; a) for a = 0, –1.
b. Do you have the same IA(.; a) for both distributions or not? What
is going on here?
9. For the income distributions x = (3,3,5,7) and y = (2,4,6,6), do the
following:
a. Calculate the generalized entropy measure and IGE(x; a) and
IGE(y; a) for a = 1,0,1,2,3,4.
b. Plot the values of a on the horizontal axis and the values of IGE(x; a)
and IGE(x; a) on the vertical axis.
c. Join the points, and check if they intersect. If they intersect, then
report at what value of a they intersect, and explain why.
10. Are the following statements true, false, or uncertain?
a. The second Theil measure is subgroup consistent.
b. The arithmetic mean is higher than the harmonic mean but less
than the geometric mean.
c. The sum of the decomposition weights of the generalized entropy
measure is always less than 1.
11. How is the generalized Lorenz curve GL(p) derived from a cdf? Draw
this process and explain. What value does the generalized Lorenz
curve take at p = 1?
12. Suppose an inequality measure is given by I(x) = (x̄—e(x))/x̄, where
e(x) is one of the equally distributed equivalent income functions used
by Atkinson (namely, a general mean with a parameter less than one).
146
a. Which equivalent income function is the lower income

standard?
b. Show that if the lower income standard grows at a faster rate than
the upper income standard, then inequality will fall.
c. Suppose the mean income grows at a rate of 3 percent. Under
what circumstances will the Atkinson index fall? When will the
Gini index fall?
13. Because of economic growth, the income distribution changes as fol-
lows over time: (1,1,1,1), (1,1,1,2), (1,1,2,2), (1,2,2,2), (2,2,2,2).
a. Explain the relevance of this example to the development literature.
b. Can unambiguous inequality comparisons be made between these
distributions?
c. How does the Gini coefficient change over time in this example?
14. Provide an example illustrating that the Gini coefficient violates
subgroup consistency. Explain why it does.
15. Country A has a more equal income distribution than Country B
such that Country A’s Lorenz curve dominates that of Country B.
a. What should be the relationship between these two countries in
terms of generalized Lorenz?
b. What does this finding say about welfare and inequality?
16. Why should a poverty measure be sensitive to the distribution of
income among the poor?
17. Suppose that the incomes in a population are given by x = (4,2,10)
and the poverty line is z = 6.
a. Find the number of people who are poor.
b. Find the headcount ratio PH.
c. Find the (normalized) poverty gap measure PG.
d. Find the squared poverty gap measure PSG.
e. If the income of person 2 falls by one unit so that the new distribu-
tion is y = (4,1,10), what happens to PH, PG, and PSG?
f. If person 2 gives person 1 a unit of income, resulting in distribution
u = (5,1,10), what happens to PH, PG, and PSG? Explain.
18. One of the big problems in evaluating poverty levels is arriving at a
single poverty line that represents the cutoff level between the poor
and the nonpoor. Many people believe that a poverty line must be
arbitrary to some extent. But if this is so, and if changing the pov-
erty line reverses poverty judgments, then all our conclusions about
poverty might be ambiguous. To solve this problem, we might make
147
comparisons not only for a single poverty line but also for a range
of poverty lines. Consider the three distributions from the previous
example: x = (4,2,10), y = (4,1,10), u = (5,1,10).
a. If z = 6 is the poverty line, does x or y have more poverty accord-
ing to the headcount ratio? Will this determination be reversed at
some other poverty line? Explain. Does x or y have more poverty
according to the poverty gap measure? Will this determination be
reversed at some other poverty line? Explain.
b. If z = 6 is the poverty line, does x or u have more poverty accord-
ing to the headcount ratio? Will this determination be reversed at
some other poverty line? Explain. Does x or u have more poverty
according to the poverty gap measure? Will this determination be
reversed at some other poverty line? Explain.
c. Do you think unambiguous comparisons with variable poverty
lines might be made in practice? If not, why not? If so, why?
19. Which inequality measure is the Sen-Shorrocks-Thon (SST) poverty
index based on?
a. Explain why the SST index is not subgroup consistent and provide
a counterexample to illustrate your point.
b. Which inequality measure is the Foster-Greer-Thorbecke (FGT)
index PSG(x; z) based on? Show that the measure is subgroup con-
sistent.
20. Why should a measure of poverty satisfy scale invariance (homoge-
neity of degree 0 in incomes and the poverty line)? Which poverty
measures satisfy scale invariance?
21. Suppose instead of the PSG(x; z) measure one were to use the
PMG(x; z) measure.
a. What is the main constructive difference between these two
measures?
b. What would be the advantages and disadvantages of using the
PMG(x; z) measure?
22. Why do inequality decompositions have a between-group term but
poverty decompositions do not?
23. Suppose inequality decreases without growth of mean income. What
may likely happen to poverty? Suppose growth of mean income
occurs without a change in inequality. What may likely happen to
poverty? Explain.
148
24. Suppose that the per capita poverty gap measure is used with a rela-
tive poverty line that sets z = αμ for some α > 0. When does one
distribution have a lower level of relative poverty for all α > 0? (Hint:
Think Lorenz.)
25. We have already shown that the poverty measures are different from
each other and differ in their sensitivity to a distribution. Please pro-
vide certain examples with illustrative distributions and poverty lines
such that
a. The SST index rises, but the three FGT indices fall.
b. The headcount ratio rises, but the SST index, poverty gap mea-
sure, and squared gap measure fall.
c. The poverty gap measure rises, but the headcount ratio, SST, and
squared gap measures fall.
d. The squared gap measure rises, but the headcount ratio, poverty
gap measure, and SST measure fall.
Notes
1. For further discussion on the use of consumption expenditure data ver-

sus income data, see Atkinson and Micklewright (1983) and Grosh and
Glewwe (2000).
2. For a more detailed discussion of some of these issues, see Deaton (1997).
3. For the concept and a more detailed discussion about the principle, see
Pigou (1912, 24–25); Dalton (1920); Atkinson (1970); Dasgupta, Sen,
and Starrett (1973); and Rothschild and Stiglitz (1973).
4. For further discussion of the concept, see Foster and Shorrocks (1991).
–
5. Going forward in this book, we will use the notation WA(x) and x inter-
changeably. They both denote the mean of distribution (x).
6. The measure was originally proposed by Sen (1976b) and thus we
named the income standard after him. See also Foster and Sen (1997).
7. A related property has been developed by Zheng (2007a). Called unit
consistency, it has a weaker requirement than the scale invariance
property. The unit consistency property requires that if one distribu-
tion is more unequal than another distribution, then just changing the
unit of measurement keeps the former distribution more unequal than
the latter. The property can be formally stated as follows: for any two
distributions x and x', if I(x) < I(x'), then I(cx) < I(cx') for any c > 0.
149
For example, if the elements of two distributions are converted from

Indian rupees to U.S. dollars, then the direction of inequality between
any two distributions should not change if the inequality measure satis-
fies unit consistency. An inequality measure that satisfies scale invari-
ance also satisfies unit consistency, but the converse is not necessarily
true. A class of decomposable inequality measures satisfying unit con-
sistency has been developed by Zheng (2007a). In this book, however,
we focus on relative inequality measures satisfying the scale invariance.
8. For a more in-depth theoretical discussion of the transfer sensitivity
property, see Shorrocks and Foster (1987).
9. A geographical interpretation of the residual term can be found in
Lambert and Aronson (1993), where the residual term is shown to be
an effect of the re-ranking effect. The inequality of a distribution is
computed in three steps: (a) within-group inequalities are computed
in each subgroup; (b) the groups are ranked by their mean incomes
and a concentration curve representing between-group inequalities is
constructed; and (c) the Lorenz curve is constructed. The difference
between the Lorenz curve of the distribution in the third step and the
concentration curve from the second step is known as the residual term.
10. The Lorenz curve was developed by Max Lorenz (1905).
11. Interested readers, who may desire to have further theoretical under-
standing of the properties and their interrelationship, should see Zheng
(1997) and Chakravarty (2009).
12. A related but weaker property has been developed by Zheng (2007b).
See note 8.
13. This axiom is also known in the literature as strong transfer (see Zheng
2000). However, to keep the terminologies comparable across sections,
we prefer to use the term transfer principle.
14. A weaker version of this property exists that is known in the literature
as weak transfer (see Chakravarty 1983), which can be stated as follows:
if distribution x' is obtained from distribution x by a regressive transfer
between two poor people while the poverty line is fixed at z and the
number of poor does not change, then P(x'; z) > P(x; z). If distribu-
tion x" is obtained from another distribution x by a progressive transfer
between two poor people while the poverty line is fixed at z and the
number of poor does not change, then PS(x"; z) < P(x; z). Note that this
property is different from the weak transfer principle that we define in
this book.
150
15. Previously, Sen (1976b) proposed the index PS(x; z) = PH[PIG + (1 − PIG)-
IGini(xq)], where xq is the income distribution of the poor only. This mea-
sure was modified later by Thon (1979) and Shorrocks (1995).
16. For a more elaborated discussion on various formulations of the SST
index, see Xu and Osberg (2003).
17. Rawls’s welfare function maximizes the welfare of society’s worse-off
member. “Social and economic inequalities are to be arranged ... to the
greatest benefit of the least advantaged...” (Rawls 1971, 302).
18. For an in-depth discussion on poverty ordering, see Atkinson (1987),
Foster and Shorrocks (1988), and Ravallion (1994).
19. Note that the poverty deficit curve and the generalized Lorenz curve
have an interesting relationship. They are based on the area under-
neath the cdf and the quantile function, where a quantile function is
an inverse of a cdf. See figure 2.7.
20. For various approaches to measuring pro-poor growth for a fixed poverty
line, see Kakwani and Son (2008).
21. For a discussion on the poverty-growth-inequality triangle, see
Bourguignon (2003).
22. The growth-redistribution decomposition becomes a bit more compli-
cated when there is interregional migration. For such decomposition
with change in population, see Huppi and Ravallion (1991). An appli-
cation of their method can be found in table 30 of chapter 3.
References
Atkinson, A. B. 1970. “On the Measurement of Inequality.” Journal of

Economic Theory 2 (1970): 244–63.
———. 1987. “On the Measurement of Poverty.” Econometrica 55 (4): 749–64.
Atkinson, A. B., and J. Micklewright. 1983. “On the Reliability of Income
Data in the Family Expenditure Survey 1970–1977.” Journal of the Royal
Statistical Society, Series A (146): 33–61.
Bourguignon, F. 2003. “The Growth Elasticity of Poverty Reduction: Explaining
Heterogeneity across Countries and Time Periods.” In Inequality and Growth:
Theory and Policy Implications, edited by T. Eicher and S. Turnovsky, 3–26.
Cambridge, MA: Massachusetts Institute of Technology.
Chakravarty, S. R. 1983. “A New Index of Poverty.” Mathematical Social
Sciences 6: 307–13.
151
———. 2009. Inequality, Polarization and Poverty: Advances in Distributional

Analysis. New York: Springer.
Clark, S., R. Hemming, and D. Ulph. 1981. “On Indices for the Measurement
of Poverty.” The Economic Journal 91 (362): 515–26.
Dalton, H. 1920. “The Measurement of the Inequality of Incomes.” The
Economic Journal 30: 348–61.
Dasgupta, P., A. Sen, and D. Starrett. 1973. “Notes on the Measurement of
Inequality.” Journal of Economic Theory 6 (2): 180–87.
Deaton, A. 1997. The Analysis of Household Surveys: A Microeconometric
Approach to Development Policy. Baltimore: World Bank.
Deaton, A., and S. Zaidi. 2002. “Guidelines for Constructing Consumption
Aggregates for Welfare Analysis.” Living Standards Measurement Study
Working Paper 135, World Bank, Washington, DC.
Foster, J. E., J. Greer, and E. Thorbecke. 1984. “A Class of Decomposable
Poverty Measures.” Econometrica 52 (3): 761–66.
Foster, J. E., and A. Sen. 1997. On Economic Inequality. 2nd ed. Oxford,
U.K.: Oxford University Press.
Foster, J. E., and A. F. Shorrocks. 1988. “Poverty Orderings.” Econometrica
56 (1): 173–77.
———. 1991. “Subgroup Consistent Poverty Indices.” Econometrica 59 (3):
687–709.
Foster, J. E., and M. Székely. 2008. “Is Economic Growth Good for the Poor?
Tracking Low Incomes Using General Means.” International Economic
Review 49 (4): 1143–72.
Gini, C. 1912. “Variabilità e mutabilità.” Reprinted in Memorie di metodolog-
ica statistica, edited by E. Pizetti and T. Salvemini. Rome: Libreria Eredi
Virgilio Veschi (1955).
Grosh, M., and P. Glewwe, eds. 2000. Designing Household Survey
Questionnaires for Developing Countries: Lessons from 15 Years of the Living
Standards Measurement Study. Washington, DC: World Bank.
Huppi, M., and M. Ravallion. 1991. “The Sectoral Structure of Poverty dur-
ing an Adjustment Period: Evidence for Indonesia in the Mid-1980s.”
World Development 19 (12): 1653–78.
Kakwani N., and H. H. Son. 2008. “Poverty Equivalent Growth Rate.”
Review of Income and Wealth 54 (4): 643–55.
Kanbur, R. 2006. “The Policy Significance of Inequality Decompositions.”
Journal of Economic Inequality 4 (3): 367–74.
152
Lambert, P., and J. R. Aronson. 1993. “Inequality Decomposition Analysis

and the Gini Coefficient Revisited.” The Economic Journal 103 (420):
1221–27.
Lorenz, M. O. 1905. “Methods of Measuring the Concentration of Wealth.”
Publications of the American Statistical Association 9 (70): 209–19.
Pigou, A. C. 1912. Wealth and Welfare. London: Macmillan.
Ravallion, M. 1994. Poverty Comparisons. Chur, Switzerland: Harwood
Academic Press.
Rawls, J. 1971. A Theory of Justice. Cambridge, MA: Harvard University Press.
Rothschild, M., and J. E. Stiglitz. 1973. “Some Further Results on the
Measurement of Inequality.” Journal of Economic Theory 6 (2): 188–204.
Sen, A. K. 1976a. “Poverty: An Ordinal Approach to Measurement.”
Econometrica 44 (2): 219–31.
———. 1976b. “Real National Income.” Review of Economic Studies 43 (1):
19–39.
Shorrocks, A. F. 1980. “The Class of Additively Decomposable Inequality
Measures.” Econometrica 48 (3): 613–25.
———. 1983. “Ranking Income Distributions.” Economica 50 (197): 3–17.
———. 1995. “Revisiting the Sen Poverty Index.” Econometrica 63 (5):
1225–30.
Shorrocks, A. F., and J. E. Foster. 1987. “Transfer Sensitive Inequality
Measures.” Review of Economic Studies 54 (3): 485–97.
Thon, D. 1979. “On Measuring Poverty.” Review of Income and Wealth 25
(4): 429–39.
Watts, H. W. 1968. “An Economic Definition of Poverty.” Discussion paper,
Institute for Research on Poverty, University of Wisconsin, Madison.
Xu, K. and L. Osberg. 2003. “The Social Welfare Implications,
Decomposability, and Geometry of the Sen Family of Poverty Indices.”
Canadian Journal of Economics 35 (1): 138–52.
Zheng, B. 1997. “Aggregate Poverty Measures.” Journal of Economic Surveys
11 (2): 123–62.
———. 2000. “Poverty Orderings.” Journal of Economic Surveys 14 (4):
427–66.
———. 2007a. “Unit-Consistent Decomposable Inequality Measures.”
Economica 74 (293): 97–111.
———. 2007b. “Unit-Consistent Poverty Indices.” Economic Theory 31 (1):
113–42.
153
Chapter 3
How to Interpret ADePT Results
In this chapter, we discuss how to interpret tables and graphs generated by

the ADePT analysis program. The chapter is organized in six sections:
• In the first section, we discuss how to interpret results at the country

level, decomposing across rural and urban areas.
• In the second and third sections, we move into analyses at a more
disaggregated level: across subnational regions in the second section
and across various population subgroups—such as household charac-
teristics, employment situation, and so forth—in the third section.
• In the fourth and fifth sections, we perform sensitivity and domi-
nance analyses. These are useful for policy evaluation, because results
in the first two sections are based on many assumptions, such as
choice of poverty line and selection of methodologies for measuring
poverty and inequality.
• It is always important to check how robust these results are with
respect to the assumptions. For example, we may assume the poverty
line to be a certain level of income or per capita expenditure and find
poverty decreasing over time. Then how can we be sure that poverty
has not increased for other possible poverty lines?
• Insights revealed in the first five sections may be helpful when prepar-
ing any report on poverty and inequality.
• In the final section, we discuss some advanced analyses.
155
Tables and graphs in this chapter were generated by ADePT’s Poverty

and Inequality modules using the Integrated Household Survey of Georgia
dataset for 2003 and 2006. Calculations assumed the equivalence scale
parameter is 1, which implies that every household member is assumed to
be adult equivalent. Hence, per capita expenditure was calculated by divid-
ing the total expenditure by the number of household members regardless of
their age and gender. Calculations assumed the economy-of-scale parameter
is 1. This implies that no economies of scale exist when two or more indi-
viduals share a household. (Other scale choices are, of course, possible, and
these parameters can be changed in ADePT.)
Consumption expenditures are in lari (or GEL, the Georgian national
currency) per month. Many tables use one or two poverty lines of GEL 75.4
and GEL 45.2 per month. In the first case, if a household fails to meet a
monthly consumption expenditure of GEL 75.4 for each member in that
household, then the household (and each member in the household) is
identified as poor. In the second case, a household is identified as poor if the
household fails to meet a per capita expenditure of GEL 45.2 per month.
Tables may have an occasional small numerical inconsistency. To
improve readability, ADePT displays data with a limited number of decimal
places by rounding the underlying raw data. This process can result in values
that appear incorrect, such as 29.9 + 1.0 = 31.0 (as opposed to 29.9 + 1.0 =
30.9, or 29.9 + 1.1 = 31.0). Spreadsheets generated by ADePT (the sources
for tables in this chapter) include raw data, which are visible in the formula
bar when a cell is selected.
Rounding numbers also affects how we present some of the results.
Certain poverty and inequality measures are traditionally reported in
decimals. However, this presentation does not provide us enough power to
differentiate between numbers. For example, the Gini coefficient of 0.26
and the Gini coefficient of 0.34 both may read as 0.3. Similarly, the FGT2
poverty index, or the squared poverty gap index, may take reasonable low
values in decimals such as 0.019 or 0.024. Again, these numbers may be
significantly different. Therefore, to improve readability, we normalize all
poverty and inequality figures in a 0–100 scale.
The text in this chapter has numerous references to table cells. To help
you quickly find data in tables, numbers and letters in brackets reference
table cells by row and column. For example, [3,E] refers to the cell in row 3,
column E.
156
Chapter 3: How to Interpret ADePT Results
Analysis at the National Level and Rural/Urban

Decomposition
While preparing a report on poverty and inequality, one would first be inter-
ested in results at the national level. This part of the chapter contains seven
tables with results at the national level. We then decompose the results
across urban and rural areas.
Income Distribution across the Population
Initially, understanding income distribution across the population is impor-

tant. A distribution’s density function is the percentage of population that
falls within a range of per capita expenditure. Figure 3.1 graphs the per
capita expenditure density function for urban Georgia. The vertical axis
shows probability density function of consumption expenditures. The hori-
zontal axis is per capita expenditure or any other equivalent achievement.
Figure 3.1: Probability Density Function of Urban Georgia
Urban
0.008
Probability density function
0.006
0.004
0.002
Median
0
0 200 400 600 800
Welfare aggregate
2003 2006
Source: Based on ADePT Poverty and Inequality modules using Integrated Household Survey of
Georgia 2003 and 2006.
157
In figure 3.1, the solid curve is urban Georgia’s density function for 2003,
and the dotted curve is the density function of urban consumption expendi-
ture distribution for 2006. The median is an important income standard that
can be found in the diagram. It is indicated by the corresponding vertical
lines: solid line for 2003 and dotted line for 2006.
A density function can also be useful for understanding a distribution’s
skewness. As can be seen from figure 3.1, the density functions for both years
are positively skewed. However, an important change from 2003 to 2006 is
that more people mass around the distribution’s median in 2006. We can
also see that the density functions for both years are unimodal. When more
than one mode exists, a society is considered to be polarized by consumption
expenditure or income.
Standard of Living and Inequality across the Population
Table 3.1 reports the mean and median per capita consumption expenditure
and their growth over time, and the inequality across the population using
the Gini coefficient. It also decomposes them across rural and urban areas
and across two years: 2003 and 2006. Table rows denote three geographical
regions: urban area, rural area, and all of Georgia (row 3). Per capita con-
sumption expenditure is measured in lari per month.
Columns A and B report the mean per capita consumption expenditure
for 2003 and 2006, respectively. Column C reports the percentage change
or growth in per capita expenditure over the course of these three years. The
average per capita expenditure of the urban area in 2003 is GEL 128.9 [1,A],
which is larger than the average rural per capita expenditure of GEL 123.5
[2,A]. The mean urban per capita expenditure in 2006 is GEL 127.3 [1,B],
Table 3.1: Mean and Median Per Capita Consumption Expenditure, Growth, and the Gini
Coefficient
Mean Median Gini coefficient

2003 2006 Growth 2003 2006 Growth Change
(GEL) (GEL) (%) (GEL) (GEL) (%) 2003 2006 (%)
Region A B C D E F G H I
1 Urban 128.9 127.3 −1.2 108.4 101.1 −6.8 33.5 35.6 2.2
2 Rural 123.5 124.8 1.0 101.5 105.3 3.7 35.3 35.1 −0.3
3 Total 126.1 126.0 −0.1 104.7 103.3 −1.4 34.4 35.4 0.9
Source: Based on ADePT Poverty and Inequality modules using Integrated Household Survey of Georgia 2003 and 2006.
158
which fell by 1.2 percent [1,C]. The mean rural per capita expenditure, in
contrast, increased by 1.0 percent to GEL 124.8 in 2006 [2,B]. Georgia’s
overall per capita consumption expenditure in 2003 is GEL 126.1 [3,A],
which fell by 0.1 percent to GEL 126.0 in 2006 [3,B].
Columns D, E, and F report the median per capita expenditures for
2003 and 2006 and their growth rates. The percentage changes in medians
or median growths are much larger than the mean per capita expenditure
growth. The rural median growth is 3.7 percent [2,F], whereas the urban
median “growth” is –6.8 percent [1,F]. The overall change in median is
–1.4 percent [3,F].
Columns G, H, and I use the Gini coefficient to capture inequality in
the distribution. The rural Gini coefficient has marginally fallen from 35.3
[2,G] to 35.1 [2,H], while the urban Gini coefficient over these three years
increased from 33.5 in 2003 [1,G] to 35.6 in 2006 [1,H]. The overall Gini
coefficient changed by 0.9 from 34.4 [3,G] to 35.4 [3,H]. (Gini coefficient
is reported on a scale from 0 to 100 in this chapter, rather than from 0 to 1.)
Lessons for Policy Makers
Note that the mean and the median, two different measures of standard of
living, are differently sensitive to the distribution of per capita consumption
expenditure. Mean is more sensitive to extreme values, whereas median is
more robust to extreme values. For example, if the only change in the dis-
tribution of per capita expenditure is at the highest quintile or the lowest
quintile, the change would be reflected by the mean, but the median would
not change. In contrast, in certain situations, when changes occur in the
middle of the distribution, mean per capita expenditures may remain unal-
tered, but the median may reflect the change.
It is important to analyze and understand the growth in both these
measures of central tendency. However, changes in different measures of
central tendency do not provide enough information about the change
in the overall distribution. They do not tell us how the spread or inequal-
ity within the distribution changes over time, which can be captured by
an inequality measure. In the above exercise, rural mean and median per
capita expenditure increased, but rural inequality marginally fell. On the
contrary, the urban inequality has increased over these three years from
33.5 in 2003 [1,G] to 35.6 in 2006 [1,H], while the mean and median
have fallen.
159
Overall Poverty
Table 3.2 examines the performance of groups of people considered poor.

It analyzes poverty in Georgia by decomposing across rural and urban areas
using three different poverty measures: headcount ratio, poverty gap measure,
and squared gap measure. These three poverty measures belong to the FGT
(Foster-Greer-Thorbecke) family of poverty measures. Table rows denote
three geographic regions: urban, rural, and all of Georgia (rows 3 and 6). The
variable is monthly per capita consumption expenditure in lari. There are
two poverty lines: GEL 75.4 per month and GEL 45.2 per month.
Columns A and B report headcount ratios for 2003 and 2006, respec-
tively. A region’s headcount ratio is the proportion of the population that
is poor compared to that region’s total population. When the poverty line
is GEL 75.4 per month, then the urban headcount ratio in 2003 is 28.1
percent [1,A]. This means that 28.1 percent of the population in the urban
area belongs to households that cannot afford the per capita consumption
expenditure of GEL 75.4 per month. The urban headcount ratio for 2006 is
30.8 percent [1,B]. Column C reports the change in urban headcount ratios
over the course of these three years, which is an increase of 2.7 percentage
points [1,C].
In contrast, the rural headcount ratio decreased by 0.5 percentage point
from 31.6 percent [2,A] in 2003 to 31.1 percent [2,B] in 2006. Overall,
Georgia’s poverty headcount has increased by 1.0 percentage point from
29.9 percent [3,A] to 31.0 percent [3,B]. Similarly, for the poverty line of
Table 3.2: Overall Poverty

percent
Headcount ratio Poverty gap measure Squared gap measure

2003 2006 Change 2003 2006 Change 2003 2006 Change
Poverty line = GEL 75.4
1 Urban 28.1 30.8 2.7 8.6 9.3 0.7 3.9 4.0 0.1
2 Rural 31.6 31.1 −0.5 10.7 10.9 0.2 5.2 5.5 0.3
3 Total 29.9 31.0 1.0 9.7 10.1 0.4 4.6 4.8 0.2
4 Urban 8.9 9.3 0.4 2.4 2.4 0.0 1.0 1.0 −0.1
5 Rural 11.4 12.1 0.7 3.6 4.0 0.3 1.7 1.9 0.2
6 Total 10.2 10.7 0.5 3.0 3.2 0.2 1.4 1.4 0.1
160
GEL 45.2 per month, Georgia’s headcount ratio increased from 10.2 percent
in 2003 [6,A] to 10.7 percent in 2006 [6,B]. The rural headcount ratio in this
case increased from 11.4 percent [5,A] to 12.1 percent [5,B]. This change
implies that the proportion of extreme poor (per capita expenditure below
GEL 45.2) in the rural area increased, but the proportion of nonextreme
poor (per capita expenditure between GEL 45.2 and GEL 75.4) decreased.
Columns D, E, and F analyze the poverty gap measure in 2003 and 2006.
The poverty gap measure lies between a minimum of 0 and a maximum of
100, where the minimum is when no one in a region is poor and the maxi-
mum is when everyone has zero consumption expenditure and the poverty
line is positive. When the poverty line is GEL 75.4, the urban area’s poverty
gap measure is 8.6 in 2003 [1,D], which increases by 0.7 to 9.3 in 2006 [1,E].
Likewise, the rural area’s poverty gap measure increases by 0.2 from 10.7 in
2003 [2,D] to 10.9 in 2006 [2,E]. The total increase in poverty gap measure
is 0.4 from 9.7 [3,D] to 10.1 [3,E]. When the poverty line is GEL 45.2, the
overall poverty gap measure increases by 0.2 from 3.0 in 2003 [6,D] to 3.2
in 2006 [6,E].
Columns G, H, and I analyze the squared gap measure. The squared gap
measure also lies between a minimum of 0 and a maximum of 100, where
the minimum is when no one in a region is poor and the maximum is when
everyone has zero consumption expenditure and the poverty line is positive.
This measure is sensitive to inequality across the poor. Column I shows
that the rural area’s squared gap measure when the poverty line is GEL 75.4
increased by 0.3 from 5.2 in 2003 [2,G] to 5.5 in 2006 [2,H]. For the rural
area it increased by 0.1 point from 3.9 [1,G] to 4.0 [1,H]. A similar pattern
of changes is visible for the lower poverty line.
Consider the situation when the poverty line is GEL 75.4. From column C,
one can see that the headcount ratio increased in the urban area by 2.7 per-
centage points and it decreased in the rural area by 0.5 percentage point. In
other words, the rural area performed better than the urban area in reducing
the proportion of poor people.
However, when we look at the poverty gap numbers, we see a different
scenario. It turns out, in fact, from column F that the poverty gaps for both
regions have registered increases, with the urban area registering a larger
increase (0.7 point increase in the urban area compared with 0.2 point
161
increase in the rural area). Thus, although the number of poor in the rural
area decreased, the same is not true when deprivation is measured in terms
of the average relative shortfall. Column F still reflects that the increase in
the rural poverty gap is lower than that of its urban counterpart. But col-
umn I shows that the increase in the squared gap measure is larger in the
rural area (0.3) than in the urban area (0.2), which implies that inequality
among the rural poor has been sufficiently high that despite a fall in the
headcount ratio, the increase in the squared gap measure is larger than that
in the urban area.
The change in the rural area’s headcount ratio is quite different when
the poverty line is GEL 45.2 per month. The increase in rural poverty is
much higher than the increase in urban poverty by all three measures. In
fact, the squared gap measure slightly decreases for the urban area. We con-
clude from this result that the situation for the rural area’s extreme poor has
actually worsened in 2006 compared with 2003.
Distribution of Poor across Rural and Urban Areas
Table 3.3 analyzes the distribution of population and poor people across
rural and urban areas. Table rows denote three geographic regions: urban,
rural, and all of Georgia (rows 3 and 6). The variable is per capita consump-
tion expenditure in l per month. There are two poverty lines: GEL 75.4 per
month and GEL 45.2 per month.
Columns A, B, and C analyze the headcount ratio, that is, the popula-
tion percentage that is poor. Columns A and B report the headcount ratio
Table 3.3: Distribution of Poor in Urban and Rural Areas

percent
Headcount ratio Distribution of the poor Distribution of population

Poverty Line = GEL 75.4
1 Urban 28.1 30.8 2.7 45.6 48.6 3.0 48.5 48.9 0.3
2 Rural 31.6 31.1 −0.5 54.4 51.4 −3.0 51.5 51.1 −0.3
3 Total 29.9 31.0 1.0 100.0 100.0 0.0 100.0 100.0 0.0
Poverty Line = GEL 45.2
4 Urban 8.9 9.3 0.4 42.4 42.3 −0.1 48.5 48.9 0.3
5 Rural 11.4 12.1 0.7 57.6 57.7 0.1 51.5 51.1 −0.3
6 Total 10.2 10.7 0.5 100.0 100.0 0.0 100.0 100.0 0.0
162
for the years 2003 and 2006, respectively, while column C reports the differ-
ence across these two years. Columns D, E, and F report the distribution of
poor people across rural and urban areas, with the number in the cell being
the proportion of poor people located in that region. Another way of seeing
this is as the region’s percentage contribution to poverty, or the headcount
ratio times the share of the region’s overall population divided by the overall
headcount ratio. Columns G, H, and I provide the population distribution
across rural and urban areas, or the percentage of the overall population
residing in that region.
The headcount ratio for the urban area’s population in 2003 is 28.1
percent [1,A]. In other words, 28.1 percent of the urban area popula-
tion is poor. The headcount ratio increased for urban Georgia in 2006 to
30.8 percent [1,B].
Of all poor people in Georgia in 2003, 45.6 percent [1,D] reside in
urban areas. The share of all poor people living in urban areas increases to
48.6 percent in 2006 [1,E]. This represents an increase of 3.0 percentage
points [1,F]. The shares of rural and urban area population do not change
much over the course of the three years. But when the poverty line is GEL
75.4 per month, the share of poor in urban areas increases in 2006 because
of the increase in headcount ratio.
This exercise has a very useful policy implication because the headcount
ratio does not provide any information about where most poor people live.
A region may have a lower headcount ratio, but if that region is highly
populated, then the number of poor may be high. Thus, policies should focus
on regions with high headcount ratios as well as regions with larger shares
of poor.
Composition of the FGT Family of Indices
Table 3.4 analyzes the composition of poverty figures reported in table 3.2.
Table rows denote three geographic regions: urban, rural, and all of Georgia
(rows 3 and 6). The variable is per capita consumption expenditure in lari
per month. There are two poverty lines: GEL 75.4 Lari per month and GEL
45.2 Lari per month.
163
Table 3.4: Composition of FGT Family of Indices by Geography
Headcount ratio Income gap Poverty gap GE(2) among Squared gap
(%) ratio measure the poor measure
Region A B C D E
2003
1 Urban 28.1 30.5 8.6 4.6 3.9
2 Rural 31.6 33.7 10.7 5.9 5.2
3 Total 29.9 32.3 9.7 5.3 4.6
2006
4 Urban 30.8 30.1 9.3 4.1 4.0
5 Rural 31.1 34.9 10.9 6.4 5.5
6 Total 31.0 32.6 10.1 5.3 4.8
2003
7 Urban 8.9 26.8 2.4 4.0 1.0
8 Rural 11.4 31.8 3.6 5.3 1.7
9 Total 10.2 29.7 3.0 4.7 1.4
2006
10 Urban 9.3 25.7 2.4 3.3 1.0
11 Rural 12.1 32.7 4.0 5.7 1.9
12 Total 10.7 29.7 3.2 4.7 1.4
The headcount ratio reports the proportion of people within a region

who are poor. The poverty gap measure and the squared gap measure can be
broken down as follows:
• The poverty gap measure is the headcount ratio multiplied by the

income gap ratio divided by 100.
• The income gap ratio is the average per capita expenditure shortfall
from the poverty line divided by the poverty line.
The squared gap (PSG) can be decomposed into three factors: headcount
ratio (PH), income gap ratio (PIG), and generalized entropy measure (GE)
for α = 2 among the poor, such that PSG = PH [P2IG + 2(1 − PIG)2 IGE (x; 2)].
These measures make possible a richer set of information for policy
analysis. An improvement in the poverty gap measure may result from a
reduction in the number of poor or a reduction in the average normalized
gap among the poor. Similarly, an improvement in the squared coefficient of
variation may result from a decrease in the number of poor, a decrease in the
164
average normalized gap among the poor, or a decrease in inequality among

the poor in terms of the generalized entropy measure.
For the GEL 75.4 per month poverty line, the poverty gap measure
for Georgia increased from 9.7 in 2003 [3,C] to 10.1 in 2006 [6,C]. This
increase comes from both a headcount ratio increase from 29.9 percent
[3,A] to 31.0 percent [6,A] and an income gap ratio increase from 32.3 [3,B]
to 32.6 [6,B]. However, the urban poverty gap measure increase derives
from an increase in the headcount ratio and a reduction in the income gap
ratio. In contrast, the rural poverty gap measure increase was a result of
an increase in the income gap ratio because the rural headcount ratio fell
slightly between 2003 and 2006.
Some interesting results are also evident when the poverty line is set
at GEL 45.2 per month. The urban poverty gap measure does not change
because an increase in the number of poor has been offset by an income
gap ratio decrease. In fact, the total poverty gap measure increase from 3.0
in 2003 [9,C] to 3.2 in 2006 [12,C] was caused solely by an increase in the
headcount ratio from 10.2 percent [9,A] to 10.7 percent [12,A], because the
income gap ratio remained unchanged at 29.7 [9,B] and [12,B].
The squared gap measure depends on another component: inequality among

the poor. Surprisingly, inequality among the poor does not change between
2003 and 2006 for both the higher and the lower poverty lines. For both
poverty lines and both years, inequality among the poor is higher in the
rural area. Thus, not only does the number of rural poor increase when the
poverty line is GEL 45.2, but also the average normalized shortfalls and
inequality across the poor go up.
Quantile Incomes and Quantile Ratios
Besides analyzing poverty, one must understand the situation of the rela-
tively poor population compared to the rest of the population. Table 3.5
reports five quantile per capita expenditures (PCEs) and certain quantile
ratios of per capita consumption expenditure for Georgia and its rural
and urban areas. It compares two different periods: 2003 and 2006. Table
rows denote three geographic regions: urban, rural, and all of Georgia
165
Table 3.5: Quantile PCEs and Quantile Ratios of Per Capita Consumption Expenditure
Percentile
Quantile ratio
10th 20th 50th (median, 80th 90th
(GEL) (GEL) GEL) (GEL) (GEL) 90-10 80-20 90-50 50-10
2003
1 Urban 47.4 64.1 108.4 182.1 229.6 79.3 64.8 52.8 56.3
2 Rural 42.2 58.8 101.5 173.1 230.0 81.6 66.0 55.9 58.4
3 Total 44.8 61.4 104.7 177.0 229.8 80.5 65.3 54.4 57.3
2006
4 Urban 46.7 61.2 101.1 174.0 231.3 79.8 64.8 56.3 53.8
5 Rural 41.0 58.5 105.3 175.9 229.1 82.1 66.8 54.0 61.1
6 Total 43.8 59.8 103.3 175.0 230.5 81.0 65.8 55.2 57.6
Note: PCE = per capita expenditure.
(rows 3 and 6). Per capita consumption expenditure is measured in lari

per month.
Columns A through E denote quantile PCE for five percentiles. Column
A denotes the quantile PCE at the 10th percentile, column B denotes the
quantile PCE at the 20th percentile, and so forth. Columns F through I
report the quantile ratios based on the quantile PCE reported in the first
five columns. Column F, for example, reports the 90/10 ratio, computed as
(quantile PCE at the 90th percentile – quantile PCE at the 10th percentile) /
quantile PCE at the 90th percentile. The larger the 90/10 ratio, the larger is
the gap between these two percentiles.
In 2003, the quantile PCE at the 10th percentile of Georgia is GEL 44.8
[3,A], implying that 10 percent of the Georgian population lives with per
capita consumption expenditure less than 44.8. Similarly, 20 percent of the
Georgian population lives with per capita consumption expenditure less than
61.4 [3,B]. In contrast, 10 percent of the Georgian population lives with per
capita expenditure more than GEL 229.8 [3,E], which is the 90th percentile.
The corresponding 90/10 quantile ratio using these two quantile PCEs
is 80.5 [3,F], which means that the gap between the two percentiles is
80.5 percent of the quantile PCE at the 90th percentile. Stated another
way, the quantile PCE at the 90th percentile is 100 / (100 – 80.5) = 5.1
times larger than the 10th percentile. Likewise, the quantile PCE at the
80th percentile of Georgia is GEL 177.0 [3,D], which is nearly three times
larger than the quantile PCE at the 20th percentile [3,B]. The correspond-
ing 80/20 measure is 65.3 [3,G]. Inequality between the quantile PCE at
166
the 90th percentile per capita expenditure and the quantile PCE at the
10th percentile is larger in the rural area (81.6 [2,F]) than in the urban area
(79.3 [1,F]) in 2003. The 90/10 measure increases for Georgia and both its
urban and rural areas in 2006 [4,F] and [5,A].
This table is helpful in holistically understanding inequality across the per

capita consumption expenditure distribution. The mean and median are
measures of a distribution’s central tendency and the distribution’s size, while
the Gini coefficient is a single measure of the overall distribution that does
not provide any information about which part of the distribution changed.
The four additional quantile PCEs reported in table 3.5 provide infor-
mation about different parts of the distribution. For example, the Gini
coefficient analysis in table 3.1 shows that inequality in the rural area has
decreased, whereas inequality in the urban area has increased. Which part
of the distribution is responsible for such changes? The Gini coefficient does
not provide an answer to this question. A decrease in inequality in the rural
area has not been obtained by increasing the income of the poorest because
the quantile PCE at the 10th percentile in the rural area fell to GEL 41.0 in
2006 [5,A] compared to GEL 42.2 in 2003 [2,A]. The quantile PCE at the
80th percentile increased from GEL 173.1 in 2003 [2,D] to GEL 175.9 in
2006 [5,D]. In other words, even though the Gini coefficient fell, inequality
between the quantile PCEs at the 80th percentile and the 20th percentile
increased in the rural area: from 66.0 in 2003 [2,G] to 66.8 in 2006 [5,G],
according to the 80/20 measure.
Partial Means and Partial Mean Ratios
Table 3.6 reports two lower partial means, two upper partial means, and two
partial mean ratios, based on the partial means between two periods: 2003
and 2006. Table rows denote three geographic regions: urban, rural, and all
of Georgia (rows 3 and 6). Per capita consumption expenditure is measured
in lari per month.
Columns A and B report two lower partial means (LPM), columns C and
D report two upper partial means (UPM), and columns E and F report partial
mean ratios. The first partial mean ratio, for example, reports the 90/10
partial mean ratio, computed as (90th percentile UPM – 10th percentile
167
Table 3.6: Partial Means and Partial Mean Ratios
Lower partial mean Upper partial mean

Partial mean ratio
10th percentile 20th percentile 90th percentile 80th percentile
(GEL) (GEL) (GEL) (GEL) 90-10 80-20
Region A B C D E F
2003
1 Urban 34.5 45.2 319.5 261.8 89.2 82.7
2 Rural 29.0 39.9 321.3 259.1 91.0 84.6
3 Total 31.5 42.3 320.4 260.5 90.2 83.8
2006
4 Urban 34.5 44.3 347.7 273.5 90.1 83.8
5 Rural 27.8 39.0 317.0 258.2 91.2 84.9
6 Total 30.8 41.6 332.0 265.7 90.7 84.4
LPM) / 90th percentile UPM). The larger the 90/10 ratio, the larger is the
gap between these two partial means.
A lower partial mean is the average per capita expenditure of all people
below a specific percentile cutoff. An upper partial mean is the mean per
capita expenditure above a specific percentile. A partial mean ratio captures
inequality between a lower partial mean and an upper partial mean.
It is evident from the table that the average per capita expenditure of the
urban Georgian population’s poorest 20 percent is only GEL 45.2 in 2003
[1,B], whereas the average income of the population’s richest 20 percent is
GEL 261.8 [1,D]. The corresponding 80/20 partial mean ratio is 82.7 [1,F],
which means that the gap between the two partial means is 82.7 percent
of the 80th upper partial mean. Stated another way, the mean per capita
expenditure of the population’s richest 20 percent is 100 / (100 – 82.7) =
5.8 times larger than the mean per capita expenditure of the population’s
poorest 20 percent. Likewise, in rural areas, the mean per capita expendi-
ture of the population’s richest 20 percent (GEL 259.1 [2,D]) is 6.5 times
larger than the mean per capita expenditure of the population’s poorest
20 percent (GEL 39.9 [2,B]) in 2003. The corresponding 80/20 partial mean
ratio is 84.6 [2,F].
In table 3.5, we reported different percentiles of a distribution. For example,

the 10th percentile for Georgia in 2003 is GEL 44.8 [3,A], meaning that
168
10 percent of the Georgian population lives with a per capita expendi-

ture less than GEL 44.8. But what is the average income of these people?
Similarly in table 3.5, 10 percent of the Georgian population has a per
capita expenditure more than GEL 229.8 [3,E], which is the 90th percen-
tile for Georgia, but we do not know exactly how rich this group is. Partial
means are useful for answering this question, and the partial mean ratios tell
us the difference in the average per capita expenditures between a poorer
and a richer group.
Distribution of Population across Quintiles
Table 3.7 analyzes the population distribution in Georgia and its rural and
urban areas across five quintiles of per capita consumption expenditure.
It compares two time periods: 2003 and 2006. Table rows denote three
geographic regions: urban, rural, and all of Georgia (row 1). Per capita
consumption expenditure is measured in lari per month. Each of the five
columns denotes a quintile. Column A denotes the lowest, or first, quintile,
column B denotes the second quintile, and so forth.
All cells in row 1 have a value of 20, obtained by dividing Georgia’s
entire population into five equal groups in terms of per capita expenditure.
Each group contains 20 percent of the population. The fifth quintile con-
tains the richest 20 percent of the population, the fourth quintile consists
of the second-richest 20 percent of the population, and so on, and the first
quintile consists of the poorest 20 percent of the population.
Table 3.7: Distribution of Population across Quintiles

percent
Quintile
First Second Third Fourth Fifth
Region A B C D E
1 Total 20.0 20.0 20.0 20.0 20.0
2003
2 Urban 18.1 19.6 20.4 20.8 21.1
3 Rural 21.8 20.4 19.6 19.2 19.0
2006
4 Urban 19.0 21.6 20.6 19.2 19.7
5 Rural 21.0 18.5 19.4 20.8 20.3
169
Rows 2 and 3 report the population distribution in urban and rural areas
for 2003 using the national quintiles. Consider the value 18.1 [2,A] in the
urban row. This value implies that 18.1 percent of the total urban popula-
tion falls in the first quintile. The next cell is 19.6 [2,B], meaning that
19.6 percent of the total urban population falls in the second quintile. Similarly,
21.1 percent [2,E] of the total urban population falls in the fifth quintile.
The picture is slightly different for the rural area, where 19.0 percent
[3,E] of the total rural population falls in the fifth quintile and 21.8 per-
cent [3,A] falls in the lowest quintile. In 2006, the urban population share
in the first two quintiles increased to 19.0 percent [4,A] and 21.6 percent
[4,B], respectively, but the rural population share in the same two quintiles
decreased to 21.0 percent [5,A] and 18.5 percent [5,B], respectively. In
contrast, the rural population share in the two highest quintiles increased,
[3,D] and [3,E] compared with [5,D] and [5,E], but the urban population
share in the two highest quintiles decreased, [2,D] and [2,E] compared with
[4,D] and [4,E].
This table is helpful in understanding the population’s mobility across dif-

ferent consumption expenditure levels in different regions. A single welfare
measure—inequality or poverty—cannot reflect this mobility.
Analysis at the Subnational Level
Analyses in the previous section concentrate at the national level and

across rural and urban areas. For better policy implementation, we need to
understand the results at a more disaggregated level, such as across subna-
tional or geographic regions, or across population groups having different
characteristics.
In this section, we conduct subnational analysis, and in the next section,
we conduct analysis across other population subgroups. Some tables here are
similar to tables discussed in the previous section, and we occasionally refer
to those tables.
During the analysis across population subgroups, we assume the poverty
line to be the same across all subgroups. However, in the ADePT program
different poverty lines can be used for different subgroups in the analyses.
170
Standard of Living and Inequality
Table 3.8 results from calculating the mean and median per capita consump-
tion expenditure, and the Gini coefficient, for Georgia’s subnational regions.
Columns A and B report the mean per capita consumption expenditure for
years 2003 and 2006, respectively. Column C reports the percentage change
or growth in per capita expenditure over the course of these three years.
The mean per capita expenditure decreases for some regions (such as
Kakheti [1,C], Tbilisi [2,C], and Imereti [9,C]) and increases for others
(such as Shida Kartli [3,C], Kvemo Kartli [4,C], and Samtskhe-Javakheti
[5,C]). Imereti registers the steepest fall (7.0 percent [9,C]) in mean per
capita consumption expenditure, from GEL 150.3 in 2003 [9,A] to GEL
139.9 in 2006 [9,B]. In contrast, Kvemo Kartli reflects the highest increase
in mean per capita expenditure, 16.1 percent [4,C]. It increased from GEL
93.5 in 2003 [4,A] to GEL 108.5 in 2006 [4,B].
Columns D, E, and F report median per capita expenditures and their
growth. Although the change in overall median is −1.4 percent [11,F] (much
larger than the change in overall mean), changes in subnational regions are
mixed. For Kvemo Kartli, the growths of mean and median are almost the
same [4,C] and [4,F]. For Samtskhe-Javakheti, the growth in mean [5,C] is
three times larger than the growth of median [5,F]. In contrast, the growth
Table 3.8: Mean and Median Per Capita Income, Growth, and the Gini Coefficient across
Subnational Regions
Mean Median Gini coefficient

2003 2006 Growth 2003 2006 Growth Change
(GEL) (GEL) (%) (GEL) (GEL) (%) 2003 2006 (%)
1 Kakheti 107.9 102.2 −5.2 92.7 80.4 −13.2 34.4 38.5 4.0
2 Tbilisi 144.5 143.1 −0.9 122.2 111.4 −8.8 32.1 36.4 4.3
3 Shida Kartli 122.9 125.6 2.3 98.7 101.7 3.0 36.6 35.9 −0.7
4 Kvemo Kartli 93.5 108.5 16.1 81.0 94.1 16.2 32.6 32.7 0.1
5 Samtskhe-Javakheti 116.5 121.5 4.3 98.8 100.3 1.5 32.9 31.1 −1.8
6 Ajara 107.8 101.8 −5.6 91.6 83.3 −9.0 33.9 34.4 0.4
7 Guria 134.3 125.6 −6.5 113.9 101.3 −11.1 33.9 35.0 1.1
8 Samegrelo 117.2 125.1 6.7 97.0 109.5 12.8 34.1 32.3 −1.9
9 Imereti 150.3 139.9 −7.0 128.6 122.4 −4.8 33.0 32.9 −0.1
10 Mtskheta-Mtianeti 113.0 123.6 9.3 103.7 96.7 −6.7 33.5 37.4 3.9
11 Total 126.1 126.0 −0.1 104.7 103.3 −1.4 34.4 35.4 0.9
171
of median in Samegrelo [8,F] is twice as large as the growth of mean per

capita consumption expenditure [8,C]. The most interesting pattern can be
seen for Mtskheta-Mtianeti, where the mean grows by 9.3 percent [10,C],
but the median falls by 6.7 percent [10,F].
Columns G, H, and I analyze inequality within subnational regions
using the Gini coefficient, which lies between 0 and 100. Although the
overall Gini coefficient has increased by 0.9 [11,I], a mixed picture is
found across subnational regions. In Tbilisi and Kakheti, inequality rises by
4.3 percent [2,I] and 4.0 percent [1,I], respectively. In Samtskhe-Javakheti,
inequality falls by 1.8 percent [5,I], while in Kvemo Kartli and Imereti, the
Gini coefficient changes by a meager 0.1 [5,I] and [9,I], going up and down,
respectively.
Headcount Ratio and the Distribution of Poor
Table 3.9 analyzes the headcount ratio of Georgia by population subgroup,

where each subgroup is classified by subnational regions—such as Kakheti,
Ajara, and Imereti—which could be states or provinces. The poverty line
for this table is GEL 75.4 per month (we use only one poverty line here, but
the analysis could be conducted for any number of poverty lines).
Table 3.9: Headcount Ratio by Subnational Regions, 2003 and 2006

percent
Headcount ratio Distribution of the poor Distribution of population

1 Kakheti 38.9 46.2 7.3 12.6 13.8 1.3 9.7 9.3 −0.4
2 Tbilisi 20.9 25.2 4.3 17.1 20.4 3.3 24.6 25.2 0.6
3 Shida Kartli 35.2 30.8 −4.5 8.3 7.2 −1.1 7.0 7.2 0.2
4 Kvemo Kartli 44.4 35.1 −9.3 16.8 12.2 −4.6 11.3 10.8 −0.5
5 Samtskhe-Javakheti 30.0 24.4 −5.7 4.6 3.8 −0.8 4.6 4.8 0.2
6 Ajara 37.1 44.6 7.5 10.7 13.7 2.9 8.7 9.5 0.8
7 Guria 25.3 34.4 9.2 2.7 3.5 0.7 3.2 3.1 −0.1
8 Samegrelo 33.5 29.4 −4.1 11.8 9.0 −2.8 10.5 9.5 −1.1
9 Imereti 20.6 23.0 2.3 12.1 13.4 1.3 17.5 18.0 0.5
10 Mtskheta-Mtianeti 34.3 35.2 0.9 3.3 3.1 −0.2 2.9 2.7 −0.2
11 Total 29.9 31.0 1.0 100.0 100.0 n.a. 100.0 100.0 n.a.
Note: n.a. = not applicable.
172
Table rows list subnational regions. Columns A, B, and C analyze

headcount ratios. Columns D, E, and F outline the distribution of poor
people across the subgroups, with the number in the cell being the pro-
portion of all poor people in the country that are included in that sub-
group. Another way of seeing this is the percentage contribution of the
subgroup to overall poverty, or the headcount ratio times the population
share in that group, divided by the overall headcount ratio. Columns G,
H, and I depict the population distribution in subnational regions, or the
percentage of the population that resides in that region. Row 11 shows
that the overall headcount ratio increases from 29.9 percent in 2003
[11,A] to 31.0 percent in 2006 [11,B], reflecting a 1.0 percentage point
(rounded) increase.
In cell [1,A], we find that in 2003, 38.9 percent of the population in
Kakheti is poor. In other words, the headcount ratio for this population
subgroup is 38.9 percent. Cell [1,B] is 46.2, the headcount ratio for the
same population subgroup in 2006. Thus, the headcount increased by
7.3 percentage points [1,C] over the course of these three years. In row 4, we
see that Kvemo Kartli’s headcount ratio decreased by 9.3 percentage points,
from 44.4 percent [4,A] to 35.1 percent [4,B]. The headcount ratio also fell
between 2003 and 2006 in other regions, such as Shida Kartli [3,C] and
Samtskhe-Javakheti [5,C].
Cell [1,D] is 12.6, meaning that of all poor people in Georgia in 2003,
12.6 percent can be found in Kakheti. The share of all poor living in
Kakheti increases to 13.8 percent in 2006 [1,E], an increase of 1.3 percentage
points.
Now compare Kvemo Kartli and Imereti. Clearly, Kvemo Kartli’s pov-
erty headcount ratio (44.4 percent [4,A]) is more than twice as large as
Imereti’s poverty headcount ratio (20.6 percent [9,A]) in 2003. However,
the share of all poor people is only around 40 percent larger in Kvemo
Kartli (16.8 percent in Kvemo Kartli [4,D], compared with 12.1 percent
in Imereti [9,D]). This is due to the different population shares of the two
regions as given in the table’s final columns. The population share living
in Imereti in 2003 is 17.5 percent [9,G], while the Kvemo Kartli share
is only 11.3 percent [4,G]. Therefore, a policy maker should take into
account a region’s population share in addition to the headcount ratio,
because a region may have a lower headcount ratio because of a higher
number of poor.
173
Poverty Gap Measure and Subnational Contribution

to Overall Poverty
Table 3.10 analyzes Georgia’s poverty gap measure across subnational regions.
The poverty line is GEL 75.4 per month. Table rows list subnational regions.
Columns A, B, and C analyze poverty gap measures for 2003, 2006, and the
changes over time. Columns D, E, and F report the percentage contribution
of the subnational regions to the overall poverty gap measure. Columns G,
H, and I depict the population distribution of the subnational regions, or the
percentage of the overall population that resides in each region.
The overall poverty gap measure increases from 9.7 in 2003 [11,A] to
10.1 in 2006 [11,B], reflecting a 0.4 point increase [11,C]. For Kakheti,
the poverty gap measure in 2003 is 13.4 [1,A]. The poverty gap measure
for the same population subgroup in 2006 is 17.8 [1,B]. Thus, the poverty
gap measure increased by 4.4 points [1,C] over three years. The poverty gap
measure in Kvemo Kartli decreased by 3.5 points, from 15.4 in 2003 [4,A] to
11.9 in 2006 [4,B]. The poverty gap measure also fell between 2003 and 2006
in other regions, such as Samegrelo [8,C] and Mtskheta-Mtianeti [10,C].
Kakheti’s contribution to the overall poverty gap measure is 13.4 percent
[1,D]. Its contribution increased to 16.3 percent in 2006 [1,E], an increase
of 2.9 percentage points [1,F].
Table 3.10: Poverty Gap Measure by Subnational Regions
Contribution to Distribution of
Poverty gap measure overall poverty (%) population (%)
1 Kakheti 13.4 17.8 4.4 13.4 16.3 2.9 9.7 9.3 −0.4
2 Tbilisi 5.5 7.3 1.8 14.0 18.2 4.2 24.6 25.2 0.6
3 Shida Kartli 11.7 10.9 −0.8 8.5 7.8 −0.7 7.0 7.2 0.2
4 Kvemo Kartli 15.4 11.9 −3.5 18.1 12.8 −5.3 11.3 10.8 −0.5
6 Ajara 12.8 14.6 1.8 11.5 13.7 2.2 8.7 9.5 0.8
7 Guria 8.3 10.6 2.3 2.8 3.3 0.5 3.2 3.1 −0.1
8 Samegrelo 11.0 8.8 −2.2 12.0 8.2 −3.8 10.5 9.5 −1.1
9 Imereti 6.1 7.5 1.4 11.1 13.4 2.4 17.5 18.0 0.5
10 Mtskheta-Mtianeti 13.1 11.7 −1.4 3.9 3.1 −0.8 2.9 2.7 −0.2
11 Total 9.7 10.1 0.4 100.0 100.0 n.a. 100.0 100.0 n.a.
174
Now compare Guria and Imereti. Clearly, Guria’s poverty gap measure
(8.3 [7,A]) is larger than Imereti’s poverty gap measure (6.1 [9,A]) in 2003.
But Guria’s contribution is only 2.8 percent [7,D], whereas Imereti’s contri-
bution is 11.1 percent [9,D]. The contribution of subnational regions to the
overall poverty gap and the share of poor in each region are quite different.
The share of poor in each of Kakheti and Ajara is almost identical in 2006
(9.3 percent for Kakheti [1,H], compared with 9.5 percent in Ajara [6,H]),
but their contributions to the total poverty gap measure are quite different
(16.3 percent in Kakheti [1,E], compared with 13.7 percent in Ajara [6,E]).
Thus, the average normalized shortfall of per capita expenditure from the
poverty line is much higher in Kakheti, and that is not captured by the
headcount ratio analysis.
Squared Gap Measure and Subnational Contribution

to Overall Poverty
Table 3.11 analyzes Georgia’s squared gap measure across subnational

regions. The poverty line is GEL 75.4 per month. Table rows list subna-
tional regions. Columns A, B, and C analyze the squared gap measure for
2003, 2006, and the difference over time. Columns D, E, and F report the
Table 3.11: Squared Gap Measure by Subnational Regions
Contribution to overall Distribution of

Squared gap measure poverty (%) population (%)
2003 2006 Change (%) 2003 2006 Change 2003 2006 Change
1 Kakheti 6.6 9.4 2.7 14.0 18.2 4.2 9.7 9.3 −0.4
2 Tbilisi 2.1 3.0 0.9 11.4 15.9 4.6 24.6 25.2 0.6
3 Shida Kartli 6.0 5.5 −0.6 9.3 8.2 −1.1 7.0 7.2 0.2
4 Kvemo Kartli 7.8 6.2 −1.7 19.4 13.9 −5.5 11.3 10.8 −0.5
6 Ajara 6.4 6.8 0.5 12.1 13.6 1.5 8.7 9.5 0.8
7 Guria 3.7 4.6 0.9 2.6 3.0 0.4 3.2 3.1 −0.1
8 Samegrelo 5.2 3.7 −1.4 11.9 7.4 −4.5 10.5 9.5 −1.1
9 Imereti 2.7 3.6 0.9 10.3 13.6 3.3 17.5 18.0 0.5
10 Mtskheta-Mtianeti 6.8 5.9 −0.8 4.2 3.3 −0.9 2.9 2.7 −0.2
11 Total 4.6 4.8 0.2 100.0 100.0 n.a. 100.0 100.0 n.a.
175
percentage contribution of the subnational regions to the overall squared

gap measure. Columns G, H, and I depict the population distribution of
the subnational regions, or the percentage of the overall population that
resides in each region. Row 11 shows that the overall squared gap mea-
sure increased from 4.6 in 2003 [11,A] to 4.8 in 2006 [11,B], reflecting a
0.2 point increase [11,C].
The squared gap measure for Kakheti is 6.6 in 2003 [1,A]. The squared
gap measure for the same population subgroup is 9.4 in 2006 [1,B]. Thus,
the squared gap measure increased by 2.7 points in three years [1,C]. The
squared gap measure in Kvemo Kartli decreased by 1.7 points, from 7.8 in
2003 [4,A] to 6.2 in 2006 [4,B]. The squared gap measure also fell between
2003 and 2006 in other regions, such as Samegrelo [8,C] and Mtskheta-
Mtianeti [10,C]. Kakheti’s contribution to the overall squared gap measure
in 2003 is 14.0 percent [1,D]. The contribution increased to 18.2 percent in
2006 [1,E], an increase of 4.2 percentage points [1,F].
Comparing the contribution of subnational regions to the overall squared

gap measure to the contribution to the overall squared gap measure and
the share of poor in each region, we see they are not necessarily the same.
Tbilisi’s contribution to overall poverty in 2006 is larger than Kakheti’s
contribution when poverty is measured by headcount ratio and poverty
gap measure, but Tbilisi’s contribution is lower in 2006 (3.0 [2,B]) than
that of Kakheti (9.4 [1,B]) when poverty is measured using the squared gap
measure. This finding may reflect that inequality across the poor, captured
by the squared normalized shortfalls, is much higher in Kakheti, and that
is not captured by the analysis based on headcount ratio or poverty gap
measure.
Quantile Incomes and Quantile Ratios
In addition to analyzing poverty, understanding how a population’s poor

segment compares to the rest of the population is important. Table 3.12
reports quantile per capita expenditure for five percentiles and certain
quantile ratios of per capita consumption expenditure for Georgia’s sub-
national regions in 2003. Each of the first five columns denotes a quantile
PCE. Column A denotes the quantile PCE at the 10th percentile, column
176
Table 3.12: Quantile PCE and Quantile Ratio of Per Capita Consumption Expenditure, 2003
Quantile PCE
50th
Quantile ratio
10th 20th percentile 80th 90th
percentile percentile (median, percentile percentile 90-10 80-20 90-50 50-10
(GEL) (GEL) GEL) (GEL) (GEL) (%) (%) (%) (%)
1 Kakheti 37.8 52.6 92.7 150.4 191.1 80.2 65.0 51.5 59.2
2 Tbilisi 56.0 74.3 122.2 202.8 252.9 77.9 63.3 51.7 54.2
3 Shida Kartli 38.6 55.9 98.7 169.8 228.4 83.1 67.1 56.8 60.9
4 Kvemo Kartli 34.3 48.3 81.0 126.5 165.1 79.2 61.8 51.0 57.7
5 Samtskhe-Javakheti 43.0 61.2 98.8 160.5 190.2 77.4 61.9 48.0 56.5
6 Ajara 37.8 53.1 91.6 146.5 203.3 81.4 63.7 54.9 58.7
7 Guria 47.7 64.0 113.9 189.1 241.9 80.3 66.1 52.9 58.1
8 Samegrelo 41.2 56.2 97.0 160.7 208.5 80.2 65.0 53.5 57.5
9 Imereti 54.0 74.1 128.6 211.6 267.0 79.8 65.0 51.8 58.0
10 Mtskheta-Mtianeti 33.9 52.5 103.7 162.0 200.1 83.1 67.6 48.2 67.3
11 Total 44.8 61.4 104.7 177.0 229.8 80.5 65.3 54.4 57.3
Note: PCE = per capita expenditure.
B denotes the quantile PCE at the 20th percentile, column C denotes the
median, column D denotes the quantile PCE at the 80th percentile, and
column E denotes the quantile PCE at the 90th percentile.
Columns F through I report the quantile ratios based on the quantiles
reported in the first five columns. Column G, for example, reports the 80/20
ratio, computed as (quantile PCE at the 80th percentile – quantile PCE at
the 20th percentile) / quantile PCE at the 80th percentile. The larger the
80/20 ratio, the larger is the gap between these two percentiles.
In 2003, the quantile PCE at the 10th percentile of Kakheti is 37.8 [1,A],
which implies that 10 percent of the population in Kakheti lives with per
capita consumption expenditure less than GEL 37.8. Similarly, 20 percent
of Kakheti’s population lives with per capita consumption expenditure less
than GEL 52.6 [1,B]. In contrast, 10 percent of people in Kakheti live with
per capita expenditure more than GEL 191.1 [1,E], the quantile PCE at the
90th percentile. The corresponding 90/10 measure using these two quantile
PCEs is 80.2 [1,F], meaning that the gap between the two quantile PCEs is
80.2 percent of the quantile PCE at the 90th percentile. Described another
way, the quantile PCE at the 90th percentile is 100 / (100 – 80.2) = 5.1
times larger than the quantile PCE at the 10th percentile.
Likewise, the quantile PCE at the 80th percentile of Kakheti is
GEL 150.4 [1,D], nearly three times larger than the quantile PCE at the
177
20th percentile per capita expenditure [1,B]. It is evident that Shida Kartli
has a lower quantile PCE at the 10th percentile than Samegrelo but a larger
quantile PCE at the 90th percentile. As a result, the 90/10 quantile ratio of
Shida Kartli [3,F] is higher than the 90/10 quantile ratio of Samegrelo [8,F].
This table is helpful in holistically understanding inequality across the per

capita consumption expenditure distribution. The mean and median measure
a distribution’s central tendency and measure. The Gini coefficient is a single
measure of the overall distribution, but it does not provide any information
about which part of the distribution has changed. The four additional quan-
tile PCEs reported in the table provide further information about different
parts of the distribution.
Partial Means and Partial Mean Ratios
Table 3.13 reports two lower partial means, two upper partial means, and
two partial mean ratios for Georgia’s subnational regions in 2003. Columns
A and B report the two lower partial means, columns C and D report the two
upper partial means, and columns E and F report the partial mean ratios.
The first of the partial mean ratios, for example, reports the 90/10 partial
Table 3.13: Partial Means and Partial Mean Ratios for Subnational Regions,
2003
Lower partial mean Upper partial mean Partial mean ratio (%)
p10 p20 p20 p10 90-10 80-20
Region A B C D E F
1 Kakheti 25.6 35.9 222.3 276.0 90.7 83.8
2 Tbilisi 44.1 54.9 286.7 348.8 87.3 80.9
3 Shida Kartli 26.2 37.1 263.3 331.0 92.1 85.9
4 Kvemo Kartli 23.9 32.3 186.7 230.9 89.6 82.7
5 Samtskhe-Javakheti 30.5 41.5 234.4 294.8 89.6 82.3
6 Ajara 26.2 36.5 222.4 273.4 90.4 83.6
7 Guria 35.9 45.8 275.3 337.2 89.4 83.4
8 Samegrelo 30.8 40.1 241.1 302.9 89.8 83.4
9 Imereti 39.8 52.4 299.1 362.1 89.0 82.5
10 Mtskheta-Mtianeti 25.0 34.7 222.7 265.7 90.6 84.4
11 Total 31.5 42.3 260.5 320.4 90.2 83.8
178
mean ratio, computed as (90th percentile UPM – 10th percentile LPM) /

90th percentile UPM. The larger the 90/10 partial mean ratio, the larger is
the gap between these two partial means.
A lower partial mean is the average per capita expenditure of all people
below a specific percentile cutoff. An upper partial mean is the mean per
capita expenditure above a specific percentile. A partial mean ratio captures
inequality between a lower partial mean and an upper partial mean. In
table 3.5, we reported a distribution’s different quantile PCEs. For example,
the quantile PCE at the 10th percentile of Georgia in 2003 was GEL 44.8,
meaning that 10 percent of the Georgian population lives with per capita
expenditure less than GEL 44.8. However, that does not tell us the average
income of these people. Similarly, 10 percent of the Georgian population
has per capita expenditure more than GEL 229.8, Georgia’s quantile PCE
at the 90th percentile, but we do not know exactly how rich this group
is. Partial means are useful for determining these values, and partial mean
ratios tell us the difference in the average per capita expenditures between
a poorer and a richer group.
It is evident from table 3.13 that the average per capita expenditure of
the poorest 20 percent of people in Ajara is only GEL 36.5 in 2003 [6,B],
whereas the average income of the richest 20 percent of the population is
GEL 222.4 [6,C]. The corresponding 80/20 partial mean ratio is 83.6 [6,F],
meaning that the gap between the two partial means is 83.6 percent of the
80th upper partial mean. Stated another way, the mean per capita expendi-
ture of the population’s richest 20 percent is 100 / (100 – 83.6) = 6.1 times
larger than the mean per capita expenditure of the population’s poorest
20 percent. Likewise, in Shida Kartli, the mean per capita expenditure of
the population’s richest 20 percent (GEL 263.3 [3,C]) is 7.1 times larger
than the mean per capita expenditure of the population’s poorest 20 percent
(GEL 37.1 [3,B]) in 2003. The corresponding 80/20 partial mean ratio is
85.9 [3,F].
A larger inequality in terms of the quantile ratio does not necessarily trans-
late into higher inequality in terms of the partial mean ratio. In table 3.12,
we found that the 80/20 quantile ratio for Imereti (65.0) was larger than
that of Ajara (63.7), but in table 3.13 Ajara’s 80/20 partial mean ratio (83.6
[3,F]) is slightly larger than Imereti’s (82.5 [9,F]).
179
Distribution of Population across Quintiles by Subnational

Region
Table 3.14 analyzes the population distribution in subnational regions across

five quintiles of per capita consumption expenditure. Column 1 denotes the
lowest or the first quintile, column 2 denotes the second quintile, and so
forth.
entire population into five equal-sized groups in terms of per capita expen-
diture. Each group contains 20 percent of the population. The fifth quintile
contains the richest 20 percent of the population; the fourth quintile con-
sists of the second-richest 20 percent of the population, and so on, and the
first quintile consists of the poorest 20 percent of the population.
For the subnational regions, table cells report population percentage in
each quintile. Consider the value 27.6 for Kakheti [2,A]. This value implies
that 27.6 percent of Kakheti’s population lives with per capita expenditure
less than the first quintile. The next cell to the right is 20.9 [2,B], imply-
ing that 20.9 percent of Kakheti’s population falls in the second quintile.
Similarly, only 12.5 percent [2,E] of Kakheti’s population falls in the fifth
quintile.
The picture is slightly different for Imereti, where only 13.3 percent
[10,A] and 15.3 percent [10,B] of its population fall in the first and second
Table 3.14: Distribution of Population across Quintiles by Subnational

Region, 2003
percentage of population
Quintile
Region A B C D E
1 Total 20.0 20.0 20.0 20.0 20.0
2 Kakheti 27.6 20.9 20.8 18.3 12.5
3 Tbilisi 12.4 17.9 19.9 22.5 27.2
4 Shida Kartli 23.0 21.7 17.0 19.7 18.5
5 Kvemo Kartli 30.0 27.5 21.2 13.5 7.9
6 Samtskhe-Javakheti 20.1 24.0 21.6 20.2 14.1
7 Ajara 25.9 22.9 21.7 15.6 13.8
8 Guria 17.4 17.7 20.7 20.4 23.8
9 Samegrelo 23.5 19.8 20.6 21.3 14.7
10 Imereti 13.3 15.3 18.2 22.9 30.3
11 Mtskheta-Mtianeti 25.5 17.5 20.2 21.2 15.6
180
quintiles, respectively, and 30.3 percent [10,E] of its population falls in the
fifth or richest quintile. Kvemo Kartli appears to be the poorest among all
subnational regions because 30.0 percent [5,A] of its population falls in the
poorest quintile and only 7.9 percent [5,E] of its population falls in the rich-
est quintile.
This table is helpful in understanding population mobility across different

consumption expenditure levels in different regions, which a single measure
of welfare, inequality, or poverty cannot reflect.
Subnational Decomposition of Headcount Ratio
Table 3.15 decomposes poverty to explore the factors that caused a change
in headcount ratio. Table rows are divided into two categories. Rows 1
through 4 report the change in the overall poverty and three factors affect-
ing this change: total intrasectoral effect, population-shift effect, and inter-
action effect. Rows 5 through 14 report the intrasectoral effects for various
regions in Georgia.1 Column A reports the absolute change in headcount
Table 3.15: Subnational Decomposition of Headcount Ratio, Changes

between 2003 and 2006
Absolute change Percentage change

A B
1 Change in headcount ratio 1.04 100.00
2 Total intrasectoral effect 1.09 104.98
3 Population-shift effect −0.18 −17.38
4 Interaction effect 0.13 12.40
Intrasectoral effects by region
5 Kakheti 0.70 67.93
6 Tbilisi 1.06 102.38
7 Shida Kartli −0.31 −30.37
8 Kvemo Kartli −1.05 −101.76
9 Samtskhe-Javakheti −0.26 −25.06
10 Ajara 0.65 62.79
11 Guria 0.30 28.70
12 Samegrelo −0.43 −41.25
13 Imereti 0.41 39.18
14 Mtskheta-Mtianeti 0.03 2.44
181
poverty and the size of the factors contributing to this change. Column B
shows how these factors change the headcount ratio.
The change in overall headcount ratio between 2003 and 2006 is 1.04
[1,A]. This overall change of 1.04 percentage points is divided into three
different effects. The first is the total intrasectoral effect, 1.09 [2,A]. The
total population-shift effect is negative and amounts to –0.18 [3,A]. The
interaction between the intrasectoral factor and the population shift fac-
tor is 0.13 [4,A]. If we sum these three effects, we get the overall absolute
change in headcount ratio poverty: (1.09 – 0.18 + 0.13) = 1.04 [1,A].
The next column reports the proportion these effects have relative to the
overall change. The proportion of the total intrasectoral effect on the over-
all change in poverty is 104.98 percent [2,B]. This number is calculated by
dividing the total intrasectoral effect by the change in poverty: (100 × 1.09)
/ 1.04 = 104.98. The corresponding entries for the population-shift effect
and the interaction effect are calculated by the same method. For example,
(100 × –0.18) / 1.04 = –17.38 and (100 × 0.13) / 1.04 = 12.40 [4,B].
The next set of results decomposes the total intrasectoral effect across
Georgia’s regions. Column A reports the size of the intrasectoral effect, and
column B reports the intrasectoral effect as a proportion of the total change
in the overall headcount ratio. For example, the intrasectoral effect for
Kakheti is 0.70 [5,A], and its proportion of the overall poverty change is
67.93 percent [5,B], calculated as (100 × 0.70) / 1.04 = 67.93.
The intrasectoral effect of Kakheti is calculated as the change in
headcount ratio between 2003 and 2006, which is 7.3 percentage points
(reported in column C of table 3.9), multiplied by its population share in
2003 (reported in column G of table 3.9). The intrasectoral effects are nega-
tive for regions such as Shida Kartli, Kvemo Kartli, Samtskhe-Javakheti, and
Samegrelo, because the poverty headcount ratio fell in these regions. For the
rest of the regions, the intrasectoral effects are positive. The contribution of
this effect is highest for Tbilisi and lowest for Kvemo Kartli.
The total intrasectoral effect is even higher than the total change in the
overall headcount ratio. Thus, if the region-wise population shares are
kept constant, then the change in poverty is 1.09 percentage points [2,A].
However, if we keep the regional headcount ratios constant and consider
182
only the changes in regional population shares, then the poverty rate would
have fallen by 0.18 percentage point [3,A]. Thus, the intrasectoral effect
dominates and the overall headcount ratio increases. Finally, the second set
of results gives us an idea about the headcount ratio’s regional contribution
in terms of intrasectoral effect.
Poverty Analysis across Other Population Subgroups
In this section, we discuss the results when the population is divided in various
ways: household head’s characteristics, household member’s employment sta-
tus, education level, age group, demographic composition, and landownership.
Standard of Living and Inequality by Household Head’s

Characteristics
Table 3.16 reports the mean and median per capita consumption expendi-
ture and their growth over time and inequality across the population using
the Gini coefficient across various household characteristics. Table rows
denote various household characteristics. Columns A and B report the
mean per capita consumption expenditure for 2003 and 2006, respectively.
Column C reports the percentage change in per capita expenditure over
these three years. It is evident from rows 1 and 2 that the mean per capita
expenditure goes up by 1.1 percent [2,C] for female household heads but
decreases by 0.5 percent [1,C] for male household heads.
Columns D, E, and F report the median per capita expenditures for 2003
and 2006 and the growth rates between these years. Although the overall
change in median is –1.4 percent [20,F] (much larger than the change in
overall mean of –0.1 percent [20,C]), the changes in the groups with vari-
ous household characteristics are mixed. For female household heads, the
median increases by 1.5 percent [2,F], but it falls by 2.2 percent [1,F] for
male household heads. We find a mixed picture for the other household
characteristics.
Columns G, H, and I report inequality by household head’s characteris-
tics using the Gini coefficient, which lies between 0 and 100. Although the
overall Gini coefficient increases by 0.9 [20,I] in 2006, changes for different
household characteristics vary over a broad range.
183
Table 3.16: Mean and Median Per Capita Consumption Expenditure, Growth, and Gini
Coefficient, by Household Characteristics
Mean per capita Median per capita

consumption expenditure consumption expenditure Gini coefficient
2003 2006 Change 2003 2006 Change
(GEL) (GEL) (%) (GEL) (GEL) (%) 2003 2006 Change
Characteristic of
household head A B C D E F G H I
Gender
1 Male 127.2 126.6 −0.5 106.7 104.3 −2.2 33.7 34.8 1.1
2 Female 122.9 124.3 1.1 98.9 100.4 1.5 36.5 37.0 0.5
Age
3 15–19 110.8 217.8 96.6 90.0 150.7 67.3 16.2 31.7 15.5
4 20–24 188.0 223.5 18.9 131.7 188.3 43.0 40.6 35.0 −5.6
5 25–29 121.1 153.9 27.1 114.8 121.9 6.2 32.1 33.8 1.7
6 30–34 130.1 121.7 −6.5 111.4 98.1 −12.0 33.2 38.1 4.8
7 35–39 121.3 124.2 2.4 103.9 105.1 1.2 32.7 34.3 1.6
8 40–44 127.9 128.5 0.5 109.7 105.1 −4.2 33.8 35.3 1.5
9 45–49 127.6 132.7 4.0 102.9 104.4 1.4 35.7 36.2 0.5
10 50–54 121.5 120.7 −0.6 100.9 105.0 4.1 34.4 32.6 −1.8
11 55–59 134.7 132.8 −1.4 117.0 104.2 −10.9 33.5 38.0 4.5
12 60–64 130.5 123.0 −5.7 109.4 102.5 −6.4 32.3 34.3 2.0
13 65+ 122.8 121.8 −0.8 100.9 99.9 −1.0 35.1 34.8 −0.3
Education
14 Elementary or less 101.3 101.6 0.4 80.9 84.6 4.5 36.5 37.5 1.0
15 Incomplete secondary 109.5 106.7 −2.6 90.8 90.3 −0.5 34.5 33.4 −1.0
16 Secondary 116.2 118.6 2.1 97.3 99.6 2.3 33.7 34.1 0.4
17 Vocational-technical 127.7 116.3 −8.9 107.1 97.5 −9.0 34.6 34.6 0.0
18 Special secondary 134.4 127.5 −5.2 113.1 106.1 −6.2 33.9 33.0 −1.0
19 Higher education 153.7 155.1 0.9 129.7 123.7 −4.7 31.9 36.0 4.1
20 Total 126.1 126.0 −0.1 104.7 103.3 −1.4 34.4 35.4 0.9
Headcount Ratio by Household Head’s Characteristics
Table 3.17 analyzes poverty by population subgroup according to various

household head characteristics. The poverty line is set at GEL 75.4 per month.
Table rows report categories for three household head characteristics:
gender, age, and education level. Columns A, B, and C analyze the pov-
erty headcount ratios for 2003, 2006, and the change between those years.
Columns D, E, and F outline the distribution of poor people across the
subgroups, with the number in the cell being the proportion of all poor
people in the country contained in each subgroup. We can also call this
the subgroup’s percentage contribution to overall poverty, or the headcount
ratio times the population share included in that group. Columns G, H, and
184
Table 3.17: Headcount Ratio by Household Head’s Characteristics

percent
Poverty headcount ratio Distribution of the poor Distribution of population

Characteristic of
household head A B C D E F G H I
Gender
1 Male 28.4 30.0 1.6 69.6 71.5 1.9 73.3 73.6 0.3
2 Female 34.1 33.5 −0.5 30.4 28.5 −1.9 26.7 26.4 −0.3
Age
3 15–19 0 0 0 0 0 0 0 0.1 0.1
4 20–24 18.5 8.2 −10.3 0.3 0.2 −0.2 0.5 0.6 0
5 25–29 33.4 18.4 −15.0 1.3 0.7 −0.7 1.2 1.1 −0.1
6 30–34 26.9 36.2 9.3 3.3 3.1 −0.2 3.7 2.7 −1.0
7 35–39 31.6 31 −0.6 5.7 4.8 −0.9 5.4 4.7 −0.6
8 40–44 28.5 29.9 1.4 9 8.2 −0.8 9.5 8.5 −1.0
9 45–49 30.1 28.2 −1.9 11.9 10.7 −1.3 11.9 11.7 −0.2
10 50–54 32.8 31.1 −1.7 12.7 12.2 −0.5 11.6 12.2 0.6
11 55–59 26.0 30.0 4.0 7.7 11.2 3.5 8.9 11.6 2.7
12 60–64 24.2 32.4 8.2 8.7 7.6 −1.1 10.8 7.3 −3.5
13 65+ 32.1 32.4 0.2 39.2 41.4 2.2 36.5 39.6 3.1
Education
14 Elementary or less 44.2 43.1 −1.0 12.4 10.2 −2.2 8.4 7.3 −1.1
15 Incomplete secondary 38.4 38.7 0.3 12.7 10.0 −2.6 9.9 8.0 −1.9
16 Secondary 33.3 32.5 −0.7 42.9 40.1 −2.9 38.6 38.1 −0.5
17 Vocational-technical 30.2 36.5 6.3 8.4 11.7 3.4 8.3 9.9 1.7
18 Special secondary 26 26.9 0.9 9.9 11.2 1.3 11.4 12.8 1.5
19 Higher education 17.5 21.9 4.4 13.7 16.9 3.1 23.4 23.8 0.3
20 Total 29.9 31.0 1.0 100 100 n.a. 100 100 n.a.
I depict the subgroup population distributions, or the population percent-

age contained in each subgroup. Row 20 shows that overall headcount ratio
increases from 29.9 percent in 2003 [20,A] to 31.0 percent in 2006 [20,B],
reflecting a 1.0 percentage point increase [20,C] in the headcount ratio.
We see that 28.4 percent of male household heads [1,A] are poor. In
other words, the headcount ratio for this population subgroup is 28.4 percent.
The headcount ratio of the same group in 2006 is 30.0 percent [1,B]. So the
headcount ratio for the population in the male-headed household increased
by 1.6 percentage points [1,C] from 2003 to 2006.
In row 4, we find that 18.5 percent [4,A] of the population from house-
holds headed by someone in the 20–24 age group is poor. The headcount
ratio for the same population subgroup in 2006 is 8.2 percent [4,B], a
185
change of –10.3 percentage points [4,C]. In fact, headcount ratios have

also decreased for households with the head in the 25–29 age group [5,C],
35–39 [7,C], 45–49 [9,C], and 50–54 [10,C]. When subgroups are divided
according to household head’s education, we find that the headcount ratio
for the population living in the households where the head’s education is
elementary or less is 44.2 percent [14,A]. In both years, the population in
this subgroup had the highest headcount ratio.
Of all people who were poor in Georgia in 2003, 69.6 percent [1,D]
were from male-headed households. The share of all poor living in male-
headed households increased to 71.5 percent in 2006 [1,E], an increase of
1.9 percentage points [1,F]. In contrast, the share of poor in female-headed
households fell by 1.9 percentage points from 30.4 percent [2,D] in 2003 to
28.5 percent [2,E] in 2006.
There was not a large change in the population share in either male- or
female-headed households. For male-headed households, the proportion
of population increased by 0.3 percentage point from 73.3 percent [1,G]
to 73.6 percent [1,H]. For the female-headed households, the propor-
tion of population decreased by 0.3 percentage point from 26.7 percent
[2,G] to 26.4 percent [2,H]. Similarly, headcount ratios increased from
38.4 percent [15,A] to 38.7 percent [15,B] for the subgroup having house-
hold heads with incomplete secondary education. But the headcount ratio
for the subgroup having household heads with secondary education fell
from 33.3 percent [16,A] to 32.5 percent [16,B]. The shares of poor in both
groups decreased over the course of these three years: from 12.7 percent
[15,D] to 10.0 percent [15,E] for heads with incomplete secondary and
from 42.9 percent [16,D] to 40.1 percent [16,E] for heads with secondary
education.
One might wonder why the share of poor in households with heads
having incomplete secondary education decreased despite the increase in
the headcount ratio. The answer can be found if we look at columns G
and H. Notice that the population share with heads having incomplete
secondary or less decreased from 9.9 percent in 2003 [15,G] to 8.0 percent
in 2006 [15,H]. At the same time, headcount ratios for other subgroups
increased. For example, headcount ratios for the subgroups with household
heads in vocational-technical education and higher education increased by
6.3 [17,C] and 4.4 [19,C] percentage points, respectively. Thus, despite an
increase in headcount ratio, the shares of the poor population decreased for
the subgroup with heads having incomplete secondary education.
186
Population Distribution across Quintiles by Household

Head’s Characteristics
Table 3.18 analyzes the distribution of population across five quintiles of

per capita consumption expenditure by household head’s characteristics.
Column 1 denotes the lowest or first quintile, column 2 denotes the second
quintile, and so forth.
population into five equal-sized groups in terms of per capita expenditure.
Each group consists of 20 percent of the population. The fifth quintile con-
tains the richest 20 percent of the population, the fourth quintile consists
of the second-richest 20 percent of the population, and so on, and the first
quintile consists of the poorest 20 percent of the population.
Table 3.18: Distribution of Population across Quintiles by Household Head’s

Characteristics, 2003
percentage of per capita expenditure
Quintile
Characteristic of household head A B C D E
1 Total 20.0 20.0 20.0 20.0 20.0
Gender
2 Male 18.6 20.2 20.1 20.7 20.3
3 Female 23.8 19.3 19.7 18.0 19.1
Age (years)
4 15–19 0.0 27.1 51.1 17.1 4.8
5 20–24 10.5 19.9 12.1 19.1 38.4
6 25–29 23.4 15.5 13.4 26.1 21.6
7 30–34 16.5 21.5 16.8 23.4 21.8
8 35–39 21.6 19.9 20.0 17.7 20.8
9 40–44 19.5 17.5 21.1 20.9 21.0
10 45–49 21.7 19.9 19.2 19.2 19.9
11 50–54 21.4 20.5 19.9 20.0 18.2
12 55–59 16.2 18.6 18.2 24.1 22.9
13 60–64 15.5 19.5 22.4 20.9 21.8
14 65+ 21.5 21.1 20.3 18.6 18.5
Education
15 Elementary or less 32.5 23.8 16.5 15.9 11.3
16 Incomplete secondary 25.5 23.7 20.1 16.9 13.9
17 Secondary 22.3 22.4 20.3 18.5 16.5
18 Vocational-technical 20.1 18.9 20.1 21.0 19.9
19 Special secondary 17.4 18.1 19.9 22.7 21.8
20 Higher education 10.7 14.4 20.6 23.6 30.7
187
The rows below row 1 report the distribution of population by vari-

ous household head characteristics for 2003 using the national quintiles.
Consider the value 18.6 [2,A] for male household heads. This value implies
that 18.6 percent of the total population in male-headed households lives
with per capita expenditure less than the first quintile. The population
living in male-headed households is 20.2 percent in the second quintile
[2,B]. Similarly, 20.3 percent [2,E] of the population from the male-headed
households falls in the fifth quintile. The population distribution is almost
the same across all five quintiles.
The largest proportion of population living in the lowest quintile belongs
to households headed by someone who has not acquired education beyond
elementary level [15,A]. At the other extreme, the largest proportion of
population living in the highest quintile belongs to the households headed
by someone in the 20–24 age group [5,E].
This table is helpful in understanding population mobility across different

levels of consumption expenditure across different regions that a single wel-
fare, inequality, or poverty measure cannot reflect.
Headcount Ratio by Employment Category
Table 3.19 analyzes Georgia’s headcount ratio by population subgroups

according to household members’ employment category. The poverty line
is set at GEL 75.4 per month. Table rows list employment sectors (agricul-
ture, industry, government, and so on) as well as unemployed and inactive
categories to account for those not working.
Columns A, B, and C analyze poverty headcount ratios for 2003, 2006,
and the change over time. Columns D, E, and F outline the distribution of
poor people across the subgroups, with the number in the cell being the per-
centage of all poor people in the country that are located in that subgroup.
Stated another way, this is the percentage contribution of the subgroup to
overall poverty, or the headcount ratio times the population share in that
group. Columns G, H, and I depict subgroup population distribution, or the
population percentage found in that subgroup. The last row shows that overall
headcount ratio increases from 29.9 percent in 2003 [15,A] to 31.0 percent in
2006 [15,B], reflecting a 1.0 percentage point increase in the headcount ratio.
188
Table 3.19: Headcount Ratio by Employment Category

percent

Employment A B C D E F G H I
Self-employed
1 Agriculture 29.4 28.2 −1.3 23.2 20.2 −3.0 23.6 22.2 −1.4
2 Industry 20.5 32.2 11.7 0.4 0.5 0.1 0.5 0.5 −0.1
3 Trade 23.8 22.1 −1.6 2.5 1.8 −0.7 3.2 2.5 −0.7
4 Transport 19.2 28.9 9.7 0.4 0.7 0.3 0.7 0.7 0.1
5 Other services 20.7 27.8 7.2 0.7 0.9 0.2 1.0 1.0 −0.0
Employed
6 Industry 21.3 24.7 3.4 1.5 1.6 0.0 2.1 2.0 −0.2
7 Trade 19.5 24.1 4.6 1.1 1.1 0.1 1.6 1.5 −0.2
8 Transport 21.1 28.2 7.1 0.7 0.8 0.1 0.9 0.9 −0.1
9 Government 18.9 17.8 −1.1 1.4 1.1 −0.3 2.2 1.8 −0.4
10 Education 19.1 17.4 −1.7 2.1 1.7 −0.3 3.3 3.1 −0.2
11 Health care 16.7 19.1 2.5 0.6 0.7 0.1 1.1 1.2 0.0
12 Other 23.1 24.9 1.8 2.9 3.1 0.2 3.7 3.8 0.1
13 Unemployed 37.3 40.3 3.1 8.9 10.8 1.9 7.2 8.3 1.1
14 Inactive 32.9 33.7 0.8 53.6 55.1 1.5 48.8 50.7 1.8
15 Total 29.9 31.0 1.0 100.0 100.0 n.a. 100.0 100.0 n.a.
We find that 29.4 percent [1,A] of people engaged in the agricultural

sector are poor in 2003. In other words, the headcount ratio for this popula-
tion subgroup (with a household head employed in the agricultural sector) is
29.4 percent. The headcount ratio for the same population subgroup (with a
household head in the agricultural sector) fell to 28.2 percent in 2006 [1,B].
Thus, a 1.3 percentage point decrease [1,C] occurred in the headcount ratio
between the two years. We see that the headcount ratio among members
in the other services sector increased by 7.2 percentage points [5,C], from
20.7 percent [5,A] to 27.8 percent [5,B]. This headcount ratio increase from
2003 to 2006 is found in other sectors, such as employed industry [6,C],
trade [7,C], and transport [8,C].
Of all people who are poor in Georgia in 2003, 23.2 percent [1,D] are
employed in agriculture. We find that the share of all poor employed in
agriculture fell to 20.2 percent in 2006 [1,E]. This represents a decrease of
3.0 percentage points [1,F].
Contrast those results with the figures for the unemployed population
subgroup. Clearly, the poverty headcount ratio among this group in 2003
189
[13,A] is larger than the poverty headcount ratio in 2003 among the sub-
group employed in the agricultural sector [1,A]. However, if we consider
the share of all poor people who are found in these two subgroups in 2003,
this number is nearly twice as large in the agricultural sector as that among
the unemployed group. This is because of the different population shares of
the two subgroups as given in the final columns. The population share in the
agriculture subgroup in 2003 is 23.6 percent [1,G], while the share in the
unemployed subgroup is only 7.2 percent [13,G].
In row 1, the agricultural subgroup’s poverty headcount ratio falls 1.3
percentage points [1,C], while the share of poor in this subgroup falls by
3.0 percentage points [1,F]. For the other services subgroup, the headcount
ratio increased 7.2 percentage points [5,C] between 2003 and 2006, while
the share of poor in this subgroup increased by only 0.2 percentage point,
from 0.7 percent [5,D] to 0.9 percent [5,E].
One might wonder why these two ways of evaluating changes are so
different. Look at columns G and H. Notice that the population share
employed in the agricultural sector is more than 20 percent of the total
population in both 2003 [1,G] and 2006 [1,H]. In comparison, the popu-
lation share engaged in other services is only 1.0 percent in 2003 [5,G]
and 2006 [5,H]. Consequently, a change of smaller magnitude in the
headcount ratio in the agricultural sector has a larger impact on its share
of the poor and vice versa.
Headcount Ratio by Education Level
Table 3.20 analyzes poverty by education levels. The poverty line is set at
GEL 75.4 per month. Columns A, B, and C analyze poverty headcount
ratios for 2003, 2006, and the difference over time. Columns D, E, and F
outline the distribution of poor people across the subgroups, with the num-
ber in the cell being the proportion of all poor people in the country located
in that subgroup. This is the subgroup’s percentage contribution to overall
poverty, or the headcount ratio times the population share in that group.
Columns G, H, and I depict subgroup population distribution, or the popula-
tion percentage in that subgroup. Row 7 shows that the overall headcount
190
Table 3.20: Headcount Ratio by Education Level

percent

Education level A B C D E F G H I
1 Elementary or less 40.4 35.9 −4.6 6.5 5.7 −0.7 4.6 4.1 −0.5
2 Incomplete secondary 36.1 38.2 2.1 14.3 13.9 −0.5 11.5 10.9 −0.6
3 Secondary 33.2 31.9 −1.3 46.8 44.1 −2.6 40.8 41.7 0.9
4 Vocational-technical 30.0 35.0 5.0 7.7 8.5 0.7 7.5 7.3 −0.2
5 Special secondary 25.2 27.7 2.5 10.1 11.2 1.2 11.6 12.2 0.6
6 Higher education 17.6 20.9 3.4 14.6 16.6 1.9 24.1 23.8 −0.3
7 Total 29.9 31.0 1.0 100.0 100.0 n.a. 100.0 100.0 n.a.
ratio increases from 29.9 percent in 2003 [7,A] to 31.0 percent in 2006 [7,B],
reflecting a 1.0 percentage point (rounded) increase in the headcount ratio.
We find that 40.4 percent [1,A] of the population who have elementary-
level education or less are poor. In other words, the headcount ratio for this
population subgroup is 40.4 percent. The headcount ratio for the same popu-
lation subgroup fell to 35.9 percent in 2006 [1,B]. Thus, the headcount ratio
fell by 4.6 percentage points [1,C] between these three years. At the other
extreme, the headcount ratio for the subgroup with higher education increased
by 3.4 percentage points, from 17.6 percent [6,A] to 20.9 percent [6,B].
Of all people who are poor in Georgia in 2003, 6.5 percent [1,D]
have elementary education or less. The share of all poor with elementary
education or less decreased to 5.7 percent in 2006 [1,E], a decrease of 0.7
percentage point [1,F].
Clearly, the poverty headcount ratio among the population with incom-
plete secondary education in 2003 [2,A] is larger than the poverty head-
count ratio in 2003 among the higher education subgroup [6,A]. However,
if we consider the share of all poor people who are found in these two
subgroups in 2003, the number is larger for the population with higher edu-
cation because of the two subgroups’ different population shares, as given
in the table’s final columns. The population share with higher education
in 2003 is 24.1 percent [6,G], whereas the population share with incom-
plete secondary education is only 11.5 percent [2,G]. The headcount ratios
increased for the population with incomplete secondary education from
191
36.1 percent [2,A] to 38.2 percent [2,B], for vocational-technical education

from 30 percent [4,A] to 35 percent [4,B], for special secondary education
from 25.2 percent [5,A] to 27.7 percent [5,B], and for higher education from
17.6 percent [6,A] to 20.9 percent [6,B].
Headcount Ratio by Demographic Composition
Table 3.21 analyzes poverty by population subgroup, where each subgroup

is based first on the number of children 0–6 years of age in the household,
then on the household’s size. The poverty line is set at GEL 75.4 per month.
Columns A, B, and C analyze poverty headcount ratios for 2003, 2006, and
the difference over time. Columns D, E, and F outline the distribution of
poor people across the subgroups, with the number in the cell being the
proportion of poor people in the country contained in that subgroup. This is
the subgroup’s percentage contribution to overall poverty, or the headcount
ratio times the population share that falls in that group. Columns G, H, and
I depict subgroup population distribution, or the percentage of the popula-
tion in that subgroup. Row 12 shows that overall headcount ratio increased
Table 3.21: Headcount Ratio by Demographic Composition

percent

Demographic characteristic A B C D E F G H I
Number of children 0–6 years
1 None 28.8 28.5 −0.4 69.6 66.1 −3.5 72.2 72.0 −0.2
2 1 31.2 36.2 5.0 20.5 22.2 1.7 19.7 19.0 −0.7
3 2 35.5 39.9 4.5 8.3 10.3 2.0 7.0 8.0 1.0
4 3 or more 43.7 40.6 −3.1 1.5 1.3 −0.2 1.0 1.0 −0.0
Household size
5 1 25.8 24.1 −1.7 2.6 2.6 −0.0 3.1 3.4 0.3
6 2 23.1 21.0 −2.1 6.7 5.9 −0.8 8.7 8.7 −0.0
7 3 25.0 23.2 −1.8 11.1 9.9 −1.2 13.3 13.2 −0.1
8 4 24.4 26.2 1.7 19.5 18.5 −1.1 23.9 21.8 −2.1
9 5 31.9 33.8 1.8 23.0 23.0 0.0 21.6 21.1 −0.5
10 6 36.2 35.4 −0.9 19.6 19.1 −0.4 16.2 16.7 0.6
11 7 or more 39.3 43.2 4.0 17.3 20.9 3.6 13.2 15.0 1.8
12 Total 29.9 31.0 1.0 100.0 100.0 n.a. 100.0 100.0 n.a.
192
from 29.9 percent in 2003 [12,A] to 31.0 percent in 2006 [12,B], reflecting
a 1.0 percentage point (rounded) increase in the headcount ratio.
First, consider the results based on the number of children in households.
We find that 28.8 percent [1,A] of the population with no child in the
household is poor in 2003. In other words, the headcount ratio for this popu-
lation subgroup is 28.8 percent. The headcount ratio for the same population
subgroup decreased to 28.5 percent in 2006 [1,B]. Thus, the headcount ratio
decreased by 0.4 percentage point [1,C] over the course of these three years.
Headcount ratios also decreased for the population with three or more
children in the household by 3.1 percentage points from 43.7 percent
[4,A] in 2003 to 40.6 percent [4,B] in 2006. Similarly, consider the set of
results corresponding to the household size. The headcount ratio among the
population with only one member in the household in 2003 is 25.8 percent
[5,A], which falls by 1.7 percentage points to 24.1 percent in 2006 [5,B]. At
the other extreme, the headcount ratio among the people living in house-
holds with seven or more members increased by 4.0 percentage points from
39.3 percent [11,A] to 43.2 percent [11,B].
The next cell in row 1 is 69.6 [1,D], meaning that of all people who are
poor in Georgia in 2003, 69.6 percent of the population live in households
with no child. In the next column, we find that the share of poor with no
child decreased to 66.1 percent in 2006 [1,E], a decrease of 3.5 percentage
points [1,F].
Compare those results with the subgroup having three or more children.
It is evident that the headcount ratio among the subgroup with no child in
both years (28.8 percent in 2003 [1,A] and 28.5 percent in 2006 [1,B]) is
lower than the headcount ratio for the subgroup with three or more children
(43.7 percent in 2003 [4,A] and 40.6 percent for 2006 [4,B]). Note that the
share of the former subgroup to total poverty is 69.6 percent in 2003 [1,D],
which fell by 3.5 percentage points to 66.1 percent in 2006 [1,E]. The share
of the latter to total poverty is 1.5 percent in 2003 [4,D], which fell by
0.2 percentage point to 1.3 percent in 2006 [4,E]. However, in both years,
the share of poor in the former subgroup is more than 40 times higher than
that in the latter subgroup.
Note that the poverty rate among the subgroup with three or more children
is higher than the subgroup with no child. However, the population share
193
in the subgroup with no child is so large (72.2 percent in 2003 [1,G] and
72 percent in 2006 [1,H]), compared to the subgroup with three or more
children (only 1.0 percent in both years [4,G] and [4,H]), that the share of
the subgroup with no child in total poverty is high. The analysis in table
3.21 enables a policy maker to understand the origin of poverty at a more
disaggregated level. A policy maker should also focus on households with no
child, even though the headcount ratio is lowest in this subgroup. Similar
intuition should hold for the next set of results where the subgroups are
based on household size.
Headcount Ratio by Landownership
Table 3.22 analyzes poverty by population household landownership sub-

groups for 2003, 2006, and the change across those years. The poverty line
is set at GEL 75.4 per month. Columns A, B, and C analyze the poverty
headcount ratios. Columns A and B report the headcount ratio for 2003
and 2006, respectively, while column C reports the difference over time.
Columns D, E, and F outline the distribution of poor people across the
subgroups, with the number in the cell being the proportion of poor people
in the country located in that subgroup. This is the subgroup’s percent-
age contribution to overall poverty, or the headcount ratio times the
population share that lies in that group. Columns G, H, and I depict the
subgroups’ population distribution, or the population percentage found in
Table 3.22: Headcount Ratio by Landownership

percent

Size of landholding
(hectares) A B C D E F G H I
1 0 29.4 32.7 3.3 39.0 46.4 7.3 39.7 43.9 4.2
2 Less than 0.2 39.4 36.2 −3.1 12.7 11.9 −0.7 9.6 10.2 0.6
3 0.2–0.5 33.9 36.9 2.9 17.2 18.4 1.1 15.2 15.4 0.2
4 0.5–1.0 25.1 24.3 −0.8 19.5 15.4 −4.1 23.2 19.6 −3.6
5 More than 1.0 28.2 22.4 −5.8 11.5 7.9 −3.6 12.2 10.9 −1.3
6 Total 29.9 31.0 1.0 100.0 100.0 n.a. 100.0 100.0 n.a.
194
each subgroup. Row 6 shows that the overall headcount ratio increases from
29.9 percent in 2003 [6,A] to 31.0 percent in 2006 [6,B], reflecting a
1.0 percentage point (rounded) increase in the headcount ratio.
We find that 29.4 percent [1,A] of people who belong to households
with no landownership are poor in 2003. In other words, the headcount
ratio for this population subgroup is 29.4 percent. The headcount ratio for
the same population subgroup increases to 32.7 percent in 2006 [1,B]. Thus,
the headcount ratio increased by 3.3 percentage points [1,C] over these
three years. We see that the headcount ratio for the population in house-
holds with landownership of 0.5–1.0 hectare decreased by 0.8 percentage
point, from 25.1 percent [4,A] to 24.3 percent [4,B].
Of all poor people in Georgia in 2003, 39 percent [1,D] lived in house-
holds with no landownership. The share of poor with no landownership
increased to 46.4 percent in 2006 [1,E]. The headcount ratio among the
subgroup with landownership of 0.5–1.0 hectare (25.1 percent in 2003
[4,A] and 24.3 percent in 2006 [4,B]) is lower than the headcount for the
subgroup with a landownership of less than 0.2 hectare (39.4 percent in
2003 [2,A] and 36.2 percent for 2006 [2,B]). Note that the share of the
former subgroup to total poverty is 19.5 percent in 2003 [4,D], which fell
by 4.1 percentage points to 15.4 percent in 2006 [4,E]. The share of the
latter to total poverty is 12.7 percent in 2003 [2,D], which fell by only 0.7
percentage point to 11.9 percent in 2006 [2,E]. Note that despite a larger
fall in the poverty rate of 3.1 percentage points [2,C] for the subgroup with
landownership of less than 0.2 hectare, the share of poor in that subgroup
fell by only 0.7 percentage point [2,F]. One might wonder about the reason
behind this phenomenon.
The answer can be found if we look at columns G and H. Notice that the
population share with landownership of less than 0.2 hectare is 9.6 percent
in 2003 [2,G], and it increased by 0.6 percentage point to 10.2 percent in
2006 [2,H]. In contrast, the population share with landownership of 0.5–1.0
hectare fell by 3.6 percentage points, from 23.2 percent [4,G] in 2003 to
19.6 percent [4,H] in 2006. Moreover, the population share in the latter
subgroup is almost twice as high as that in the former subgroup in both
years. Thus, despite a larger fall in headcount ratio for the subgroup with
landownership of less than 0.2 hectare, its share in total number of poor did
not decrease significantly compared to the subgroup with landownership of
0.5–1 hectare.
195
Headcount Ratio by Age Groups
Table 3.23 analyzes poverty by population subgroup according to individuals’

ages. The poverty line is set at GEL 75.4 per month. Columns A, B, and C
analyze poverty headcount ratios for 2003, 2006, and the difference over
time, respectively. Columns D, E, and F outline the distribution of poor
people across the subgroups, with the number in the cell being the propor-
tion of poor people located in that subgroup. This is the subgroup’s percent-
age contribution to overall poverty, or the headcount ratio times the overall
population share that lies in that group. Columns G, H, and I depict the sub-
groups’ population distribution, or the percentage of the population that can
be found in that subgroup. Row 14 shows that the overall headcount ratio
increased from 29.9 percent in 2003 [14,A] to 31.0 percent in 2006 [14,B],
reflecting a 1.0 percentage point (rounded) increase in headcount ratio.
We see that 32.8 percent of the population in age group 0–5 years [1,A]
is poor. In other words, the headcount ratio for this population subgroup
is 32.8 percent. The headcount ratio for the same population subgroup
increased to 34.9 percent in 2006 [1,B]. Thus, the headcount ratio increased
by 2.1 percentage points [1,C] during these three years. In fact, the head-
count ratio increased among all age groups except 50–54 and 65+ years.
Table 3.23: Headcount Ratio by Age Groups

percent

Age group (years) A B C D E F G H I
1 0–5 32.8 34.9 2.1 5.9 6.2 0.2 5.4 5.5 0.1
2 6–14 33.3 34.5 1.2 14.4 12.6 −1.8 12.9 11.3 −1.7
3 15–19 33.3 33.7 0.4 9.6 9.5 −0.1 8.6 8.7 0.1
4 20–24 30.7 31.6 0.9 8.0 8.7 0.7 7.8 8.5 0.8
5 25–29 30.9 31.5 0.7 7.3 7.4 0.1 7.1 7.3 0.2
6 30–34 30.2 32.6 2.4 6.9 6.8 −0.2 6.9 6.4 −0.4
7 35–39 30.2 32.1 1.9 6.8 6.8 −0.0 6.7 6.5 −0.2
8 40–44 27.9 31.4 3.5 7.2 7.0 −0.2 7.7 7.0 −0.8
9 45–49 28.6 29.1 0.5 6.9 6.8 −0.1 7.2 7.2 −0.0
10 50–54 28.3 27.1 −1.1 5.6 5.4 −0.2 6.0 6.2 0.2
11 55–59 23.0 25.8 2.8 3.2 4.5 1.2 4.2 5.4 1.2
12 60–64 23.0 26.7 3.8 3.5 3.0 −0.5 4.5 3.4 −1.1
13 65+ 29.3 28.8 −0.6 14.7 15.5 0.8 15.0 16.6 1.6
14 Total 29.9 31.0 1.0 100.0 100.0 n.a. 100.0 100.0 n.a.
196
Between 2003 and 2006, the headcount ratios decreased for age group
50–54 years by 1.1 percentage points, from 28.3 percent [10,A] to 27.1 per-
cent [10,B], and for the age group 65+ years by 0.6 percentage point from
29.3 percent [13,A] to 28.8 percent [13,B]. In contrast, headcount ratios
increased for all other groups by 0.4 to 3.8 percentage points. For example,
the headcount ratio for age group 30–34 years increased by 2.4 percentage
points from 30.2 percent [6,A] in 2003 to 32.6 percent [6,B] in 2006.
Of all poor people in Georgia in 2003, 5.9 percent are in the age group
of 0–5 years [1,D]. The share of all poor in age group 0–5 years increased to
6.2 percent in 2006 [1,E], an increase of 0.2 percentage point. Now consider
age groups 6–14 and 65+ years. The headcount ratio among the population
in age group 6–14 years increased by 1.2 percentage points from 33.3 percent
in 2003 [2,A] to 34.5 percent in 2006 [2,B], but the headcount fell by 0.6
percentage point for age group 65+ years [13,C]. However, if we consider
the change in share of all poor people found in these two subgroups in 2003
(column F), this number went up for age group 65+ (0.8 [13,F]) and fell for
age group 6–14 years (–1.8 [2,F]).
One might ask why the share of the poor has fallen in spite of an increase in
headcount ratios. The answer can be found in columns G and H. Note that
the share of people in the age group 6–14 years decreased by 1.7 percent-
age points from 12.9 percent in 2003 [2,G] to 11.3 percent in 2006 [2,H].
In contrast, the population share in age group 65+ years increased by 1.6
percentage points from 15.0 percent in 2003 [13,G] to 16.6 percent in 2006
[13,H]. Thus, despite a decrease in headcount ratio for age group 65+ years,
its share of poor increased. A policy maker, therefore, should notice that a
decrease in headcount among the 65+ years age group did not necessarily
decrease the number of total poor in that age group.
Headcount Ratio and Age-Gender Pyramid
Until now, we have analyzed headcount ratios across individual population

subgroups. We have not analyzed the headcount ratio across two different
population subgroups simultaneously. Figure 3.2 presents one such example
using a graph known as an age-gender pyramid. The age-gender pyramid
analyzes the headcount ratios across gender and across different age groups
197
Figure 3.2: Age-Gender Poverty Pyramid
2003
95 33 30 95
90 31 38 90
85 36 31 85
80 31 27 80
75 31 29 75
30 25
Female poverty rate

70 70
Male poverty rate

Age in years 65 23 22 65
Age in years
60 23 21 60
55 25 29 55
50 28 28 50
45 26 28 45
40 31 29 40
35 29 30 35
30 29 30 30
25 31 29 25
20 34 30 20
15 35 34 15
10 32 31 10
5 31 34 5
6 5 4 2 1 0 1 2 4 5 6
Share in total population, %
Total population Poor population
simultaneously. However, it can be used to analyze other subgroups with

proper justification. As before, the variable for our analysis is per capita
consumption expenditure in lari, and the poverty line is set at GEL 75.4 per
month. The outside vertical axes denote the age of the members in years,
and the horizontal axis presents the share of the population.
The figure is divided vertically by gender: the right-hand side repre-
sents males and the left-hand side represents females. The distance from
the middle to each side in dark gray denotes the total population share in
that age group. The distance in light gray is the proportion of poor people
in that age group of the total number of poor, again for each gender. Data
are aggregated in five-year increments, and each increment is displayed as
a bar centered on the highest age in the increment. The data for ages 25 to
30 years, for example, are represented by the bar at 30 years. For those zero
to five years of age, the shares of both males and females are 2.2 percent,
and nearly 0.7 percent of both males and females in that age group reside in
198
poor households. The headcount ratio among females in that age group is
32 percent and among males it is 31 percent. The headcount ratio is highest
among male members in the 85–90 years age group: 38 percent of the males
in that age group reside in poor households. The largest headcount ratio
among females is seen in the 80–85 years age group.
Sensitivity Analyses
In this section, we perform sensitivity analysis of poverty line choice, pov-

erty measures, and inequality measures, mostly at the national level and
across urban and rural areas. In certain cases, the results are reported at the
subnational levels or across geographic regions. However, all sensitivity
analysis can be replicated at any disaggregated level.
Elasticity of FGT Poverty Indices to Per Capita Consumption
Table 3.24 provides a tool for checking the sensitivity of the three poverty
measures to consumption expenditure. The table shows the result of increas-
ing everyone’s consumption expenditure by 1.0 percent and compares those
values across two years, 2003 and 2006. There are two poverty lines: GEL
75.4 and GEL 45.2 per month.
The percentage change in poverty caused by a 1.0 percent change in the
mean or average per capita consumption expenditure is referred to as the
elasticity of poverty with respect to per capita consumption. The particular way
Table 3.24: Elasticity of FGT Poverty Indices to Per Capita Consumption Expenditure
Headcount ratio Poverty gap measure Squared gap measure

1 Urban −1.89 −1.72 0.17 −1.95 −2.03 −0.09 −2.09 −2.23 −0.14
2 Rural −1.66 −1.53 0.13 −1.71 −1.64 0.07 −1.82 −1.72 0.09
3 Total −1.77 −1.62 0.15 −1.81 −1.82 0.00 −1.93 −1.93 0.00
4 Urban −2.06 −2.35 −0.29 −2.36 −2.47 −0.12 −2.24 −2.50 −0.26
5 Rural −1.87 −1.64 0.23 −1.86 −1.78 0.07 −1.94 −1.86 0.08
6 Total −1.95 −1.94 0.01 −2.05 −2.04 0.01 −2.05 −2.06 −0.02
Note: FGT = Foster-Greer-Thorbecke.
199
in which we consider an increase in the average per capita consumption

expenditure is by increasing everyone’s consumption expenditure by the
same percentage. This type of change is distribution neutral, because the
relative inequality does not change.
The main columns denote three different sets of poverty measures:
headcount ratio, poverty gap measure, and squared gap measure. The first
two columns within each set report the elasticities for 2003 and 2006,
respectively, while the third column reports the difference between these
two years.
Let us start with the GEL 75.4 per month poverty line. The elasticity of
poverty with respect to the mean consumption expenditure for the urban
area in 2003 is –1.89 [1,A]. In other words, if the consumption expenditure
increases by 1.0 percent for everyone, then the mean per capita consump-
tion expenditure increases by 1.0 percent and the urban headcount ratio
falls by –1.89 percent, or stated differently, 1.89 percent of the population
who were living under the poverty line of GEL 75.4 will be out of poverty.
If the mean consumption expenditure is increased by 1.0 percent, then
the headcount ratio for the urban area falls by 1.72 percent in 2006 [1,B]. A
higher value implies higher sensitivity. The urban headcount ratio elasticity
is less sensitive to consumption expenditure in 2006 than in 2003 by 0.17
[1,C]. Similarly, the elasticity of poverty gap to the per capita consumption
expenditure for the urban area in 2003 is –1.95 [1,D], which increases by
–0.09 (rounded) to –2.03 in 2006 [1,E]. The elasticity of squared gap mea-
sure in 2003 is –2.09 [1,G], which increases by –0.14 to –2.23 in 2006 [1,H].
Negative elasticities mean a fall in poverty caused by an increase in con-
sumption expenditure. The higher magnitude implies higher elasticity even
though the sign is negative. Note that both the poverty gap measure and the
squared gap measure, unlike the headcount ratio, are more sensitive to con-
sumption expenditure in 2006 than in 2003. A similar pattern is seen for the
GEL 45.2 per month poverty line: the poverty gap measure and the squared
gap measure are more sensitive to the per capita consumption expenditure.
All elasticities in the rural area are lower in magnitude compared to
what we see in the urban area for both poverty lines. In other words, all
rural poverty measures are less sensitive to the per capita consumption
expenditure. The overall headcount ratio elasticity decreases slightly from
–1.95 in 2003 [6,A] to –1.94 in 2006 [6,B] for the GEL 45.2 poverty line,
but it decreases by 0.15 from –1.77 in 2003 [3,A] to –1.62 in 2006 [3,B] for
the GEL 75.4 poverty line. The elasticities of the overall poverty gap and
200
the squared gap measure did not change much between these two years for
either poverty line.
Note that poverty lines are set normatively, which is difficult to justify
exclusively. A slight change in per capita consumption expenditure may or
may not change the poverty rates by significant margins. If the distribution is
highly polarized or, in other words, if the society has two groups of people—
one group consisting of rich people and the other group consisting of extreme
poor—then a slight change in everyone’s income by the same proportion
may not affect the headcount ratio.
In contrast, if marginal poor are concentrated around the poverty line,
then a slight change in everyone’s income by the same proportion would
have a huge impact on the poverty measures. For example, in the table
the poverty measures are more sensitive to the lower GEL 45.2 per month
poverty line than the higher GEL 75.4 per month poverty line. This is
because the concentration of poor around the lower poverty line is much
larger than that around the higher poverty line. Hence, this type of analysis
may tell us about the impact of any policy on the poverty rate used by the
policy maker.
Sensitivity of Poverty Measures to the Choice of Poverty Line
Table 3.25 presents a tool for checking the sensitivity of the headcount ratio
with respect to the chosen poverty line. This exercise is similar to the exer-
cise for checking the elasticity of poverty measures to per capita consump-
tion expenditure, but it is more rigorous. It is always possible to find a certain
percentage of decrease in the poverty line that matches the increase in the
consumption expenditure for everyone by 1.0 percent. In this exercise,
we check the sensitivity of the poverty measure by changing the poverty
line in more than one direction. Thus, in the table, we ask how the actual
headcount ratio changes as the poverty line changes from its initial value,
whether it is GEL 75.4 per month or GEL 45.2 per month.
Rows denote the change in poverty line, both upward and downward.
Columns report the change in three poverty measures: the headcount ratio,
the poverty gap measure, and the squared gap measure, and their change
from actual. The variable is per capita consumption expenditure, measured
201
Table 3.25: Sensitivity of Poverty Measures to the Choice of Poverty Line, 2003
Headcount Change from Poverty gap Change from Squared gap Change from
ratio actual (%) measure actual (%) measure actual (%)
A B C D E F
1 Actual 29.9 0.0 9.7 0.0 4.6 0.0
2 +5 percent 32.6 9.0 10.7 10.7 5.1 11.4
3 +10 percent 35.3 18.0 11.7 21.7 5.6 23.3
4 +20 percent 40.5 35.2 13.9 44.3 6.8 48.5
5 −5 percent 26.9 −10.0 8.7 −10.2 4.1 −10.8
6 −10 percent 24.2 −19.1 7.7 −19.9 3.6 −21.1
7 −20 percent 19.4 −35.3 6.0 −38.1 2.7 −40.0
8 Actual 10.2 0.0 3.0 0.0 1.4 0.0
9 +5 percent 11.4 11.8 3.4 12.2 1.5 12.4
10 +10 percent 12.7 24.1 3.8 25.2 1.7 25.6
11 +20 percent 15.8 54.5 4.7 53.8 2.1 54.6
12 −5 percent 9.2 −9.9 2.7 −11.6 1.2 −11.6
13 −10 percent 8.0 −21.4 2.4 −22.4 1.1 −22.4
14 −20 percent 6.0 −40.9 1.8 −41.6 0.8 −41.4
in lari. In this table, we report the results only for 2003, but this analysis can
be conducted for any year.
Column A reports the headcount ratios for different poverty lines, and
column B reports the change in the headcount ratios from the actual pov-
erty line, which can be either GEL 75.4 per month or GEL 45.2 per month.
Rows 2 and 9, corresponding to +5 percent, denote the increase in poverty
line by 5 percent. Thus, when the poverty line is GEL 75.4, then a 5 percent
increase means the poverty line becomes GEL 79.2 and the headcount ratio
increases by 3.7 percentage points from 29.9 percent [1,A] to 32.6 percent
[2,A], or the headcount ratio increases by 9.0 percent [2,B] from its actual
level of 29.9 percent.
Similarly, if the poverty line is decreased by 10 percent (–10 percent)
from GEL 75.4, then the poverty headcount rate falls by 5.7 percentage
points from 29.9 percent [1,A] to 24.2 percent [6,A], or the headcount ratio
decreases by 19.1 percent from the actual level of 29.9 percent. The head-
count ratio is more sensitive to the change in poverty line when the actual
poverty line is GEL 45.2 than when the poverty line is GEL 75.4. In fact,
the poverty gap measure and the squared gap measure are also more sensitive
to change in poverty line when the actual poverty line is GEL 45.2 rather
than GEL 75.4.
202
This table helps us understand the robustness of a particular poverty esti-

mate. Selection of any poverty line is debatable, because it is set with nor-
mative judgment. If a change in the poverty line causes a drastic change in
a poverty measure, then a cautious policy conclusion should be drawn from
the analysis based on that particular poverty line. In contrast, if a poverty
measure does not vary much because of a change in the poverty line, then a
more robust conclusion can be drawn.
Other Poverty Measures
Table 3.26 analyzes the overall poverty for Georgia and decomposes it across
rural and urban areas using three other poverty measures not in the FGT class.
The table reports three different sets of poverty measures: the Watts index,
Sen-Shorrocks-Thon (SST) index, and Clark-Hemming-Ulph-Chakravarty
(CHUC) index (these measures are defined in chapter 2). This is a type of
sensitivity analysis, but of the poverty measurement methodology. There are
two poverty lines: GEL 75.4 per month and GEL 45.2 per month.
Columns A and B report the Watts index for both years. The Watts
index is the mean log deviation relative to the poverty line. It is evident
from row 1 that the urban Watts index increases from 12.0 in 2003 [1,A]
to 12.7 in 2006 [1,B] when the poverty line is GEL 75.4 but falls slightly
between the same years when the poverty line is GEL 45.2 [4,A] and [4,B].
Table 3.26: Other Poverty Measures
Watts index Sen-Shorrocks-Thon index CHUC index

A B C D E F G H I
1 Urban 12.0 12.7 0.7 15.7 16.8 1.1 16.6 16.5 0.0
2 Rural 15.6 16.2 0.5 19.2 19.6 0.4 22.2 22.8 0.6
3 Total 13.9 14.5 0.6 17.5 18.3 0.7 19.6 19.8 0.3
4 Urban 3.3 3.2 −0.1 4.7 4.6 0.0 5.1 4.5 −0.6
5 Rural 5.2 5.7 0.5 7.0 7.6 0.6 8.5 9.0 0.4
6 Total 4.3 4.5 0.2 5.9 6.2 0.3 6.9 6.8 −0.1
Note: CHUC = Clark-Hemming-Ulph-Chakravarty.
203
Columns D and E report the SST index, which is also based on the
headcount ratio, the income gap ratio, and the Gini coefficient across the
censored distribution of consumption expenditure. The last is obtained
by replacing consumption expenditure of all nonpoor people by the pov-
erty line. We see that when the poverty line is GEL 75.4, the SST index
for the urban region in 2003 is 15.7 [1,D], and it increases by 1.1 to 16.8
in 2006 [1,E]. Likewise, the rural region’s SST index increased by 0.4,
from 19.2 in 2003 [2,D] to 19.6 in 2006 [2,E], for the same poverty line.
The total increase in SST index is 0.7, from 17.5 in 2003 [3,D] to 18.3
in 2006 [3,E].
The final three columns report the CHUC index and its changes across
time. Unlike the SST index, the CHUC index does not reflect an increase
in poverty across all regions. In fact, urban poverty falls marginally between
2003 [1,G] and 2006 [1,H] when the poverty line is GEL 75.4. Furthermore,
when the poverty line is set at GEL 45.2, the CHUC index shows a fall in
Georgia’s overall poverty [6,I].
If these three measures, capturing different aspects of poverty and inequal-

ity among the poor, agree with the results from the measures in the FGT
class, then the poverty analysis is robust. In contrast, if these measures do
not agree with each other, the policy conclusion should be drawn with more
care. Comparing table 3.2 with table 3.26, we see that the three measures
reported in table 3.2 do not always agree with the results based on the
poverty gap measure and squared gap measure. Thus, any conclusion about
whether poverty has increased or decreased should be made cautiously.
Other Inequality Measures
Table 3.27 reports the Atkinson inequality measures and generalized entropy
measures for 2003, then decomposes them across different regions. This is
a type of sensitivity analysis for inequality measurement methodology. We
report the Gini coefficient only in the last two sections of this chapter.
However, the Gini coefficient may not be subgroup consistent (subgroup
consistency is defined in chapter 2). Rows denote results for urban and rural
population subgroups and for different geographic regions, such as Kakheti,
Tbilisi, and Shida Kartli.
204
Table 3.27: Atkinson Measures and Generalized Entropy Measures by

Geographic Regions, 2003
Atkinson measure Generalized entropy measure

A(1/2) A(0) A(−1) GE(0) GE(1) GE(2)
A B C D E F
1 Urban 9.1 17.7 34.3 19.4 18.8 22.8
2 Rural 10.1 19.8 38.9 22.0 21.0 25.6
Subnational regions
3 Kakheti 9.8 19.2 39.1 21.3 20.1 24.4
4 Tbilisi 8.3 15.9 29.8 17.3 17.3 20.8
5 Shida Kartli 11.0 21.6 44.8 24.4 22.8 28.2
6 Kvemo Kartli 8.9 17.3 33.9 19.0 18.6 24.0
8 Ajara 9.4 18.5 36.5 20.4 19.2 22.6
9 Guria 9.4 18.2 35.7 20.1 19.9 27.2
10 Samegrelo 9.5 18.3 35.3 20.2 19.7 23.7
11 Imereti 8.8 17.3 33.8 19.0 18.0 20.8
13 Total 9.6 18.8 36.8 20.8 20.0 24.2
Note: GE = generalized entropy.
Columns A, B, and C report the Atkinson measures for a = 1/2, 0, and –1,
respectively, and columns D, E, and F report the generalized entropy measures
for a = 0, 1, and 2, respectively. (For a theoretical discussion on the Atkinson
inequality measure and generalized entropy measures, please refer to chapter
2.) Intuitively, an Atkinson inequality measure of order a is the gap between
the mean achievement and the generalized mean of achievements of order
a divided by the mean achievement. Generalized mean is sensitive to inequal-
ity across the distribution, where a lower value of a reflects higher sensitivity
to inequality across the distribution. In other words, a lower value of a reflects
higher aversion toward inequality and, thus, it is also known as the inequality
aversion parameter. When everyone has identical achievement, then it does
not matter how sensitive one is toward inequality, so the generalized mean
is equal to the arithmetic mean for all a. For the analysis in table 3.27, the
inequality measures put more emphasis on the lower end of the distribution
and thus assume a < 1. The Atkinson measure lies between 0 and 1. Similarly,
if a household has equal per capita expenditure in a region, then the general-
ized entropy measure is also 0 for all a.
The Atkinson measure for a = 1/2, or A(1/2), for the urban area is
9.1 [1,A]. Intuitively, the number implies that the generalized mean of
205
order 0.5 for urban Georgia is 9.1 percent lower than Georgia’s mean per
capita expenditure in 2003. The next two cells to the right report A(0) and
A(–1) for urban Georgia, where A(0) = 17.7 [1,B] and A(–1) = 34.3 [1,C].
Therefore, A(0) is 17.7 percent lower than the mean per capita expenditure
and A(–1) is 34.3 percent lower than the mean per capita expenditure.
Columns D, E, and F report three generalized entropy measures for a = 0, 1,
and 2, denoted by GE(0), GE(1), and GE(2), respectively.
Row 2 reports the three Atkinson measures and three generalized
entropy measures for rural Georgia. Each of these six measures shows that
rural Georgia is more unequal than urban Georgia. For example, the A(1/2)
for the rural area is 10.1 [2,A], compared with 9.1 in the urban area [1,A],
and A(0) for the rural area is 19.8 [2,B], compared with 17.7 for the urban
area [1,B]. However, the difference is much larger when the two regions are
compared with respect to A(–1): 38.9 for the rural area [2,C] and 34.3 for
the urban area [1,C].
Next, we consider the results across regions. The level of inequality
of Ajara according to A(1/2) is 9.4 [8,A], which is higher than that of
Samtskhe-Javakheti at 9.0 [7,A]. This means that Ajara has larger income
inequality than Samtskhe-Javakheti. Even according to A(0), A(–1),
GE(0), and GE(–1), Ajara has higher income inequality than Samtskhe-
Javakheti. However, in terms of GE(2), which gives more weight to larger
incomes across the population, Samtskhe-Javakheti [7,F] has higher income
inequality than Ajara [8,F].
A region’s income standards reflect that region’s welfare level. However,

higher welfare does not necessarily mean more equal distribution. A high
level of inequality may be detrimental to a region’s welfare. We already
reported the Gini coefficient for that purpose. However, given that the Gini
coefficient has certain limitations, we report three Atkinson inequality mea-
sures and three generalized entropy measures to check the inequality ranking
for regions. These six inequality measures are commonly used separately
from the Gini coefficient.
Also unlike the Gini coefficient, Atkinson and generalized entropy
class inequality measures are normative measures, in which we may choose
varying degrees of inequality aversion. If these six measures agree with the
Gini coefficient, then a conclusion based on the Gini coefficient can be
206
considered robust. However, if these six measures provide different rankings

than the Gini coefficient, then a more cautious policy conclusion should be
drawn based only on Gini.
Dominance Analyses
In the previous section, we conducted some dominance analysis with respect

to the choice of poverty lines and measurement methodologies. In this sec-
tion, we perform additional dominance analyses. Note that when we analyze
sensitivity with respect to the poverty line, we do not compare the results
for all poverty lines. Similarly, when we check the sensitivity of inequal-
ity using different Atkinson and generalized entropy measures, we do not
conduct the analysis for all parameter values. The dominance tests in this
part of the chapter go beyond the sensitivity analyses. For example, accord-
ing to the dominance analyses in this section, we can say that poverty has
unambiguously risen for all poverty lines, or inequality has risen, no matter
which inequality measure is used to assess it.
Poverty Incidence Curve
A poverty incidence curve is the distribution function of the welfare indi-

cator across the population. The poverty incidence curve is useful while
performing a dominance analysis of the headcount ratio with respect to the
poverty line. In this dominance exercise, the welfare indicator is per capita
consumption expenditure, assessed by lari. The horizontal axis of figure 3.3
denotes per capita consumption expenditure. The height of the poverty
incidence curve at any per capita consumption expenditure denotes the
proportion of people having less than that per capita expenditure.
Therefore, the link between the poverty incidence curve and the head-
count ratio is clear. The height of the poverty incidence curve is the head-
count ratio when the poverty line is set at a particular per capita consumption
expenditure. For a poverty line, a larger height denotes a larger headcount
ratio or a larger share of the population having per capita expenditure below
the poverty line. If the poverty incidence curve of a distribution lies to the
right of the poverty incidence curve of another distribution, then the former
distribution is understood to have an unambiguously lower headcount ratio
or the former distribution has lower headcount ratios for all poverty lines.
207
Figure 3.3: Poverty Incidence Curves in Urban Georgia, 2003 and 2006
Urban
1.0
0.8
0.6
0.4
0.2
0
0 160 320 480 640 800
Welfare indicator
2003 2006
Note: The red vertical line is the poverty line.
Figure 3.3 graphs the poverty incidence curves for urban Georgia in
2003 and 2006. The vertical axis reports the headcount ratio. The solid
line denotes the poverty incidence curve for 2003, while the dashed line
denotes the poverty incidence curve for 2006. We saw earlier that the urban
headcount ratio is higher in 2006 for both poverty lines: GEL 75.4 and GEL
45.2. What about other poverty lines? Can we say that poverty has unam-
biguously fallen for any poverty line? Figure 3.3 suggests that we may not
be able to. If we set the hypothetical poverty line somewhere between GEL
320 and GEL 480, then the headcount ratio would have been lower in 2006
than that in 2003.
Although such a poverty line seems very high and unlikely to be set at that
value, the main point of the exercise is clear. When two poverty incidence
curves cross, then an unambiguous judgment cannot be made. The crossing
208
may take place at a much lower level, as happened in the rural area. We
have already seen that the headcount ratio showed an increase in 2006
when the poverty line is set at GEL 75.4 but showed a decrease when the
poverty line is set at GEL 45.2. Given the infinite number of possible pov-
erty lines, it would be cumbersome to check them all one by one. Instead,
the poverty incidence curve is a convenient way of checking for dominance
(if two poverty incidence curves never cross). If dominance does not hold,
then the graph can tell us which part is responsible for the ambiguity.
Poverty Deficit Curve
A poverty deficit curve is useful while performing a dominance analysis of the

poverty gap measure with respect to the poverty line. In this dominance
exercise, the welfare indicator is per capita consumption expenditure,
assessed by lari. In figure 3.4, the horizontal axis denotes the welfare indica-
tor or per capita consumption expenditure. The height of the poverty den-
sity curve is proportional to the poverty gap measure, so that a larger height
Figure 3.4: Poverty Deficit Curves in Urban Georgia, 2003 and 2006
Urban
300
200
Total deficit
100
0
0 160 320 480 640 800
Welfare indicator
2003 2006
209
for a poverty line denotes a larger poverty gap measure. If a distribution’s

poverty deficit curve lies to the right of another distribution’s poverty deficit
curve, then the former distribution is understood to have an unambiguously
lower poverty gap measure, or the former distribution has lower poverty gap
measures for all poverty lines.
Figure 3.4 graphs the poverty deficit curves of urban Georgia for 2003
and 2006. The vertical axis reports total deficit, which is directly propor-
tional to the poverty gap measure for the corresponding poverty line. The
solid line denotes the poverty deficit curve for 2003, while the dashed line
denotes the poverty deficit curve for 2006. We saw earlier that the urban
poverty gap measure is higher in 2006 for both poverty lines: GEL 75.4 and
GEL 45.2. What about other poverty lines? Can we say that poverty has
unambiguously fallen for any poverty line? The graph suggests that we may
not be able to. If we set the hypothetical poverty line to about GEL 320,
then the poverty gap measure would have been lower in 2006 than in 2003.
Although such a poverty line seems very high and unlikely to be set at that
value, the main point of the exercise is clear. When two poverty deficit
curves cross, then an unambiguous judgment cannot be made based on the
poverty gap measure. Given the infinite number of possible poverty lines, it
would be cumbersome to check them all one by one. Instead, the poverty
deficit curve is a convenient way of checking for dominance (if two poverty
deficit curves never cross). If dominance does not hold, then the graph can
tell us which part is responsible for the ambiguity.
Poverty Severity Curve
A poverty severity curve is useful when performing a dominance analysis of

the squared gap measure with respect to the poverty line. In this dominance
exercise, the welfare indicator is the per capita consumption expenditure,
assessed by lari. In figure 3.5, the horizontal axis denotes the welfare indica-
tor or the per capita consumption expenditure. The height of the poverty
severity curve is proportional to the squared gap measure, so that a larger
height for a poverty line denotes a larger squared gap. If a distribution’s pov-
erty severity curve lies to the right of another distribution’s poverty severity
curve, then the former distribution is understood to have an unambiguously
210
Figure 3.5: Poverty Severity Curves in Rural Georgia, 2003 and 2006
Rural
100
80
Total severity, '000
60
40
20
0
0 160 320 480 640 800
Welfare indicator
2003 2006
lower squared gap, or the former distribution has a lower squared gap for all
poverty lines.
Figure 3.5 graphs the poverty severity curves of rural Georgia for 2003 and
2006. The figure’s vertical axis reports total severity, which is directly propor-
tional to the squared gap measure of the corresponding poverty line. The solid
line denotes the poverty severity curve for 2003, while the dashed line denotes
the poverty severity curve for 2006. We saw earlier that the rural squared gap
measure is higher in 2006 for both poverty lines: GEL 75.4 and GEL 45.2.
What about the other poverty lines? Can we say that poverty has unambigu-
ously fallen for any poverty line? The figure suggests that we may not be able
to. One of the poverty severity curves does not lie below another poverty
severity curve for all poverty lines. When two poverty severity curves cross,
then an unambiguous judgment cannot be made based on the squared gap
211
measure. Given the infinite number of possible poverty lines, it would be

cumbersome to check them all one by one. Instead, the poverty severity
curve is a convenient way of checking for dominance (if two poverty sever-
ity curves never cross). If dominance does not hold, then the graph can tell
us which part is responsible for the ambiguity.
Growth Incidence Curve
Figure 3.6 graphs the growth incidence curve of Georgia’s per capita con-
sumption expenditure. The vertical axis reports the growth rate of consump-
tion expenditure between 2003 and 2006, and the horizontal axis reports
the per capita consumption expenditure percentiles. We earlier reported the
growth rate of mean per capita consumption expenditure and found that the
overall growth rate was slightly negative. We also compared the median and
four other quantile incomes.
Figure 3.6: Growth Incidence Curve of Georgia between 2003 and 2006
Urban
3
2
Annual growth rate %
–1
–2
–3
1 10 20 30 40 50 60 70 80 90 100
Expenditure percentiles
Growth-incidence 95% confidence bounds
Growth at median Growth in mean
Mean growth rate
212
However, that analysis does not give us the entire picture, so we perform
this dominance analysis through a growth incidence curve that graphs the
growth rate of per capita consumption expenditure for each percentile of the
population. The height of a growth incidence curve for a particular percentile
of population is the per capita consumption expenditure growth of that per-
centile. In fact, a growth incidence curve assesses how the quantile incomes
change over time. If the growth rates of the lower quintiles are larger than
the growth rates of the upper quintiles, then the growth is said to be pro-poor.
The dotted-dashed straight line denotes the growth in mean per capita
expenditure, which is negative in this case. It is not necessary that the entire
population received an equal share of this growth. It is evident from the fig-
ure that the per capita expenditure growth rate for the population’s higher
percentiles between 2003 and 2006 is much larger and more positive than
that for their lower percentile counterparts. Even though growth has been
mixed throughout the quantile incomes, the lowest quantile income has a
large negative growth. Given that the growth rate was negative, this means
that the population’s poorer section had a proportionally larger decrease in
its per capita expenditure.
What we can state by looking at the figure is that the quantile ratios—
such as 90/10, 80/20, or 90/50—increased between 2003 and 2006. The
shaded area around the growth incidence curve reports the 95 percent con-
fidence bounds. Can we say something about poverty? Yes, we can. For an
absolute poverty line, the headcount ratio between 2003 and 2006 should
not fall because the per capita expenditures of the population’s lower per-
centile decreased. Thus, growth in Georgia between 2003 and 2006 was not
pro-poor.
Lorenz Curve
Figure 3.7 graphs the Lorenz curve of urban Georgia’s per capita expenditure
for 2003 and 2006. The vertical axis reports the share of total consumption
expenditure, and the horizontal axis reports the percentile of per capita
expenditure. A Lorenz curve graphs the share of total consumption expendi-
ture spent by each percentile of the population. Thus, the height of a Lorenz
curve for a particular percentile is the share of total consumption expenditure
spent by that percentile out of the region’s total consumption expenditure.
The Lorenz curve’s height is 1 when the percentile is 1. In other words, the
share of the total consumption expenditure spent by the entire population is
213
Figure 3.7: Lorenz Curves of Urban Georgia, 2003 and 2006
Urban
1.0
0.8
Lorenz curve
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1.0
Cumulative population proportion
2003, Gini=33.49 2006, Gini=35.65

Line of equality
100 percent. The diagonal straight line denotes the situation of perfect
equality: each person has the same per capita expenditure.
As inequality increases, the Lorenz curve bows out, and the area between
the Lorenz curve and the line of perfect equality increases. The area between
a Lorenz curve and the line of perfect equality is proportionally related to
the Gini coefficient: it is twice the Gini coefficient. If a distribution’s Lorenz
curve lies completely to the right of another Lorenz curve, then the former
distribution has unambiguously lower inequality, and any Lorenz-consistent
measure—such as the Gini coefficient, the Atkinson class of indices, and
the generalized entropy measures—ranks the former distribution as less
unequal. If the Lorenz curves of two distributions cross, we cannot unam-
biguously rank those two distributions, even when one is ranked as more
unequal than another by all the Lorenz-consistent measures we discussed
earlier. Therefore, the Lorenz curve provides an opportunity to conduct a
sensitivity analysis for the reported inequality measures.
214
The solid line represents the Lorenz curve for 2003, while the dotted
line corresponds to 2006. It is evident that the dotted curve lies nowhere to
the left of the solid curve. This implies that the inequality in urban Georgia
unambiguously increased in 2006 compared with 2003. If these two curves
had crossed, then the reported inequality measures would not have neces-
sarily agreed with each other.
Standardized General Mean Curve
Dominance in terms of the Lorenz curves is not very common. Therefore,

for inequality comparisons, we need to rely on various measures we cov-
ered earlier. We reported the Atkinson measures and generalized entropy
measures in addition to the Gini coefficient. The Gini coefficient is not
subgroup consistent, which means that if inequality in one region increases
but remains the same in another region, the overall inequality may fall. We
also showed in chapter 2 that each generalized entropy for a < 1 is a mono-
tonic transformation of the Atkinson inequality measures, and for a ≠ 1 it
is a monotonic transformation of the general means. However, we report the
Atkinson measures and the generalized entropy measures for only certain
values of parameter a. This exercise should be understood as a dominance
analysis of the Atkinson measures and the generalized entropy measures.
Figure 3.8 graphs the standardized general mean curve of Georgia’s
per capita expenditure for 2003 and 2006. The vertical axis reports the
standardized general mean of per capita expenditure, where standardiza-
tion is done by dividing the general mean of per capita expenditures by
their mean. The horizontal axis reports parameter a, which is the degree
of generalized mean and also known as the degree of a society’s aversion
toward inequality.
The general mean of a distribution tends toward the maximum and the
minimum per capita expenditure in the distribution when a tends to ∞
and – ∞, respectively. Given that the largest per capita expenditure in any
distribution is usually several times larger than the mean per capita expendi-
ture, allowing a to be very large prevents meaningful analysis. Therefore, we
restrict a to between – 5 and 5, which we consider large enough. The height
of a standardized general mean curve for a particular value of parameter a
is the general mean per capita expenditure divided by the mean per capita
expenditure. The height of any standardized general mean curve is 1 at a = 1.
215
Figure 3.8: Standardized General Mean Curves of Georgia, 2003 and 2006
Total
3.0
2.4
Generalized mean
1.8
1.2
0.6
0
–5 –4 –3 –2 –1 0 1 2 3 4 5
Alpha
2003 2006
The solid line represents Georgia’s standardized general mean curve in

2003, while the dashed line represents Georgia’s standardized general mean
curve in 2006. If a standardized general mean curve lies completely above
another standardized general mean curve to the left of a = 1 and completely
below to the right of a = 1, then every Atkinson inequality measure and
generalized entropy measure for a ≠ 1 agree that the former distribution has
lower inequality than the latter. It is evident from the figure that for larger
values of parameter a, inequality in 2006 has worsened. However, for a
less than 1, inequality has not significantly deteriorated. The standardized
general mean curve is a convenient way of verifying the robustness of the
Atkinson inequality measures and the generalized entropy measures.
Advanced Analysis
In this chapter’s final section, we discuss certain advanced analysis methods.

These techniques require knowledge of regression analysis. We assume read-
ers have the required background.
216
Consumption Regression
Table 3.28 analyzes determinants of the variable used for measuring welfare
(the per capita consumption expenditure in this case). Rows denote the set
of regressors (such as logarithm of household size, share of children in the
age group of 0–6 years, share of male adults, share of elderly) and a set of
Table 3.28: Consumption Regressions
2003 2006
Urban Rural Urban Rural
Coef SE Coef SE Coef SE Coef SE
Factors A B C D E F G H
Household characteristics
1 Log of household size −0.093 0.06 −0.010 0.05 −0.001 0.06 0.051 0.05
2 Log of household size squared −0.020 0.03 −0.078*** 0.02 −0.102*** 0.03 −0.114*** 0.02
3 Share of children age 0–6 years (dropped) (dropped) (dropped) (dropped)
4 Share of children age 7–16 years −0.252*** 0.09 0.223** 0.09 0.249** 0.10 0.076 0.09
5 Share of male adults −0.064 0.10 0.254*** 0.09 0.477*** 0.11 0.251*** 0.10
6 Share of female adults −0.004 0.10 0.453*** 0.10 0.592*** 0.11 0.435*** 0.10
7 Share of elderly (age ≥60 years) −0.124 0.11 0.462*** 0.10 0.488*** 0.12 0.355*** 0.10
Characteristics of household head
8 Log of household head’s age −0.063 0.05 0.076 0.05 −0.318*** 0.05 0.210*** 0.05
Regions
9 Kakheti (dropped) (dropped) (dropped) (dropped)
10 Tbilisi 0.446*** 0.05 (dropped) 0.258*** 0.05 (dropped)
11 Shida Kartli 0.182*** 0.06 0.147*** 0.03 −0.050 0.06 0.182*** 0.03
12 Kvemo Kartli 0.061 0.06 0.075** 0.03 −0.023 0.06 0.183*** 0.03
13 Samtskhe-Javakheti −0.115* 0.06 0.185*** 0.03 0.231*** 0.07 0.163*** 0.04
14 Ajara 0.226*** 0.06 −0.035 0.04 0.103* 0.06 0.067* 0.04
15 Guria −0.077 0.08 0.250*** 0.03 0.030 0.08 0.131*** 0.04
16 Samegrelo 0.112** 0.06 0.194*** 0.03 −0.007 0.06 0.238*** 0.03
17 Imereti 0.270*** 0.05 0.529*** 0.03 0.208*** 0.05 0.381*** 0.03
18 Mtskheta-Mtianeti −0.060 0.07 0.164*** 0.03 0.020 0.08 0.144*** 0.04
sland
19 0 ha (dropped) (dropped) (dropped) (dropped)
20 Less than 0.2 ha 0.121*** 0.03 0.162*** 0.05 0.104*** 0.03 0.166*** 0.04
21 0.2–0.5 ha 0.180*** 0.04 0.356*** 0.04 0.138*** 0.05 0.193*** 0.03
22 0.5–1.0 ha 0.255*** 0.05 0.478*** 0.04 0.125* 0.07 0.365*** 0.03
23 More than 1.0 ha 0.021 0.09 0.565*** 0.05 0.192** 0.08 0.484*** 0.04
Gender of household head
24 Male (dropped) (dropped) (dropped) (dropped)
25 Female −0.073*** 0.02 −0.002 0.02 −0.101*** 0.02 −0.027 0.02
Education of household head
26 Elementary or less (dropped) (dropped) (dropped) (dropped)
27 Incomplete secondary −0.067 0.07 0.034 0.03 0.226*** 0.07 0.086*** 0.03
28 Secondary 0.021 0.06 0.105*** 0.03 0.179*** 0.06 0.196*** 0.03
29 Vocational-technical 0.118* 0.06 0.147*** 0.04 0.225*** 0.07 0.255*** 0.04
30 Special secondary 0.156*** 0.06 0.217*** 0.03 0.269*** 0.06 0.322*** 0.04
31 Higher education 0.289*** 0.06 0.274*** 0.03 0.441*** 0.06 0.477*** 0.04
(continued)
217
Table 3.28: Consumption Regressions (continued)
2003 2006
Coef SE Coef SE Coef SE Coef SE
Factors A B C D E F G H
Employment status of household head
Self-employed
32 Agriculture (dropped) (dropped) (dropped) (dropped)
33 Industry −0.028 0.09 0.430*** 0.09 −0.122 0.11 0.208* 0.11
34 Trade 0.082 0.05 0.275*** 0.05 0.056 0.06 0.193*** 0.06
35 Transport 0.026 0.08 0.311*** 0.07 −0.039 0.08 0.311*** 0.07
36 Other services 0.072 0.07 0.340*** 0.08 −0.099 0.07 0.033 0.09
Employed
37 Industry −0.043 0.06 0.127** 0.06 −0.036 0.06 0.140** 0.06
38 Trade −0.094 0.06 0.144 0.09 −0.024 0.07 0.115 0.10
39 Transport −0.021 0.06 0.212*** 0.08 −0.174** 0.07 0.282*** 0.08
40 Government −0.041 0.06 0.227*** 0.06 0.012 0.07 0.277*** 0.08
41 Education −0.037 0.06 0.054 0.06 −0.029 0.07 0.045 0.07
42 Health care −0.041 0.08 0.085 0.15 −0.039 0.08 0.279* 0.15
43 Other −0.120** 0.05 0.150*** 0.04 −0.022 0.06 0.005 0.05
44 Unemployed −0.376*** 0.05 −0.138** 0.06 −0.325*** 0.05 −0.066 0.06
45 Inactive −0.219*** 0.04 −0.117*** 0.02 −0.169*** 0.05 −0.067*** 0.02
Other
46 Constant 4.851*** 0.21 3.425*** 0.19 5.328*** 0.20 2.976*** 0.20
47 Number of observations 4,525 7,106 4,112 6,773
48 Adjusted R2 0.18 0.20 0.17 0.16
Note: Coef = coefficient. ha = hectare. SE = standard error, sland = area of land owned.
*** p < 0.01, ** p < 0.05, * p < 0.1.
dummy variables (such as regional dummies, gender dummies, dummies for

household head education, and dummies for household head employment
status). Columns report regression coefficients (Coef) and standard errors
(SE) of four different ordinary least square regressions, where the depen-
dent variable, or the regressand, is the logarithm of per capita consumption
expenditure. The four regression results correspond to the urban and rural
areas for 2003 and 2006.
Each regression result has two columns. The first column reports regres-
sion coefficients and the second reports standard errors of the coefficients.
A regression coefficient of any regressor indicates the change in the regres-
sand caused by a one-unit increase in that regressor. The standard error
of a regressor indicates the reliability of its coefficient. Standard errors are
always positive, and a higher standard error indicates lower reliability of the
coefficient.
218
Rows 46, 47, and 48 report the intercept term, number of observations,
and adjusted R-squares (R2), respectively. The intercept term, or constant
term, denotes the level of the consumption expenditure logarithm not
explained or determined by any regressors, or adjusted R-square denoted
power of prediction of all regressors, or the model’s goodness-of-fit. If the
adjusted R-square is 1, then the regressors predict the regressand with com-
plete accuracy. If a regressor’s p-value is less than 0.01, then *** is added
to the coefficient. If the p-value is less than 0.05, then ** is added to the
coefficient. Finally, if the p-value is less than 0.1, then * is added to the
coefficient. P-values denote regressors’ significance level.
Note that all variables in the regions, sland, gender of household head, edu-
cation of household head, and employment status of household head categories
are dummy variables or binary variables. They take a value of only 0 or 1.
A binary variable coefficient denotes the change in regressand when
the dummy variable’s value changes from 0 to 1. For example, consider the
coefficient of the regressor Female in the household head gender category for
urban regression in 2003. The coefficient is –0.073 [25,A], implying that the
per capita expenditure logarithm for a member in a female-headed household
is 0.073 units lower than that of a male-headed household. The regressor’s
standard error is 0.02 [25,B] with a p-value less than 0.01 (indicated by ***
after the regressor), and thus the coefficient is highly significant. The coef-
ficient of the same regressor for urban regression in 2006 is –0.101 [25,E]
with a p-value of less than 0.01, implying that the per capita consumption
expenditure gap between female- and male-headed households increased
over the three-year period.
The table provides a detailed analysis of the determinants of per capita

consumption expenditure. If we focus on column A, it is evident that vari-
ables such as the share of children age 7–16 years [row 43], female-headed
households [row 25], and household head unemployed [row 44] and inactive
employment [row 45] status have significant negative effects on per capita
consumption expenditure for the urban area in 2003.
In contrast, the variables such as 0.5–1.0 hectare of landholding [row 22],
household head higher education [row 31], and living in Imereti [row 17]
have a significant positive impact on per capita expenditure for both urban
and rural areas in both years. Hence, the analysis summarized in table 3.28
219
provides a tool to understand per capita consumption expenditure determi-

nants and to develop appropriate poverty eradication policies.
Changes in the Probability of Being in Poverty
Table 3.29 analyzes changes in the probability of being in poverty using a

probit regression model based on the consumption regression presented in
table 3.28. Rows denote changes in values for various variables—such as
change from having no children 0–6 years old to having one child, change
Table 3.29: Changes in the Probability of Being in Poverty

percent
2003 2006
Variables A B C D
Demographic event, child born in the family
1 Change from having no children 0–6 years old to having 1 child 2.0 18.0 31.5 17.8
2 Change from having no children 0–6 years old to having 2 children 4.7 33.1 57.9 33.5
Land acquisition event
3 Change from “0 ha” to “less than 0.2 ha” −18.9 −15.3 −16.6 −18.0
4 Change from “0 ha” to “0.2–0.5 ha” −27.4 −33.6 −21.8 −20.9
5 Change from “0 ha” to “0.5–1.0 ha” −37.4 −44.5 −19.8 −38.5
6 Change from “0 ha” to “over 1.0 ha” −3.5 −51.9 −29.5 −49.6
Change of household head (following divorce, migration, and so forth)
7 Change from “Male” to “Female” 13.0 0.2 18.4 3.9
Education event: change in household head’s education
8 Change from “Elementary or less” to “Incomplete 10.5 −4.5 −28.0 −10.3
secondary”
9 Change from “Elementary or less” to “Secondary” −3.3 −13.6 −22.4 −23.0
10 Change from “Elementary or less” to “Vocational-technical” −17.6 −18.8 −27.8 −29.5
11 Change from “Elementary or less” to “Special secondary” −23.0 −27.2 −33.0 −36.7
12 Change from “Elementary or less” to “Higher education” −40.3 −33.7 −51.4 −51.8
Sector of employment event: household head’s sector of employment
13 Change from “Agriculture” to “Industry” 5.7 −53.1 25.3 −28.1
14 Change from “Agriculture” to “Trade” −15.6 −36.5 −10.6 −26.2
15 Change from “Agriculture” to “Transport” −5.2 −40.5 7.7 −40.3
16 Change from “Agriculture” to “Other Services” −13.8 −43.8 20.3 −4.7
17 Change from “Agriculture” to “Industry” 8.8 −17.7 7.1 −19.4
18 Change from “Agriculture” to “Trade” 19.6 −20.0 4.8 −16.0
19 Change from “Agriculture” to “Transport” 4.2 −28.7 36.8 −37.0
20 Change from “Agriculture” to “Government” 8.4 −30.7 −2.3 −36.4
21 Change from “Agriculture” to “Education” 7.5 −7.8 5.7 −6.4
22 Change from “Agriculture” to “Health Care” 8.3 −12.0 7.8 −36.7
23 Change from “Agriculture” to “Other” 25.3 −20.9 4.3 −0.7
24 Change from “Agriculture” to “Unemployed” 87.4 20.7 72.1 9.7
25 Change from “Agriculture” to “Inactive” 48.2 17.4 35.6 9.8
Source: Based on consumption regression presented in table 3.28.
220
from owning 0 hectare of land to > 1 hectare of land, and change from
male-headed household to female-headed household. Columns report the
percentage changes in the probability of being in poverty for rural and urban
areas and across 2003 and 2006.
Recall from our discussion about table 3.28 that the interpretation of
dummy or binary variables is different from that of continuous variables.
A dummy variable, unlike a continuous variable, may take only a value of
either 0 or 1. Table 3.28 described how the probability of being in poverty
changes as values of certain variables change.
The probability of being in poverty in 2003 increased by 2.0 percent
[1,A] if an individual moved from an urban household with no children in
the 0–6 years age group to an urban household with one child in the same
age group, all other factors being identical. The probability of being in pov-
erty in 2003 is increased by 18.0 percent [1,B] if an individual moved from
a rural household with no children in the 0–6 years age group to a rural
household with one child in the same age group, all else being identical. In
the urban area, the increase in the probability of being in poverty in 2006
for the same reason is 31.5 percent [1,C].
Similarly, in 2003 if an individual moved from a male-headed urban
household to a female-headed urban household, all else being identical,
then the probability of being in poverty increased by 13.0 percent [7,A].
If an individual moved from a male-headed rural household to a female-
headed rural household, all else being identical, then the probability of
being in poverty increased by only 0.2 percent [7,B]. The largest increase in
the probability of being in poverty in 2003 in the urban area occurred when
an individual moved from a household where the head is employed in the
agricultural sector to a household where the head is unemployed [24,A], all
else being identical.
The table provides a detailed analysis of how the probability of being in pov-
erty changes when some of the crucial determinants of poverty are adjusted.
Note that if the household head’s education in the urban area in 2006
increased from elementary education or less to secondary education, all else
remaining equal, then the probability of a member in that household being
in poverty fell by 22.4 percent [9,C]. Similarly, in rural Georgia for both
years, if the household head transferred from the agricultural sector to any
221
other employment sector, all else being equal, then the probability of being
in poverty fell. Hence, this analysis provides a tool to better understand the
source of poverty and what type of policy would be more efficient in terms
of eradicating poverty.
Growth and Redistribution Decomposition of Poverty Changes
Table 3.30 decomposes the change in poverty into a change in the mean per
capita consumption expenditure and a change in distribution of consump-
tion expenditure around the mean, following Huppi and Ravallion (1991).
Table rows denote three regions—urban, rural, and all of Georgia—for two
different poverty lines. The per capita consumption expenditure is measured
in lari per month. Poverty lines are set at GEL 75.4 (poor) and GEL 45.2
(extremely poor) for each household and household member. For simplicity
in this table, we present the decomposition for headcount ratio only, but the
technique is equally applicable to other poverty measures in the FGT class.
Columns A and B report the headcount ratio of the three regions for
years 2003 and 2006, respectively, and column C reports the changes over
time. Columns D, E, and F decompose the change in the headcount ratio
between 2003 and 2006 into three different terms. Column D reports the
effect of growth on poverty, referred to as the growth effect. Column E reports
the effect of redistribution on poverty and is called the redistribution effect.
Column F reports the interaction term and is referred to as the interaction
effect, following Huppi and Ravallion (1991).
Table 3.30: Growth and Redistribution Decomposition of Poverty Changes,

Headcount Ratio
percent
2003 2006 Actual change Growth Redistribution Interaction

Region A B C D E F
1 Urban 28.1 30.8 2.7 0.6 1.9 0.1
2 Rural 31.6 31.1 −0.5 −0.7 −0.1 0.3
3 Total 29.9 31.0 1.0 0.0 1.0 0.0
4 Urban 8.9 9.3 0.4 0.3 0.0 0.1
5 Rural 11.4 12.1 0.7 −0.2 1.0 0.0
6 Total 10.2 10.7 0.5 0.0 0.5 0.0
222
It is evident from the table that the overall headcount ratio in 2003 is
29.9 percent [3,A], which increased to 31.0 percent in 2006 [3,B]. These
numbers can be verified from table 3.2. The actual change in the overall
headcount ratio is 1.0 percentage point (rounded) [3,C]. The actual change
is broken down into three components: growth effect, redistribution effect,
and interaction effect. By looking at the corresponding figures in columns D,
E, and F, we see that the change is caused mainly by redistribution rather
than growth. We can verify from table 3.1 that growth in mean is negligible
compared to change in inequality in terms of the Gini coefficient.
The picture is slightly different for the urban and rural areas. The urban
headcount ratio rose by 2.7 percentage points from 28.1 percent [1,A] to
30.8 percent [1,B], with both growth effect and redistribution effect being
positive. The urban redistribution effect [1,E] is more than three times
larger than the urban growth effect [1,D]. For the rural area, the headcount
ratio fell from 31.6 percent [2,A] to 31.1 percent [2,B]. In this case, both the
growth effect [2,D] and the redistribution effect [2,E] are negative.
The appendix contains additional tables and figures that may be helpful
in understanding concepts and results in terms of the data for Georgia in
2003 and 2006.
Note
1. For technical details, see Huppi and Ravallion (1991).
Reference
Huppi, M., and M. Ravallion. 1991. “The Sectoral Structure of Poverty dur-
ing an Adjustment Period: Evidence for Indonesia in the Mid-1980s.”
World Development 19 (12): 1653–78.
223
Chapter 4
Frontiers of Poverty Measurement
As conditions change and policy concerns evolve, there is a steady demand

from countries and institutions for new tools to evaluate poverty. In this
chapter, we briefly discuss frontier technologies that are, at the time of this
writing, in various stages of being implemented in the ADePT software.
Most are refinements of the traditional approach to poverty measurement,
but some elaborate on related concepts of inequality and income standards.
Ultra-Poverty
Our first enhancement builds on a theme that originally led to the con-
struction of poverty measures beyond the headcount ratio, namely, that
within the poor population important differences exist in the nature of
poverty. The headcount ratio P0 ignores these differences by valuing each
poor person equally without regard to the depth of poverty. Measures like
the poverty gap P1 reflect the depth of poverty among the poor, while oth-
ers like the FGT (Foster-Greer-Thorbecke) index P2 take into account its
distribution by emphasizing those with the largest gaps. The measurement of
ultra-poverty carries this differentiation one step further by focusing on the
poorest of the poor.
People who are most impoverished according to some well-defined cri-
terion are often the subject of special concern. The poverty experienced
by this group is often called “extreme” or “acute.” Here we use the term
ultra-poverty to describe the condition of poorest poor. Who are the ultra-
poor and how can their poverty be measured? The answer depends on the
225
underlying concept of poverty and the availability of data. The traditional

monetary approach to poverty would suggest focusing on people more deeply
income deprived. A second chronic poverty approach might define the
ultra-poor as those who are more persistently deprived. If many different
achievements are being measured, those who are more multiply deprived may
be the ultra-poor. Alternatively, deprivation that is more spatially concen-
trated might be associated with ultra-poverty. The discussion here focuses on
the first of these: ultra-poverty as deep deprivation in income.
In addition to the usual poverty line z that signifies the minimum accept-
able level for the population under consideration, we now assume that an
even lower ultra-poverty line zu < z is used to identify a more deeply deprived
group called the ultra-poor. One method of evaluating ultra-poverty is to
apply a traditional poverty measure P to the income distribution x given
the lower line zu. The resulting level P(x;zu) could then be used to evaluate
ultra-poverty in a way entirely analogous to the way poverty is evaluated
using P(x;z) at the usual poverty line. Indeed, the pair P(x;z) and P(x;zu)
could be used in concert to gauge the extent to which poverty and ultra-
poverty change across time and space.
A difficulty with this approach is that, aside from the special case of the
headcount ratio, the levels of ultra-poverty and poverty obtained are not
directly related to each other. For example, P1(x;zu) identifies fewer people
than P1(x;z), but because zu is smaller than z, the normalized gaps of the
ultra-poor are also reduced in P(x;zu). The ultra-poverty line zu is playing
two roles here: the cutoff by which the set of ultra-poor people is identified
and the standard against which shortfalls are evaluated in the aggregation.
An alternative would be to use the ultra-poverty line zu in the first role and
the standard poverty line z in the second. Ultra-poverty would be measured
commensurate with overall poverty figures and would allow a straightfor-
ward calculation of the importance of the ultra-poor in a country’s overall
poverty.
Hybrid Poverty Lines
It was argued above that an absolute poverty line zu may not be sustainable
when a large change occurs in the size of the income distribution. A similar
observation applies when comparing two countries at very different levels of
development using an absolute line. The problem is that when the income
226
Chapter 4: Frontiers of Poverty Measurement
standard varies a great deal, it seems reasonable that the poverty line should
reflect this change, at least to some extent. Yet an absolute poverty line, by
definition, is fixed and independent of any changes in the income standard.
Stated differently, when an income standard (such as the mean) changes by
1 percent, an absolute poverty line changes by 0 percent, so that the income
elasticity of the poverty line is zero.
An alternative approach uses a relative poverty line zr, defined as a fixed
proportion of a given income standard. For example, 60 percent of median
income is a relative poverty line used in the European Union. For relative
poverty lines, if a country’s income standard changes by 1 percent, then
the poverty line also changes by 1 percent, implying that the poverty line’s
income elasticity is one. An argument against this approach is that it makes
the poverty line too sensitive to changes in the income standard.
Several approaches have been explored to negotiate the landscape
between the extremes of absolute and relative poverty lines. Foster (1998)
suggests a hybrid poverty line that is a weighted geometric mean of rela-
tive and absolute poverty lines. In symbols, the poverty line is z = zrrza1-r,
where 0 ≤ r ≤ 1. Note that r can be interpreted as the income elasticity of
the hybrid poverty line, because when zr’s income standard rises by 1 percent,
the relative component zr rises by 1 percent, which, in turn, increases the hybrid
poverty line z by r percent.
On the one hand, if parameter r is set to zero so the entire weight is
given to the absolute component, then the hybrid poverty line becomes the
absolute poverty line where the elasticity is zero. On the other hand, if r is
one so the full weight is on the relative component, then the hybrid poverty
line becomes the relative poverty line and the elasticity is one. If 0 < r < 1,
then the hybrid poverty line will lie between the absolute and relative lines
and have an elasticity between zero and one.
How is the elasticity to be set? One approach is to estimate the param-
eter using data on existing poverty lines and income levels. Foster and
Székely (2006) regress poverty lines on private consumption per capita for
92 household surveys across 18 countries and find an elasticity of 0.36. A
second approach is to select “reasonable” values and check for robustness.
Madden (2000), for example, analyzed Irish poverty using the 1987 and
1994 Irish Household Budget Surveys for two intermediate values of the
parameter, 0.5 and 0.7, and found that results were similar for the two.
Finally, by interpreting r as the extent to which society believes the
poor should share in growth, we can view the selection of r as a normative
227
decision requiring political discourse to obtain a solution. Regardless of the

method for choosing r, the resulting tools allow a useful decomposition of
poverty into an absolute poverty group (those below the absolute poverty
line) and a hybrid or relative group (those above the absolute but below
the hybrid poverty line). This is analogous to the above decomposition
into the ultra-poor and the poor above the ultra-poverty line and likewise
could be helpful in guiding differential policy responses for the two groups.
Atkinson and Bourguignon (2000) combine absolute and relative pov-
erty lines in a different way. When mean income is low enough that za > zr,
they suggest that the absolute poverty line would be appropriate and hence
the income elasticity of the poverty line is zero in this region. However,
when incomes are high enough for zr > za, the relative poverty line should
apply, yielding a unitary income elasticity of the poverty line. The hybrid
poverty line of a country is then the maximum of the absolute poverty line
and the relative poverty line, or z = max{za,zr}. Atkinson and Bourguignon
use data on poverty lines and mean incomes to calibrate the absolute and
relative lines.
Ravallion and Chen (2011) argue that an income elasticity of one is
implausible and posit a weak relativity axiom that requires poverty to fall if
all incomes rise by the same proportion. They then provide the alternative
hybrid poverty line formula z = max{za,f+zr}, where f > 0 is interpreted as
the fixed cost of social inclusion. They set the three parameters of their
proposed formula with the aid of data. Although the line of Atkinson
and Bourguignon (2000) does not satisfy the weak relativity axiom for the
standard scale invariant poverty measures, the lines of Foster (1998) and
Ravallion and Chen (2011) do.
Categorical and Ordinal Variables
The previous analysis applies to any cardinal welfare indicator, where cardi-
nality requires that values convey more information than just more or less.
Nonmonetary examples of cardinal variables might include calories, years of
schooling, or hectares of land. Many other variables are more appropriately
interpreted as ordinal, because their values are only indicators of order.
Others might be categorical and have no values or underlying ordering at
all. Examples of ordinal variables include self-reported health and subjec-
tive well-being. Categorical variables include sanitation facilities or the
228
floor materials in a house. What can be done if we want to evaluate the size,
spread, or base of such a welfare indicator?
Allison and Foster (2004) describe ways of comparing distributions of
self-reported health in terms of spread and, in the process, provide new
approaches to evaluating size and base for this ordinal variable. The main
tools are dominance rankings. Changes in size and poverty are evaluated
using first-order stochastic dominance. Changes in spread are twin first-
order dominance movements away from the median category. To calculate a
mean, an inequality measure, or an FGT poverty index for a > 0, one must
cardinalize the ordinal variable, and hence the comparisons obtained are
not generally meaningful (because a different cardinalization could reverse
the ranking).
The headcount ratio, however, is identical for all cardinalizations and
thus is an appropriate tool for measuring poverty when the variable is
ordinal or even categorical. Of course, the headcount ratio provides no
information at all about depth. Bennett and Hatzimasoura (2011) provide
one approach to evaluating depth with an ordinal variable, based on a
reinterpretation of the poverty gap as “average headcount ratios” across
different poverty lines.
Chronic Poverty
Returning to the case of income, we saw how poor people can differ
from one another in policy-relevant ways. For example, poor people with
deeper income shortfalls are distinct from those just below the poverty
line. Time is a second dimension for differentiating among the poor:
persistent poverty is different from temporary poverty. Persistent poverty is
usually termed chronic poverty, and there are two main ways of identifying
and measuring it:
• The components approach of Jalan and Ravallion (2000) identifies

as chronically poor someone whose average income across several
periods is below the poverty line. This method rules out people whose
incomes temporarily dip below the line in a given period, but who,
on average, earn more than poverty line income. Chronic poverty
can then be measured by applying a standard poverty measure to the
average incomes distribution.
229
The use of average income implies that each period’s income is a

perfect substitute for any other period’s income. Alternative methods
that allow for imperfect income substitution across periods have been
proposed: see Calvo and Dercon (2009) or Foster and Santos (2006).
• In the spells method, exemplified by Foster (2009), the chronically
poor are those whose incomes are frequently below the poverty line,
say, in two of four periods. People with fewer poverty spells are not
chronically poor—their spells are censored out when chronic poverty
is measured. Aggregation proceeds as in the standard FGT case, but
now data on spells, normalized gaps, and squared normalized gaps are
collected in matrices.
The dimension-adjusted FGT indices are simply the means of the
respective censored matrices. For example, the dimension-adjusted
headcount ratio is the number of spells experienced by chronically
poor people divided by the maximum number of spells that could be
experienced by everyone. This approach assumes there is no income
substitution across periods, and, indeed, incomes are never aggre-
gated as they are in the components approach. It also presumes that
poverty spells have the same value, no matter the period or person.
Either approach to measuring chronic poverty allows the separate iden-

tification of chronic and transient poor and a corresponding decomposition
of poverty into chronic and transient components. This can be particularly
useful for tracking chronic poverty across subgroups for better targeting of
the policy mix.
Note that chronic poverty measurement increases data requirements
substantially. Panel data linked across periods at the individual or household
level are needed to undertake this form of measurement; it is not enough to
have multiple data cross-sections. Given the relative scarcity of panel data,
substantial efforts are being devoted to find novels ways of constructing virtual
panels from cross-sectional data. See, for example, Dang and others (2011).
Multidimensional Poverty
There is interest in developing and applying poverty measures that are

multidimensional in that shortfalls from multiple welfare indicators are
230
used to identify the poor and measure poverty. Several reasons exist for
this interest:
• Sen’s capability approach has received greater acceptance as a way

of conceptualizing well-being and poverty. According to Sen (1999),
poverty is seen as capability deprivation. Because many capabilities
exist, an accurate assessment of someone’s poverty requires a simul-
taneous assessment of multiple dimensions.
• The number of datasets that would support a multidimensional
assessment has increased.
• Strong demand comes from countries, international organizations,
and nongovernmental organizations for instruments that measure
poverty multidimensionally. For example, since 2009 the official
poverty measure in Mexico has been multidimensional, reflecting
shortfalls in income and several other “social” dimensions as required
by the relevant law (CONEVAL 2011). More recently, Colombia
elected to supplement its official income poverty measure with a
multidimensional poverty measure that is also used to coordinate
social policy among its ministries and the presidency (Angulo, Diaz,
and Pardo 2011).
The World Development Report 2000/2001: Attacking Poverty (World

Bank 2000) expressed the generally accepted idea that poverty is inherently
multidimensional. But as emphasized by Ferreira (2011), less agreement
exists on how to measure poverty when it has many constituent welfare
indicators. One way is to examine the nature of the indicators and how
they relate to one another. Some variables—such as earned and unearned
income—are easy to combine into a single aggregate. Others—such as
health and employment outcomes—are not. It is helpful to distinguish
between these cases.
When the variables can be meaningfully aggregated into a composite
welfare indicator for each person, the distribution of the composite indica-
tor could be evaluated using traditional poverty measurement methods. An
aggregate cutoff could be chosen to identify who is poor, and their poverty
could be measured using a poverty measure. In this way, the multidimen-
sional case could be converted to the single dimensional environment,
where well-known methods apply.
231
However, just because combining variables into one indicator is fea-

sible does not necessarily mean it is the best way to proceed. Aggregate
analysis conceals deprivations in individual variables that are compensated
by higher levels in other dimensions. If missing deprivations are policy rel-
evant, a more disaggregated approach may be needed. In India, for example,
aggregate consumption is expanding and poverty headcount ratios are fall-
ing, yet a high prevalence of malnutrition persists among children. Because
of this situation, shifting focus from shortfalls in aggregate consumption to
shortfalls in food consumption, or to shortfalls in consumption of food by
children, may be natural, if the data allow. When an aggregate welfare indi-
cator conceals policy-relevant information, a lower level of aggregation may
be preferable, even when full aggregation is feasible.
Now consider the case where all the key variables cannot be meaning-
fully aggregated into a single composite welfare indicator or where, for
policy reasons, complete aggregation is not desirable (such as where depriva-
tions in a certain variable are important to track). In this case, alternative
approaches must be explored. One option is to limit consideration to a sub-
set of the welfare indicators that can be aggregated and to drop the rest. This
approach has the benefit of expediency but ignores key poverty components.
Let us suppose instead that all variables must be used and that two or more
welfare indicators remain after aggregation. How can poverty be measured?
Many recent papers have considered this question, including Tsui (2002);
Bourguignon and Chakravarty (2003); Alkire and Foster (2007, 2011);
Massoumi and Lugo (2008); and Bossert, Chakravarty, and D’Ambrosio
(2009). As with chronic poverty measurement, the aggregation step used by
each is based directly on traditional, single-dimensional poverty measures,
appropriately expanded to account for many dimensions. For the identifica-
tion step, all begin with a cutoff in each dimension—which might be called
a deprivation cutoff. If the variable is below its respective cutoff, the person
is considered to be deprived in that dimension. Most of these papers then
adopt the union approach to identification, whereby anyone who is deprived
in even a single dimension is identified as poor. Some also discuss the
intersection approach, which is at the other extreme where someone must be
deprived in every dimension to be identified as being poor.
As noted by Alkire and Foster (2011), the union approach often identi-
fies a very large group of poor, whereas the intersection approach often iden-
tifies a vanishingly narrow slice, and this becomes particularly evident when
the number of dimensions expands. They propose an intermediate approach
232
to identifying the poor that depends on a simple measure of the breadth or

multiplicity of deprivation the person experiences. In this approach, every
deprivation has a value. The overall breadth of deprivation experienced by
a person is measured by summing the values of deprivations experienced. A
poverty cutoff is selected, and if the breadth of deprivation is above or equal
to the poverty cutoff, then the person is identified as being poor. The union
approach is obtained at one extreme where the poverty cutoff is very low,
while the intersection approach arises at the other where the cutoff is very
high. An intermediate poverty cutoff identifies as poor those who are suf-
ficiently multiply deprived. This is the dual cutoff approach to identification
suggested by Alkire and Foster (2011).
For aggregation, Alkire and Foster (2011) extend the FGT class of indi-
ces to the multidimensional context. They do this by constructing three
matrices analogous to the vectors used in the FGT definitions, except that
now each person has a vector containing information to be aggregated into
the overall measure. The matrices are censored in that the data of anyone
who is not poor are replaced by a vector of zeroes. The censored deprivation
matrix g0 contains deprivation values (when a person is deprived in a dimen-
sion and poor) or zeroes (when the person is not deprived in the dimension
or not poor). The adjusted headcount ratio M0 = m(g0) is its mean. The mea-
sure can be equivalently expressed as M0 = HA, where H is the population
percentage identified as poor and A is the average breadth of deprivation
they experience. Analogous definitions yield the adjusted poverty gap M1
and the adjusted FGT M2, as part of a family Ma of measures where a ≥ 0.
The methodology of Alkire and Foster combines a dual cutoff identification
approach and an adjusted FGT index.
The adjusted headcount ratio has several properties that make it
particularly attractive in practical applications. It can be used when
the underlying data are ordinal or even categorical. Its interpretation as
H × A is similar to the interpretation of PG, the traditional poverty gap,
because PG = PH × PIG, where PH is the traditional headcount ratio and
PIG is the average normalized income gap of the poor. M0 augments the
information in H using A, which is a measure of breadth rather than
depth. It is decomposable by population subgroup. It can dig down
into the aggregate numbers to understand the key deprivations that are
behind the measured poverty level. Related examples can be found in
Alkire and Foster (2011) and the recent Human Development Reports
of the United Nations Development Programme, which implemented
233
the approach across 109 countries as the Multidimensional Poverty Index

(MPI) (see also Alkire, Foster, and Santos 2011).
Measuring poverty in a multidimensional environment is challenging,
and the dual cutoff, adjusted headcount ratio methodology has been subject
to intense scrutiny. See, for example, Ravallion (2011) and the rejoinders by
Alkire and Foster (2011) and Alkire, Foster, and Santos (2011). Ravallion
(2011) offers an alternative approach that evaluates each dimension sepa-
rately using a traditional single-dimensional method to generate a dashboard
of dimension-specific deprivation measures. This approach provides useful
information about who is deprived in a given dimension and the extent of
their deprivation, and by using headcount ratios, it can also deal with ordi-
nal or categorical variables.
However, the approach provides no answer to the central question of
identification: Who is poor? In addition, the dimension-specific indices
reflect only the marginal distributions of the separate dimensions and hence
ignore their joint distribution. The methodology of Alkire and Foster relies
importantly on the joint distribution to determine the extent to which an
individual is multiply deprived. Their proposal is a first attempt at a practical
methodology for measuring poverty multidimensionally. Given the demand
for measurement methods that capture the multidimensional nature of pov-
erty, we can expect greater use of these and related methods in the future.
Multidimensional Standards
How should a society measure progress? Per capita income or expenditure

is well suited for indicating the resources available to an average member
of the society. However, there are at least two substantive critiques of this
measure as the sole indicator of progress. First, as noted previously in the
discussion of income standards, per capita income or expenditure thor-
oughly ignores the distribution of resources among the population. Other
possibilities, such as the Atkinson or Sen income standards, are sensitive
to the distribution and might well be preferable as an indicator of societal
progress.
Second, monetary resources are not the only resources important to a
person’s well-being. Without a more complete picture of the capabilities
available to people, or at least of the levels of achievement in the vari-
ous domains, we may be seeing only a partial view of progress. To be sure,
234
income and other welfare indicators are often positively correlated, for
both individual and country-level data, which may suggest that the
nonincome indicators are not needed. But as emphasized by Sen (1999),
notable exceptions to these regularities exist. A proper measure of progress
should convey empirical realities in all eventualities, including excep-
tional cases. Correlation does not justify dropping important dimensions
in assessing progress.
The Human Development Index (HDI) of the United Nations
Development Programme was designed as an alternative to income per
capita that includes health and education achievements in a country
(thus addressing the second critique). The underlying structure of the
traditional HDI is straightforward, even if the details of its construction
are not. A country’s achievements in income, health, and education are
summarized in three indicators that are normalized to lie between zero
and one. The traditional HDI is a simple mean of these components. The
precise construction of the indicators—including the choice of “goalposts”
for normalizing a variable and its specific transformation—can affect the
HDI’s picture of development across countries. As an example, the income
indicator used in the HDI is based on a logarithm of income per capita; if
the untransformed variable were used instead, the ranking at the upper end
would more closely follow the income ranking of these countries.
This lack of robustness is indicative of the challenge of constructing
component indicators that can be meaningfully combined into a composite
indicator. One alternative to combining dimensions into an overall indica-
tor is to provide a dashboard of dimensional indicators. If indicators were not
being combined, normalizing goalposts and special transformations would
not be needed; the variables could be presented in their original, more
comprehensible forms. In particular, one could dispense with the log trans-
formation, because average income itself would rank countries the same
way—within this dimension.
However, many good reasons exist for using a composite indicator rather
than a vector of components. A single numerical indicator is more salient
and easier to track. A comprehensive measure emphasizes the point that we
are more interested in overall progress than progress in any given dimension.
Moreover, given that the aggregation formula is decomposable, it invites
further analysis to identify which components are driving the overall results.
The success of the HDI would have been unlikely if only a dashboard of
dimensional indicators had been provided.
235
The mean is just one way of combining dimensions to get a measure of

progress. Other forms are possible. Foster, McGillivray, and Seth (2010,
2013) use the weighted additive form of the traditional HDI but allow the
weights to vary from the HDI’s case of equal weights. They examine the
robustness of HDI comparisons to variations in weights and derive condi-
tions under which the original ranking is preserved.
A second aggregation formula can be found in the “new” HDI that
appeared in the Human Development Report 2010 (UNDP 2010). Instead
of aggregating by using an arithmetic mean, the new HDI has adopted a
geometric mean. Under this approach, component indicators are viewed
as imperfect substitutes rather than the perfect substitutes implicit in an
additive form. The rates of trade-off across dimensions now depend on the
component levels, with indicators having lower relative levels being valued
more highly. This approach rewards balanced development in which no one
dimension lags too far behind or moves too far ahead of the rest.
In the Human Development Report 2010, the relation between the old
and the new methodology is presented in a figure in the statistical annex
(UNDP 2010, 217 figure T1.1). Although the old and new HDI rankings
have a positive relationship, the ranks are not perfectly positively associ-
ated. The new HDI values tend to be lower than the old HDI values, mainly
because the income component had been normalized with respect to a much
larger value, in addition to applying a geometric mean instead of the tradi-
tional arithmetic mean.
By focusing purely on average achievements in a country, the HDI is also
subject to the first critique of per capita income—that it ignores inequal-
ity across people. In a multidimensional setting, there are more ways for
a concern for inequality to be incorporated into a measure. One aspect is
inequality within each dimension. Hicks (1997), for example, uses the Sen
(or Gini-discounted) mean to evaluate the distribution of each component,
then averages across dimensions. Greater inequality with dimensions lowers
the Sen mean and hence the overall measure.
Noting that the resulting measure is not subgroup consistent, Foster,
López-Calva, and Székely (2005) propose an alternative class of distri-
bution-sensitive measures. A general mean with fixed parameter a < 1 is
applied to each component, thereby discounting for within-dimension
inequality using an Atkinson inequality measure. To ensure that the overall
measure is subgroup consistent, they aggregate across dimensions using the
same general mean (having the same fixed parameter a < 1). The resulting
236
formula can be viewed as a general mean of the matrix of individual welfare

indicators and is an example of what might be called a multidimensional
standard—which generalizes the notion of an income standard from a vector
(of one welfare indicator across many people) to a matrix (of several welfare
indicators across many people).
The approach has another advantage besides subgroup consistency:
measures in this class are path independent, in that one obtains the same
overall value whether one aggregates within each dimension and then
across dimensions (as defined above) or one aggregates across dimensions
for each person (analogous to a utility function) and then across people (as
with a traditional individualistic social welfare function). The latter order
of aggregation is more traditional in welfare economics, because it builds
up from the individual. However, the alternate definition is easier to derive
empirically, because the data need not be linked at the individual level.
This convenient property was used in the construction of the Inequality-
Adjusted Human Development Index (IA-HDI), which has been reported
in the Human Development Reports since 2010 (see Alkire and Foster 2010
for a more extensive discussion). It is a member of the Foster, López-Calva,
and Székely (2005) class using the geometric mean (or a = 0).
The second aspect of multidimensional inequality concerns association
across dimensions and is perhaps best explained using terminology from
statistics. The distribution of welfare indicators across people can be sum-
marized in the joint distribution, which indicates the prevalence of combina-
tions of welfare indicators across the population. Each joint distribution has
associated with it a marginal distribution for each welfare indicator, which
indicates the prevalence of the various levels of a welfare indicator in the
population. Two different joint distributions may have the same marginal
distribution; the association or correlation between indicators can be very
different even when the distribution within each indicator is the same.
For example, suppose two societies have the same marginal distributions
of achievements, and the well-being is measured by two dimensions: income
and education. In the first society the indicators are highly positively cor-
related, meaning that one with higher income has higher education. This
may be due to a failure of governance in providing free public education.
As a result, people with low income are unable to obtain higher levels of
education. Now suppose in a second country, the marginal distributions of
two societies are the same, but the correlation is much lower. This may have
happened because the government arranged public provision of education.
237
To detect the difference between these two situations, we need to use a

measure that is sensitive to association between dimensions.
Seth (2009) extended the method of Foster and others to a class of mul-
tidimensional standards that are sensitive to both forms of inequality: the
welfare indicators of each person are first aggregated using a general mean
of order b < 1; then these personal aggregates are aggregated using a general
mean of order a < 1 to obtain the overall measure. Note that when a is
equal to b, the measure belongs to the Foster and others class and is neutral
to the second form of inequality. When a is not equal to b, the measure is
sensitive to association among dimensions. For the detailed methodology,
see Seth (2012). This second form of inequality has also been discussed
in the poverty measurement literature (see Tsui 2002; Bourguignon and
Chakravarty 2003; Alkire and Foster 2007, 2011).
Given a multidimensional standard s incorporating one or both notions
of inequality, it is then straightforward to define a multidimensional
inequality measure as the percentage shortfall of s from the overall mean
achievement, namely, I = (m − s)/m. It should be noted, though, that many
assumptions are needed to construct s, which can make multidimensional
inequality I hard to measure in practice. Key among these are assumptions
pertaining to the cardinalization and comparability of the component
indicators; changing the way a variable is measured and altering its value
vis-à-vis other variables can change the rankings provided by s and the
inequality measure. Particularly vexing is the case where one or more of
the variables are ordinal, so that the cardinal form of each variable must,
by definition, be arbitrary. One way forward is to restrict consideration
to multidimensional versions of stochastic dominance (see Atkinson and
Bourguignon 1982). However, the case that addresses this issue—first-order
dominance—is precisely the case where the first form of inequality must
be ignored. Further work is needed to construct robust multidimensional
standards and practical indicators of multidimensional inequality.
Inequality of Opportunity
The previous section examined the general case where several welfare indi-
cators contribute to a person’s well-being. We now return to the simpler
case of a single welfare indicator, but where other variables provide infor-
mation on relevant characteristics or “identities” of the individuals. Recent
238
work has moved from evaluations of inequality across all people to measures
of inequality across groups of people, with the goal of isolating forms of
inequality that are particularly objectionable or policy relevant.
Roemer (1998) divides identity variables into circumstances, which
are unrelated to actions taken by the person and hence the person is not
accountable for such circumstances, and efforts, which are under the per-
son’s control. He argues that inequality across groups of people defined by
circumstances is particularly objectionable. For example, income inequal-
ity across racial groups or across groups defined by the education levels of
one’s parents should be of special concern because it reflects an underlying
inequality of opportunity. Ferreira and Gignoux (2008) implement this
approach by applying Theil’s second inequality measure, or the mean log
deviation, to a smoothed distribution defined by replacing each income in
a group with the group mean. In other words, inequality of opportunity is
measured as a between-group inequality term. This general approach can
be applied for different circumstance variables, and hence ways of defining
groups, to obtain different inequality of opportunity measures conditional
on that choice.
Stewart (2002) contends that group inequalities, which she calls horizon-
tal inequalities, can be more important than overall or vertical inequalities.
But rather than invoking a normative notion of equal opportunity, she uses
an empirical argument: horizontal inequalities, such as those across ethnic
groups, tend to be more closely linked to conflict than are vertical inequali-
ties. Stewart emphasizes that many possible dimensions of achievements
could be evaluated. The horizontal inequalities in a given dimension for a
configuration of groups can be measured and monitored using the associated
between-group inequality term.
The World Bank’s Human Opportunity Index (HOI) is another group
inequality measure that uses an opportunity interpretation of group inequal-
ities. Here the focus is on the provision of social services, so the underlying
distribution is taken to be dichotomous, with a zero being posted for all
people without access to the service and a one for those having access. The
overall mean of this variable then corresponds to the coverage rate for the
social service. The aim is to go beyond the mean coverage to account for
differential coverage rates across population subgroups, where the groups
are defined using circumstantial variables. An inequality measure is applied
to the smoothed distribution (which replaces a person’s actual value with
the group’s coverage rate) to obtain a measure of inequality of opportunity.
239
The HOI is the overall coverage rate discounted by the inequality of oppor-
tunity or, equivalently, a distribution-sensitive income standard applied
directly to the smoothed distribution.
The inequality measure used in the original HOI was the relative mean
deviation, a rather crude inequality measure that ignores transfers on either
side of the mean (see de Barros and others 2009). However, it is easy to
consider other inequality measures that generate between-group inequality
measures that are sensitive to differential coverage across subgroups on the
same side of the mean. For example, if we use the Atkinson inequality mea-
sure based on the geometric and arithmetic means, the resulting HOI will
evaluate the smoothed distribution using the geometric mean. Note that
every different social service can lead to a different picture of a population’s
opportunity to access social services. An overall view may require aggregat-
ing access to services at the individual level or aggregating HOIs into an
overall index. In addition, the measure is dependent on the particular cir-
cumstances selected to define population subgroups. These implementation
challenges are worthwhile because the measures can help reveal inequalities
that are especially salient and unjust.
Polarization
The term polarization describes a situation where a population spreads apart

into well-defined extremes of high and low and loses observations in the
middle. It is related to inequality in that a regressive transfer from low
incomes to high incomes (across the middle) increases both polarization and
inequality. However, the process of observations coming closer together at
the extremes and thereby raising polarization entails progressive transfers
that lower inequality. The two concepts go in different directions for this
form of transformation.
The concept of polarization is not the same as the concept of inequality
and requires its own measurement approach. Several polarization measures
have been proposed over the past two decades, but the two most frequently
cited are those of Foster and Wolfson (1992, 2010) and Esteban and Ray
(1994). The Foster-Wolfson polarization measure first divides the entire
population into two groups: one with achievements larger than the median
achievement and the other with achievements below the median. The
polarization measure is the difference between the between-group inequality
240
and the within-group inequality (as measured by the Gini coefficient) times
the ratio of mean to median (where the ratio of mean to median is a measure
of skewness of the distribution).
Foster and Wolfson (1992, 2010) also propose dominance orderings
based on polarization curves that can determine whether unambiguous
increases in polarization have taken place. First-order polarization occurs
when there are first-order stochastic dominant movements away from the
median. Second-order polarization occurs when there are second-order
dominant movements away from the median. The Foster-Wolfson polar-
ization measure is related to the area below the second-order polarization
curve. This approach has been extended by Zhang and Kanbur (2001) and
Chakravarty and D’Ambrosio (2010).
In contrast to the Foster-Wolfson approach, in which two groups of
observations are endogenously defined using the median as the dividing
line, Esteban and Ray (1994) assume that several groups of observations are
exogenously given, each around its own pole. Their polarization measure
rises when the groups pull apart from one another, or when observations
within a group become more tightly clustered together. The measure is
challenging to implement in practice because no clear way is given for
dividing an overall distribution into relevant clusters. These and other
practical problems of implementation are addressed in Duclos, Esteban,
and Ray (2004).
References
Alkire, S., and J. E. Foster. 2007. “Counting and Multidimensional

Poverty Measurement.” Working Paper 7, Oxford Poverty and Human
Development Initiative, University of Oxford.
———. 2010. “Designing the Inequality-Adjusted Human Development
Index (HDI).” Working Paper 37, Oxford Poverty and Human
Development Initiative, University of Oxford.
———. 2011. “Counting and Multidimensional Poverty Measurement.”
Journal of Public Economics 95 (7–8): 476–87.
Alkire, S., J. E. Foster, and M. E. Santos. 2011. “Where Did Identification
Go?” Journal of Economic Inequality 9 (3): 501–5.
Allison, R. A., and J. E. Foster. 2004. “Measuring Health Inequality Using
Qualitative Data.” Journal of Health Economics 23 (3): 505–24.
241
Angulo, R. C., B. Y. Diaz, and R. Pardo. 2011. “Multidimensional

Poverty Index (MPI-Colombia) 1997–2010.” Department of Planning
(Departamento Nacional de Planeación), Bogota.
Atkinson, A. B., and F. Bourguignon. 1982. “The Comparison of Multi-
Dimensioned Distributions of Economic Status.” Review of Economic
Studies 49 (2): 183–201.
———. 2000. “Poverty and Inclusion from a World Perspective.” In
Governance, Equity, and Global Markets, edited by J. E. Stiglitz and P.-A.
Muet, 151–66. Oxford: Oxford University Press.
Bennett, C. J., and C. Hatzimasoura. 2011. “Poverty Measurement with
Ordinal Data.” IIEP-WP-201114, Institute for International Economic
Policy, Elliott School of International Affairs, The George Washington
University, Washington, DC.
Bossert, W., S. R. Chakravarty, and C. D’Ambrosio. 2009. “Multidimensional
Poverty and Material Deprivation.” Working Paper 129, Society for the
Study of Economic Inequality (ECINEQ), Palma de Mallorca, Spain.
Bourguignon, F., and S. R. Chakravarty. 2003. “The Measurement of
Multidimensional Poverty.” Journal of Economic Inequality 1 (1): 25–49.
Calvo, C., and S. Dercon. 2009. “Chronic Poverty and All That: The
Measurement of Poverty over Time.” In Poverty Dynamics: Interdisciplinary
Perspectives, edited by A. Addison, D. Hulme, and R. Kanbur, 29–58.
Oxford: Oxford University Press.
Chakravarty, S. R., and C. D’Ambrosio. 2010. “Polarization Orderings of
Income Distributions.” Review of Income and Wealth 56 (1): 47–64.
CONEVAL (Consejo Nacional de Evaluación de la Política de Desarrollo
Social). 2011. “Coneval Presents 2010 Poverty Levels for Each
Municipality.” Press Release No. 015, Mexico, D.F., December 2.
Dang, H.-A., P. Lanjouw, J. Luoto, and D. McKenzie. 2011. “Using
Repeated Cross-Sections to Explore Movements in and out of Poverty.”
Policy Research Working Paper 5550, World Bank, Washington, DC.
de Barros, R. P., F. H. G. Ferreira, J. R. Molinas, M. Vega, and
J. S. Chanduvi. 2009. Measuring Inequality of Opportunities in Latin
America and the Caribbean. Washington, DC: World Bank.
Duclos, J.-Y., J. M. Esteban, and D. Ray. 2004. “Polarization: Concepts,
Measurement, Estimation.” Econometrica 72 (6): 1737–72.
Esteban, J. M., and D. Ray. 1994. “On the Measurement of Polarization.”
242
Ferreira, F. 2011. “Poverty Is Multidimensional. but What Are We Going to

Do about It?” Journal of Economic Inequality 9 (93): 493–95.
Ferreira, F., and J. Gignoux. 2008. “The Measurement of Inequality of
Opportunity: Theory and an Application to Latin America.” Policy
Research Working Paper 4659, World Bank, Washington, DC.
Foster, J. E. 1998. “Absolute versus Relative Poverty.” American Economic
Review 88 (2): 335–41.
———. 2009. “A Class of Chronic Poverty Measures.” In Poverty Dynamics:
Interdisciplinary Perspectives, edited by A. Addison, D. Hulme, and
R. Kanbur, 59–76. Oxford: Oxford University Press.
Foster, J. E., L. López-Calva, and M. Székely. 2005. “Measuring
the Distribution of Human Development: Methodology and an
Application to Mexico.” Journal of Human Development and Capabilities
6 (1): 5–25.
Foster, J. E., M. McGillivray, and S. Seth. 2010. “Rank Robustness of
Composite Indices: Dominance and Ambiguity.” Paper presented at the
31st General Conference of the International Association for Research
in Income and Wealth, St. Gallen, Switzerland, August 22–28.
———. 2013. “Composite Indices: Rank Robustness, Statistical Association,
and Redundancy.” Econometric Reviews 32 (1): 35–56.
Foster, J. E., and M. E. Santos. 2006. “Measuring Chronic Poverty.” Paper
presented at the 11th Annual Meeting of the Latin American and
Caribbean Economic Association (LACEA), Instituto Tecnológico
Autónomo de México, Mexico D. F., November 2–4.
Foster, J. E., and M. Székely. 2006. “Poverty Lines over Space and Time.”
Unpublished manuscript, Vanderbilt University, Nashville, TN.
Foster, J. E., and M. C. Wolfson. 1992. “Polarization and the Decline of
the Middle Class: Canada and the U.S.” Unpublished manuscript,
Vanderbilt University, Nashville, TN.
———. 2010. “Polarization and the Decline of the Middle Class: Canada
and the U.S.” Journal of Economic Inequality 8 (2): 247–73.
Hicks, D. A. 1997. “The Inequality-Adjusted Human Development Index:
A Constructive Proposal.” World Development 25 (8): 1283–98.
Jalan, J., and M. Ravallion. 2000. “Is Transient Poverty Different? Evidence
for Rural China.” Journal of Development Studies 36 (6): 82–89.
Massoumi, E., and M. A. Lugo. 2008. “The Information Basis of Multivariate
Poverty Assessments.” In Quantitative Approaches to Multidimensional
243
Poverty Measurement, edited by N. Kakwani and J. Silber, 1–29. London:

Palgrave MacMillan.
Madden, D. 2000. “Relative or Absolute Poverty Lines: A New Approach.”
Review of Income and Wealth 46 (2): 181–99.
Ravallion, M. 2011. “On Multidimensional Indices of Poverty.” Journal of
Economic Inequality 9 (2): 235–48.
Ravallion, M., and S. Chen. 2011. “Weakly Relative Poverty.” Review of
Economics and Statistics 93 (4): 1251–61.
Roemer, J. 1998. Equality of Opportunity. Cambridge, MA: Harvard
University Press.
Sen, A. 1999. Development as Freedom. New York: Knopf.
Seth, S. 2009. “Inequality, Interactions, and Human Development.” Journal
of Human Development and Capabilities 10 (3): 375–96.
———. 2012. “A Class of Distribution and Association Sensitive
Multidimensional Welfare Indices.” Journal of Economic Inequality, doi:
10.1007/s10888-011-9210-3.
Stewart, F. 2002. “Horizontal Inequalities: A Neglected Dimension of
Development.” Working Paper 81, Queen Elizabeth House, University
of Oxford.
Tsui, K.-Y. 2002. “Multidimensional Poverty Indices.” Social Choice and
Welfare 19 (1): 69–93.
UNDP (United Nations Development Programme). 2010. Human
Development Report 2010: The Real Wealth of Nations—Pathways to
Human Development. New York: Palgrave-Macmillan.
World Bank. 2000. World Development Report 2000/2001: Attacking Poverty.
Washington, DC: World Bank.
Zhang, X., and R. Kanbur. 2001. “What Difference Do Polarisation
Measures Make? An Application to China.” Journal of Development
Studies 37 (3): 85–98.
244
Chapter 5
Getting Started with ADePT
This chapter provides basic information about installing and using ADePT.
The instructions are sufficient to perform a simple analysis. More informa-
tion is available:
• Detailed instructions for using ADePT are provided in the ADePT

User’s Guide, which you can download from http://www.worldbank
.org/adept Documentation.
• Video tutorials are available at http://www.worldbank.org/adept
Video Tutorials.
• ADePT provides online help through the Help Contents command.
• For help with using an ADePT module, see appropriate chapters in
this book or in another book in the Streamlined Analysis with ADePT
Software series.
• Module-specific instructions, and example datasets, projects, and
reports, are available at http://www.worldbank.org/adept Modules.
• Examples of datasets and projects are installed with ADePT. They
are located in the\example subfolder in the ADePT program folder.
Use the examples with the instructions in this chapter to familiarize
yourself with ADePT operations.
245
Conventions Used in This Chapter
• Windows, buttons, tabs, dialogs, and other features you see on screen
are shown in bold. For example, the Save As dialog has a Save button
and a Cancel button.
• Keystrokes are shown in small caps. For example, you may be
instructed to press the enter key.
• Menu commands use a shorthand notation. For example, Project
Exit means “open the Project menu and click the Exit command.”
Installing ADePT
System Requirements
• PC running Microsoft Windows XP (SP1 or later), Windows Vista,

Windows Server 2003 and later, or Windows 7; ADePT runs in 32-
and 64-bit environments.
• NET 2.0 or later (included with recent Windows installations) and
all updates and patches
• 80MB disk space to install, plus space for temporary dataset copies
• At least 512MB RAM
• At least 1024 × 768 screen resolution
• At least one printer driver must be installed (even if no computer is
connected).
• Microsoft® Excel® for Windows® (XP or later), Microsoft® Excel
Viewer, or a compatible spreadsheet program for viewing reports
generated by ADePT is required.
• A Web browser and Internet access are needed to download ADePT.
Internet access is needed to install program updates and to load Web-
based datasets into ADePT. Otherwise, ADePT runs without needing
Internet access.
Installation
1. Download the ADePT installer by clicking the ADePT Downloads

button at http://www.worldbank.org/adept.
2. Launch the installer, and follow the on-screen instructions.
ADePT automatically launches after installation.
246
Chapter 5: Getting Started with ADePT
Launching ADePT
1. Click the ADePT icon in the Windows® Start menu.

2. In the Select ADePT Module window, double-click the name of the
module you want to use (see arrow in screenshot below). To open a
health module, double-click Health, then click Health Financing or
Health Outcomes.
3. You now see the ADePT main window. (The example below shows
ADePT configured with the Poverty module. The lower left-hand
and upper right-hand panels will be different when another module
is loaded.)
247
• To switch to another module after launching ADePT:

a. Module Select Module...
b. In the Select ADePT Module window, double-click the name of
the module you want to use.
Overview of the Analysis Procedure
There are four general steps in performing an analysis:
1. Specify one or more datasets that you want to analyze.

2. Map dataset variables to ADePT analysis inputs.
3. Select tables and graphs.
4. Generate the report.
248
Perform each step in the ADePT main window:

1. Click Add... button to load dataset(s). 3. Select tables and/or graphs
Enter dataset year in Label column. to be included in report.
2. Map dataset variables to input variables

by selecting dataset variables in drop-down lists. 4. Click Generate button.
The next sections in this chapter provide detailed instructions for the
four steps.
Specify Datasets
Your first task in performing an analysis is to specify one or more datasets.

ADePT can process data in Stata® (.dta), SPSS® (.sav), and tab delimited
text (.txt) formats.
249
Operations in this section take place in the upper left-hand corner of the
ADePT main window.
1. Click the Add... button.

2. In the Open dataset dialog, locate and click the dataset you want to
analyze, then click the Open button. The dataset is now listed in the
Datasets tab.
Tip: While learning to use ADePT, you may want to experiment

with sample data. You can find sample datasets in the ADePT\
Example folder.
250
3. Specify a label for the dataset:

a. In the Label column, select the default label.
b. Type a label for the dataset. Recommended: Label the dataset using
the year the survey was conducted (for example, 2002). When
labels are years, ADePT can calculate differences between surveys.
c. Press enter.
4. Optional: Repeat steps 1–3 to specify each additional dataset.
Note: If more than one dataset is specified, the datasets must contain
only individual observations or household observations, not both.
• To remove a dataset: Click the dataset, then click the Remove button.
Three datasets have been specified in this example.
Note: ADePT does not alter original datasets in any way. It

always works with copies of datasets.
5. At the top of the Datasets tab

• Select Individual level if the datasets contain one observation for
each household member.
• Select Household level if the datasets contain one observation for
each household.
6. By default, the Show changes between periods option is activated.
• If you want ADePT to calculate changes between two periods,
select the periods to the right of the option. The left-hand
251
period must be earlier than the right-hand period, as shown

here:
• If you do not want ADePT to calculate changes between periods,

deactivate the Show changes between periods option.
Map Variables
ADePT needs to know which variables in the dataset(s) correspond to the

inputs to its calculations. You must manually map dataset variables to input
variables.
Operations described in this section take place on the left-hand side of
the ADePT main window. These examples show the Poverty module loaded
into ADePT, but the process is similar for the other modules.
There are two methods for mapping variables:
Method 1: In the lower input Variables tab, open the variable’s list, then
click the corresponding dataset variable, as shown for the Urban variable.
252
Method 2: In the upper dataset Variables tab, drag the variable name and
drop it in the corresponding field in the lower input Variables tab.
Note: You can also type dataset variable names in the input variable
fields. The above methods are preferred, however, because typing
may introduce spelling errors. A spelling error is indicated by the red
exclamation point next to the input variable field.
253
• To remove a mapping: Select the variable name in the input variable

field, and then press delete.
Some modules have multiple input variable tabs. The Education
module, for example, organizes variables in three tabs.
In some input variable fields, you can specify multiple dataset variables.
For example, in the ADePT Poverty module, you can specify two poverty
lines (variables or numeric constants) instead of one, and the program will
replicate all tables for each of the specified poverty lines.
In this example, the pline_u and pline_l dataset variables have been
mapped to the Poverty line(s) input variable.
The italic variable name indicates that this input variable field accepts
multiple dataset variables. When you select or drag a new input variable
to one of these fields, it is appended to the previous value rather than
replacing it.
Tip: Open the example project (Project Open Example Project) to
see the result of mapping dataset variables to input variables.
Select Tables and Graphs
After mapping variables, you are ready to select the tables and graphs you
want ADePT to generate.
Operations described in this section take place in the right-hand side of
the ADePT main window.
254
In the upper right-hand (outputs) panel, select the tables and graphs you
want to generate.
Note: If a name is gray, it cannot be selected. These tables and graphs cannot be gener-
ated because required variables have not been specified.
255
• To see a description of a table or graph: Click the name. Its description

is displayed in the Table description and if-condition tab in the
lower right-hand corner of the ADePT window.
256
Generate the Report
You are now ready to generate your report:
1. Click the Generate button.
• To stop calculating: Click the Stop button.
2. Examine items in the Messages tab. ADePT lists potential problems

in this tab.
ADePT can identify three kinds of problems:

Notification provides information that may be of interest to you.
Notifications do not affect the content of reports generated by
ADePT.
Warning indicates a suspicious situation in the data. Warnings
are issued when ADePT cannot determine whether the data pose
an impossible situation. Examples include violation of parameters,
presence of potential outliers in the data, inconsistent data, and
inconsistent category definitions. ADePT reports are not affected
by warnings.
257
Error prevents the use of a variable in the analysis. For example, a

variable may not exist in a dataset (in this case, ADePT continues
its calculations as if the variable was not specified). If ADePT can
match the problem to a particular variable field, then that field is
highlighted in the input Variables tab.
3. As needed, correct problems, then generate the report again.
Note: Notifications, warnings, and errors can negatively affect the
results ADePT produces. Carefully review messages and correct criti-
cal problems before drawing conclusions from tables and graphs.
Examine the Output
When the analysis is complete, ADePT automatically opens the results as a

spreadsheet in Excel® or Excel Viewer. The results are organized in multiple
worksheets:
• The Contents worksheet lists all the other worksheets, including

titles for tables and graphs.
• The Notifications worksheet lists errors, warnings, and notifications
that ADePT identified during its analysis. This worksheet may be
more useful than the Messages tab in the ADePT main window
because the problems are organized by dataset.
• Table worksheets display tables generated by ADePT.
Tip: ADePT formats table data with a reasonable number of
decimal places. Click in a cell to see the data with full resolution
in the formula bar.
• Figure worksheets display graphs generated by ADePT.
Working with Variables
Viewing Basic Information about a Dataset’s Variables
1. In the Datasets tab, click the dataset you want to examine.

2. Click the Variables tab.
258
• To search for a variable: In the Search field, type a few characters of

the variable name or variable label.
• To view statistics for a variable: Double-click the variable name or
variable label. This opens the MultiDataset Statistics window for
that variable.
Viewing a Dataset’s Data and Variable Details
1. In the Datasets tab, click the dataset you want to examine.

2. Click the Browse... button. This opens the ADePT Data Browser.
259
The Data Browser lists observations in rows and organizes variables in

columns.
• To see underlying data: Click the Hide Value Labels button .

• To see value labels: Click the Show Value Labels button .
• To view a variable’s statistics:
a. Click in the variable’s column.
b. Click the Show Statistics... button .
• To view detailed information about the dataset’s variables: Click the

Variable View tab in the bottom left-hand corner of the Data
Browser.
260
• To hide or show variable columns in the Data View tab: In the Variable View tab, click
the checkbox next to the variable name.
Tip: The ADePT User’s Guide describes other functions available in the Data
Browser.
Generating Variables
You can create new variables that are based on variables present in a dataset. This might be
useful for simulating the effects of changes in parameters on various economic outcomes. For
example, in the Poverty module you can model the effect of income transfers on some popula-
tion groups on the basis of poverty and inequality.
1. In the Datasets tab in the main window, click the dataset that you want to modify.
2. Click the Variables | [dataset label] tab.
3. Right-click in the table, then click Add or replace variable... in the pop-up menu.
261
4. In the Generate/Replace Variable dialog:

a. In the Expression field, define the new variable using the follow-
ing syntax:
<new_variable_name> = <expression> [if <filter_expression>],
where
• <new_variable_name> is a unique name not already in the
dataset(s).
• <expression> calculates new data for the variable (for more
information about expressions, see “Variable Expressions” sec-
tion below).
• <filter_expression> filters observations that affect the calcula-
tion (optional).
b. Optional: Activate the Apply to all datasets option.
Note: If you loaded multiple datasets but do not generate the new
variable for all datasets, you will not be able to use the new vari-
able in calculations. However, you may want to generate a new
variable differently for each dataset in the project.
c. Click the Generate button.
5. In the Information dialog, click the OK button.
The new variable will be listed in the Variables | [dataset name] tab
and in the Data Browser. If the variable was generated for all loaded data-
sets, it will appear in the drop-down lists in the input Variables tab.
When you save a project, variable expressions are saved with the project,
and the variables are regenerated when you open that project. Generating
new variables does not change original datasets.
Replacing Variables
You can replace an existing numeric variable by following the instructions

in “Generating numeric dataset variables.” But in the Generate/Replace
Variable dialog (step 4a), specify an existing variable name instead of a new
variable name.
As with generated variables, these expressions are saved with a project,
and the variables are regenerated when you open the project. Replacing
variables does not change original datasets.
262
Variable Expressions
The following operators can be used in expressions:

Operator Description
+ – * / basic mathematical operators
abs sign
= == equality check operators
^ pow sqrt exponent (e.g., x^2 is x squared), power (e.g., pow(4,2) is 42 = 16),
and square root
round truncate shortening operators
min max range operators
ceiling floor
Variable expressions can include constants, and strings can be used for
variables that are of type string.
Expression examples are as follows:
Expression Description
x=1 sets all variable x observations to 1
x=y+z sets variable x observations to y observation plus z observation
x=y=1 sets variable x observations to 1 (true) if y is 1; otherwise, sets variable x
observations to 0 (false)
x = 23 if z == . sets variable x observations to 23 if z is missing ( . ); otherwise, sets to.
x = Log(y) if z = 1 sets variable x observations to log of y observations if z is 1; otherwise, sets to.
s = “test” sets all variable x observations to the string “test”
Note: The periods ( . ) in the table above represent system-missing values. This symbol is defined in
SPSS® and is used to indicate missing data in datasets.
Another example: To simulate the impact on poverty of a 10 percent

increase in incomes of households with more than 4 members, replace the
existing income variable using this expression:
income = income*1.1 if hhsize > 4.
Deleting Variables
You can remove variables from the working copy of a dataset that ADePT
uses for its calculations. This operation does not change the original data-
set. Native variables, as well as generated and replaced variables, can be
deleted.
1. In the dataset Variables tab, right-click in the row containing the

variable you want to delete, then click Drop Variable [variable
name] in the pop-up menu.
2. In the Confirmation dialog, click the Yes button.
263
Setting Parameters
Some modules have a Parameters tab next to the input Variables tab. In
the Parameters tab, you can set ranges, weightings, and other module-
specific factors that ADePT will apply during its processing. A Parameters
tab may also have input variable fields for mapping dataset variables, as
shown in the drop-down list below.
The mechanics for setting parameters are straightforward: activate

options, set values, and select items in drop-down lists. The analytical rea-
sons for setting parameters can be found elsewhere in this book or in the
appropriate book in the Streamlined Analysis with ADePT Software series.
Working with Projects
After specifying datasets and mapping variables, you can save the con-
figuration for future use. A saved project stores links to datasets, variable
264
names, and other information related to analysis inputs. Projects do not

retain table and graph selections, corresponding if-conditions, and fre-
quencies and standard errors choices because they are related to analysis
outputs.
• To save a project:
a. Project Save Project or Project Save As...
b. In the Save As dialog, select a location and name for the project,
then click the Save button.
• To open a saved project:

a. Project Open Project...
b. In the Open dialog, locate and select the project, then click the
Open button.
ADePT supports Web-based projects and datasets.

• To open a Web-based project:
a. Project Open Web Project...
b. In the Open web project dialog, enter the project’s URL, then
click the OK button.
• To add a Web-based dataset:

a. In the Datasets tab, shift-click the Add... button.
b. In the Add Web Dataset dialog, enter the dataset’s URL, then
click the OK button.
Adding Standard Errors or Frequencies to Outputs
• To calculate standard errors: Before clicking the Generate button,

activate the Standard errors option.
Calculating tables with standard errors takes considerably more time

than calculating tables without them—possibly an order of magnitude
265
longer. A good approach is to obtain the result you want without stan-
dard errors, then generate final results with standard errors.
• To calculate frequencies: Before clicking the Generate button, activate

the Frequencies option.
Tables with frequencies show the unweighted number of observations

that were used in the calculation of a particular cell in a table. No
significant additional time is needed to calculate frequencies.
Results of standard error and frequency calculations associated with a
table are provided in separate worksheets, labeled SE and FREQ, within
the output report.
Applying If-Conditions to Outputs
The purpose of if-conditions is to include observations from a particular

subgroup of a population in the analysis. The inclusion condition is formu-
lated as a Boolean expression—a function of the variables existing in the
dataset. Each particular observation is included in the analysis if it satisfies
the inclusion condition (the Boolean expression evaluates to value true). In
many cases, the conditions we use are quite simple. Consider the following
examples:
If-condition Interpretation
urban=1 Only those observations having the value of variable urban equal to one will be
included in the analysis.
region=5 Only observations from the region with code 5 are included in the analysis.
age_yrs>=16 Only those individuals who are 16 years old or older are included in the analysis.
sland!=0 Exclude from analysis those individuals who are not landowners (given that the
variable sland denotes the area of the land owned).
266
1. In the list of tables and graphs, click the table or graph name.
2. Enter the if-condition at the bottom of the Table description and
if-condition tab (see list of operators below).
If-condition operators include the following:

Operator Description
= equal
== equal
>= greater than or equal
<= less than or equal
!= not equal
& logical AND
| logical OR
inlist(<variable>,n1,n2,n3,...) include only observations for which <variable>
has values n1,n2,n3,...
inrange(<variable>,n1,n2) include observations for which <variable> is
between n1 and n2.
!missing(<variable>) exclude observations with missing values in
<variable>.
3. Click the Set button. A table or graph having an if-condition is high-

lighted.
267
Generating Custom Tables
You can add a custom table to ADePT’s output.

1. Tools Show custom table tab.
2. In the lower left-hand panel’s Custom table tab, activate the Define
custom table option.
3. Design the table by selecting items in the drop-down lists and by

activating the options as desired.
The Custom table tab in the lower right-hand corner of the ADePT
main window displays a simple preview of your table design. This
preview enables you to interactively modify the table to suit your
needs.
4. In the upper right-hand (outputs) panel:
a. Scroll to the bottom of the list.
b. Select Custom table.
268
The custom table will be included in the report generated by ADePT.
269
Appendix
This appendix provides additional tables and figures that may be useful in
understanding the concepts and results discussed in chapters 1–3. We use
the same Integrated Household Survey dataset of Georgia for 2003 and
2006 that we used in chapter 3. Results in this appendix are reported at the
national level, with rural and urban breakdown, and at the subnational level
for 2003 only. Figures for a particular region cover both 2003 and 2006.
Income Standards and Inequality
In chapter 3, we examined income standards such as quantile incomes, par-

tial means, and the arithmetic mean. Remember that quantile incomes and
partial means, unlike arithmetic means, are not computed using the entire
per capita expenditure distribution. So the arithmetic mean is the only stan-
dard among these three that depends on the entire distribution. However, it
is not sensitive to any change in spread or inequality within the distribution.
Given that any inequality index can be constructed using a higher income
standard and a lower income standard, income standards can be used to
construct the different inequality indices presented in chapter 3.
Table A.1 shows additional income standards that are sensitive to
inequality across the entire distribution. Table rows report rural and urban
areas and subnational regions. Row 13 reports the income standard for
Georgia as a whole. The variable is per capita expenditure, assessed in lari.
271
Table A.1: General Means and the Sen Mean

lari
General mean
Sen
a=1 a=2 a=0 a = –1 a = 0.5 mean
Region A B C D E F
1 Urban 128.9 155.5 106.1 84.7 117.2 85.7
2 Rural 123.5 151.8 99.1 75.4 111.0 79.9
Subnational
3 Kakheti 107.9 131.6 87.2 65.7 97.4 70.7
4 Tbilisi 144.5 171.8 121.5 101.4 132.5 98.0
5 Shida Kartli 122.9 153.6 96.3 67.8 109.3 77.9
6 Kvemo Kartli 93.5 113.7 77.3 61.8 85.2 63.0
8 Ajara 107.8 129.9 87.9 68.5 97.7 71.2
9 Guria 134.3 166.9 109.9 86.4 121.6 88.7
10 Samegrelo 117.2 142.3 95.7 75.8 106.1 77.2
11 Imereti 150.3 178.9 124.3 99.6 137.1 100.7
13 Total 126.1 153.6 102.4 79.7 113.9 82.7
Columns A through E show the general means for five different values
of the inequality aversion parameter a : a = 1 for the arithmetic mean,
a = 2 for the Euclidean mean, a = 0 for the geometric mean, a = –1 for
the harmonic mean, and a = 0.5. From our discussions of general means
in chapter 2, we know that a distribution’s general mean decreases as a
increases. Column F lists the Sen mean.
Column A reports the mean per capita consumption expenditure when
a = 1. The other income standards, with the mean, can be used to construct
a particular inequality measure. For example, the mean can be combined
with the Euclidean mean to construct the generalized entropy measure for
a = 2. The mean and the geometric mean can be used to construct the
Atkinson inequality measure A(0) and the generalized entropy measure
GE(0). The mean and the harmonic mean are used together to compute the
Atkinson measure of inequality A(–1). The mean and the general mean for
a = 0.5 are combined to compute A(0.5). Finally, the mean and the Sen
mean can be used to compute the Gini coefficient.
For example, the mean per capita expenditure in Kakheti is GEL 107.9
[3,A], whereas the Sen mean is GEL 70.7 [3,F]. Thus, the Gini coefficient
is easily computed as 100 × (107.9 – 70.7)/107.9 = 34.4, which can be veri-
fied from table 3.8. Similarly, the mean for Tbilisi is GEL 144.5 [4,A] and
272
Appendix
the geometric mean is GEL 121.5 [4,C], so the Atkinson measure A(0) is
computed as 100 × (144.5 – 121.5)/144.5 = 15.9, which can be verified from
table 3.27.
Censored Income Standards and Poverty Measures
A distribution’s censored income standard is computed by applying income

standards to a per capita expenditure distribution that is censored at the
poverty line. In a censored distribution, the achievements of those below
the poverty line are retained, and the achievements of those above the pov-
erty line are replaced by the poverty line itself.
The censored income standards shown in table A.2 are closely related
to the poverty measures reported in chapter 3. Table rows report rural
and urban areas and subnational regions. Row 13 reports the income
standard for Georgia as a whole. The variable is per capita expenditure,
assessed in lari.
Column A shows the doubly censored mean of a distribution, where
censoring takes place at the distribution’s upper and lower ends. In a doubly
Table A.2: Censored Income Standards

lari
Doubly
censored General mean
mean a =1 a =0 a = –1 Sen mean
Region A B C D E
Poverty line = 75.4
1 Urban 54.2 68.9 66.9 62.9 63.6
2 Rural 51.6 67.4 64.5 58.6 60.9
Subnational region
3 Kakheti 46.1 65.3 61.8 54.4 57.7
4 Tbilisi 59.7 71.3 70.2 68.4 67.7
5 Shida Kartli 48.8 66.6 63.0 54.2 59.6
6 Kvemo Kartli 41.9 63.8 60.0 53.7 55.3
7 Samtskhe-Javakheti 52.7 67.9 65.3 60.4 61.8
8 Ajara 47.4 65.8 62.4 56.4 58.3
9 Guria 56.4 69.1 67.1 62.9 63.9
10 Samegrelo 50.2 67.1 64.5 59.7 60.6
11 Imereti 59.8 70.8 69.3 66.5 66.7
12 Mtskheta-Mtianeti 49.6 65.5 62.1 56.8 57.9
13 Total 52.8 68.1 65.6 60.6 62.2
273
censored distribution, people whose per capita expenditure is not less than
the poverty line are assumed to have poverty-line income, and people whose
per capita expenditure is less than the poverty line are assumed to have zero
per capita expenditure. The doubly censored mean is the mean of the doubly
censored distribution. The rest of the columns report income standards for
distributions that are censored once at the poverty line. Columns B, C, and
D show the arithmetic mean, the geometric mean, and the harmonic mean,
respectively. Column E reports the censored distribution’s Sen mean.
Those five censored income standards are related to five different poverty
measures, as explained in chapter 3. If the poverty line is denoted by z and a
censored income standard is denoted by a, then a poverty measure can be com-
puted by combining each of those five income standards and the poverty line.
The poverty line in this exercise is z = GEL 75.4. If the censored
income standard a is the doubly censored mean, then the headcount ratio is
(z – a)/z. Similarly, if the censored income standard a is the censored arith-
metic mean and the censored Sen mean, then (z – a)/z would be the poverty
gap measure and the Sen-Shorrocks-Thon (SST) index, respectively. If the
censored income standard a is the censored geometric mean, then the corre-
sponding poverty measure is the Watts index, computed as lnz – lna. Finally,
if the censored income standard a is the censored harmonic mean, then the
corresponding poverty measure is the Clark-Hemming-Ulph-Chakravarty
(CHUC) index, computed as (z – a)/z. Thus, a mere comparison of the
censored income standards for the same poverty line can provide a good
understanding for poverty comparisons.
Here is how different poverty measures can be obtained using each of
these censored income standards.
• In table 3.2, Georgia’s headcount ratio in 2003 for poverty line GEL
75.4 is 29.9. This can be obtained from table A.2 using the national
doubly censored mean of GEL 52.8 [13,A]: 100 × (75.4 – 52.8)/
75.4 = 29.9.
• In table 3.2, the national poverty gap measure is 9.7. This can be
obtained from table A.2 using the poverty line and the national
censored arithmetic mean of GEL 68.1 [13,B]: 100 × (75.4 – 68.1)/
75.4 = 9.7.
• In table 3.26, the national Watts index is 13.9. This can be obtained
from table A.2 using the poverty line and the national censored geo-
metric mean GEL 65.6 [13,C]: 100 × (ln75.4 – ln65.6) = 13.9.
274
Appendix
• In table 3.26, the national CHUC index is 19.6. This can be obtained
from table A.2 using the poverty line and the national censored har-
monic mean of GEL 60.6 [13,D]: 100 × (75.4 – 60.6)/75.4 = 19.6.
• In table 3.26, the national SST index is 17.5. This can be obtained
from table A.2 using the poverty line and the national censored Sen
mean of GEL 62.2 [13,E]: 100 × (75.4 – 62.2)/75.4 = 17.5.
Elasticity of Watts Index, SST Index, and CHUC Index to

Per Capita Consumption Expenditure
Table A.3 presents a tool for checking the sensitivity of three poverty mea-
sures to consumption expenditure: the Watts index, the SST index, and the
CHUC index. In the table, we ask what the percentage change in poverty
would be if everyone’s consumption expenditure increased by 1 percent.
Results are compared across 2003 and 2006.
The percentage change in poverty caused by a 1 percent change in the
mean or average per capita consumption expenditure is called the elasticity
of poverty with respect to per capita consumption. The particular way in which
we consider an increase in the average per capita consumption expenditure
is by increasing everyone’s consumption expenditure by the same percent-
age. This type of change is distribution neutral, because the relative inequal-
ity does not change.
Table A.3: Elasticity of Watts Index, SST Index, and CHUC Index to Per Capita Consumption
Expenditure
Watts index SST index CHUC index

A B C D E F G H I
1 Urban −2.00 −2.11 −0.11 −1.81 −1.88 −0.07 −1.78 −1.91 −0.12
2 Rural −1.76 −1.69 0.07 −1.57 −1.50 0.07 −1.48 −1.44 0.04
3 Total −1.86 −1.87 −0.01 −1.68 −1.67 0.00 −1.60 −1.63 −0.02
4 Urban −2.31 −2.49 −0.18 −2.31 −2.42 −0.12 −2.17 −2.41 −0.24
5 Rural −1.89 −1.83 0.06 −1.81 −1.73 0.08 −1.78 −1.78 0.00
6 Total −2.04 −2.06 −0.01 −2.00 −1.98 0.01 −1.93 −1.99 −0.06
Note: Change is shown between years 2003 and 2006. CHUC = Clark-Hemming-Ulph-Chakravarty; SST = Sen-Shorrocks-Thon.
275
Consumption expenditure is measured in lari per month, and the pov-

erty lines are set at GEL 75.4 and GEL 45.2 per month. For the former
poverty line, if a Georgian household is not capable of providing a monthly
consumption expenditure level of GEL 75.4 to each of its members, then
the household (and each member) is identified as poor. Columns A through
I denote three different sets of poverty measures—Watts index, SST index,
and CHUC index—each measure containing three columns. The first two
columns within each set report the elasticities for 2003 and 2006, respec-
tively, and the third column reports the difference between the two years.
Consider the results when the poverty line is GEL 75.4 per month.
Note that the elasticities are negative, meaning poverty falls because of an
increase in consumption expenditure, but the higher magnitudes imply high-
er elasticity even though signs are negative. The Watts index elasticity with
respect to the mean consumption expenditure for the urban area in 2003 is
–2.00 [1,A]. In other words, if the consumption expenditure increases by 1
percent for everyone, then the mean per capita consumption expenditure
increases by 1 percent and the urban headcount ratio falls by 2 percent.
If the mean consumption expenditure is increased by 1 percent in 2006,
then the Watts index falls to 2.11 percent [1,B]. A higher value implies
higher sensitivity. The urban elasticity of the Watts index is less sensitive
to consumption expenditure in 2003 than in 2006 by 0.11 percentage point
[1,C]. Similarly, the SST index elasticity relative to per capita consumption
expenditure for the urban area in 2003 is –1.81 [1,D], which increases by
0.07 point to –1.88 in 2006 [1,E]. The CHUC index elasticity in 2003 is
–1.78 [1,G], which decreases by –0.12 point to –1.91 in 2006 [1,H].
Because poverty lines are set normatively, they are difficult to justify exclu-
sively. A slight change in per capita consumption expenditure may or may
not change the poverty measures by significant amounts. If the distribution is
highly polarized or, in other words, there are two groups in the society—one
group of rich people and the other group of extremely poor people—then a
slight change in everyone’s income by the same proportion may not have any
impact on headcount ratio. In contrast, if there is a concentration of mar-
ginal poor around the poverty line, then a slight change in everyone’s income
by the same proportion would have a huge impact on poverty rates. Hence,
this type of analysis may tell us how policy changes impact the poverty rate.
276
Appendix
Sensitivity of Watts Index, SST Index, and CHUC Index to

Poverty Line
The exercise in table A.4 is analogous to the exercise for checking the elas-
ticity of poverty measures to per capita consumption expenditure, but it is
more rigorous. It is always possible to find a certain percentage of decrease
in the poverty line that matches the increase in the consumption expendi-
ture for everyone by 1 percent. In this exercise, we check the sensitivity of
poverty measures by changing the poverty line in more than one direction.
The table shows how the actual headcount ratio changes as the poverty
line changes from its initial level, whether GEL 75.4 per month or GEL
45.2 per month. Rows denote the change in poverty line in both upward
and downward directions. Columns report the change in three poverty mea-
sures: Watts index, SST index, and CHUC index. The variable is per capita
consumption expenditure measured in lari. This table shows results for 2003
only, but this analysis can be conducted for any year.
Columns A and B report the national Watts index for different pov-
erty lines, and column C shows the change in the index from the actual
poverty line. The rows corresponding to +5 percent denote the results for a
Table A.4: Sensitivity of Watts Index, SST Index, and CHUC Index to the Choice of Poverty Line,
2003
Change from Change from Change from

Watts index SST index CHUC index
actual (%) actual (%) actual (%)
A B C D E F
1 Actual 13.9 0.0 17.5 0.0 19.6 0.0
2 +5% 15.4 11.0 19.2 9.6 21.3 9.0
3 +10% 17.0 22.4 20.9 19.2 23.1 18.0
4 +20% 20.3 46.1 24.3 38.5 26.6 35.8
5 −5% 12.4 −10.5 15.9 −9.3 17.8 −8.9
6 −10% 11.1 −20.4 14.3 −18.4 16.1 −17.6
7 −20% 8.5 −38.8 11.2 −35.9 12.8 −34.6
8 Actual 4.3 0.0 5.9 0.0 6.9 0.0
9 +5% 4.8 12.2 6.6 11.8 7.7 11.3
10 +10% 5.4 25.3 7.3 24.2 8.5 23.2
11 +20% 6.6 54.0 8.9 51.3 10.2 48.6
12 −5% 3.8 −11.6 5.2 −11.3 6.1 −10.8
13 −10% 3.3 −22.3 4.6 −21.9 5.4 −21.1
14 −20% 2.5 −41.5 3.5 −40.8 4.2 −39.7
Source: Based on ADePT Poverty and Inequality modules using Integrated Household Survey of Georgia 2003.
Note: CHUC = Clark-Hemming-Ulph-Chakravarty; SST = Sen-Shorrocks-Thon.
277
5 percent increase in the poverty line. Thus, when the poverty line is GEL
75.4, a 5 percent increase moves the poverty line to GEL 79.2. The Watts
index increases by 1.5 points from 13.9 [1,A] to 15.4 [2,A], or by 11 percent
from its actual level of 13.9.
Similarly, if the poverty line changes by –10 percent from GEL 75.4,
then the poverty Watts index falls by 2.8 from 13.9 [1,A] to 11.1 [6,A], or
by 20.4 percent from the actual level of 13.9. This index is more sensitive
to change in the poverty line when the actual poverty line is lower at GEL
45.2. In fact, the SST index and the CHUC index are also more sensitive
to change in poverty line when the actual poverty line is GEL 45.2 rather
than GEL 75.4.
The table helps us understand how robust a particular poverty estimate is

with respect to the poverty line. Selection of any poverty line is debatable,
because it is set with normative judgment. On the one hand, if a poverty
measure changes drastically from a change in the poverty line, then a
cautious policy conclusion should be drawn from the analysis based on a
particular poverty line. On the other hand, if a poverty measure does not
vary much because of a change in the poverty line, then a more robust con-
clusion can be drawn.
Decomposition of the Gini Coefficient
Table A.5 analyzes the composition of inequality across different population

subgroups using the Gini coefficient. Unlike the decomposable inequality
measures containing a within-group term and a between-group term, the
Gini coefficient decomposition usually has three terms: a within-group
inequality term, a between-group inequality term, and an overlap term.
The within-group inequality term is a weighted average of all subgroup
inequalities. Note that the overlap term vanishes if the income rankings of
the subgroups do not overlap. However, the residual term is nonzero when
there are overlapping incomes.
Recall that the Gini coefficient lies between 0 and 1 (chapter 2 contains
a detailed description of the Gini coefficient). When every household in a
region has the same per capita expenditure, then the Gini coefficient is 0.
278
Appendix
Table A.5: Breakdown of Gini Coefficient by Geography
2003 2006
A B
1 Total 34.4 35.4
Urban and rural
2 Within-group inequality 17.2 17.7
3 Between-group inequality 1.1 0.5
4 Overlap term 16.2 17.2
Geographic regions
5 Within-group inequality 4.9 5.2
6 Between-group inequality 8.7 7.0
7 Overlap 20.8 23.2
Source: Based on ADePT Poverty and Inequality modules using Integrated

Household Survey of Georgia 2003 and 2006.
Row 1 reports the overall Gini coefficients. Subsequent rows report Gini
coefficient decompositions for two different population subgroups: rural
and urban regions and geographic regions. The first row of each set reports
the within-group inequality and the second and the third rows report the
between-group inequality and the overlap term, respectively. The overall
Gini coefficient in 2003 is 34.4 [1,A], which increases to 35.4 in 2006 [1,B].
Thus, in terms of the Gini coefficient, inequality increased in 2006.
The first set decomposes the population into rural and urban areas. The
total within-group Gini coefficient is 17.2 in 2003 and increases to 17.7 in
2006 [row 2]. However, the between-group inequality decreased from 1.1
in 2003 to 0.5 in 2006 [row 3]. The overlap term registers an increase from
16.2 to 17.2 [row 4].
The decomposition of population by geographic regions has a similar
story. The total within-group inequality increases from 4.9 in 2003 to 5.2 in
2006 [row 5], but the between-group inequality decreases from 8.7 in 2003 to
7.0 in 2006 [row 6], and the overlap term increases from 20.8 in 2003 to 23.2
in 2006 [row 7]. Note that the overlap term is larger for the decomposition
across geographic regions [row 7] than across rural and urban areas [row 4].
A possible reason could be the number of groups: as the number of groups
increases, the possibility of overlap increases.
This type of analysis is important for policy purposes and may affect policy
recommendations. Both the overall inequality and the intergroup inequality
279
may be detrimental to a nation’s welfare. Suppose there are two groups in

a region and the overall income inequality is moderate. After the groups
are decomposed into within-group and between-group terms, if the within-
group inequality is low and the between-group inequality is very high, then
the society is polarized. This might increase the possibility of social conflict,
as discussed in chapter 4. Thus, merely looking into the overall inequality
figures may not reveal this potential problem to the policy maker. The type
of analysis conducted in this table may turn out to be crucial.
Decomposition of Generalized Entropy Measures
The Gini coefficient is not decomposable in the usual way because it has
an overlap term. Thus, it is important to look at the usual decomposition
(within-group and between-group inequalities) using additively decompos-
able measures. With this objective, table A.6 analyzes the decomposition
of inequality across urban and rural areas and across geographic regions.
The analysis is based on three different types of generalized entropy (GE)
measures: the first Theil measure denoted by GE(1), the second Theil
Table A.6: Decomposition of Generalized Entropy Measures by Geography
2003 2006 Change

GE(0) GE(1) GE(2) GE(0) GE(1) GE(2) GE(0) GE(1) GE(2)
A B C D E F G H I
1 Total 20.8 20.0 24.2 21.8 21.5 27.8 1.1 1.6 3.6
Urban and rural
2 Between-group 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
inequality
3 Between as a share 0.1 0.1 0.1 0.0 0.0 0.0 −0.1 −0.1 −0.1
of total (%)
4 Within-group 20.8 19.9 24.2 21.8 21.5 27.8 1.1 1.6 3.7
inequality
Geographic regions
5 Between-group 1.3 1.2 1.2 0.8 0.8 0.8 −0.4 −0.4 −0.4
inequality
6 Between as a share 6.1 6.2 5.0 3.8 3.7 2.8 −2.3 −2.5 −2.2
of total (%)
7 Within-group 19.5 18.7 23.0 21.0 20.7 27.1 1.5 2.0 4.1
inequality
280
Appendix
measure denoted by GE(0), and the generalized entropy measure for α = 2

denoted by GE(2).
Each measure can be decomposed into a within-group inequality term
and a between-group inequality term, where the within-group inequal-
ity term is a weighted average of all subgroup inequalities. However, the
weights (except for the two Theil measures) do not necessarily add up to 1.
Chapter 2 provides a more detailed discussion of generalized entropy
measures.
Row 1 reports the three inequality indices for 2003 and 2006 and the
changes across these two years. In 2003, we see that GE(0) is 20.8 [4,A],
which increases by 1.1 (rounded) to 21.8 in 2006 [4,D]. Like GE(0), GE(1)
and GE(2) also increase between 2003 and 2006.
Now consider rows 2 through 4, which report inequalities across and
between two years for urban and rural areas. The between-group inequality
areas [row 2] appear to be negligible compared to the overall inequality [row 1]
for all three measures for both years. Given that the share of between-group
inequality is negligible, the within-group inequality [row 4] is almost equal to
the overall inequality.
For the next set of results, the entire population is divided into 10 geo-
graphic regions. Unlike the previous results, the between-group inequality
[row 5] is not negligible, but it is still much lower than the within-group
inequality [row 7]. For example, the between-group inequality in 2003 for
GE(0) is 1.3 [5,A], which is 6.1 percent of the overall inequality [6,A]. The
between-group inequality for GE(0) fell in 2006 to 0.8 [5,D], which is 3.8
percent of overall inequality [6,D]. GE(1) and GE(2) show a similar pattern.
However, the total within-group inequality increased between 2003 and
2006. The total within-group inequality for GE(0) increased from 19.5 in
2003 [7,A] to 21.0 in 2006 [7,D].
Policy recommendations might be driven by this analysis, because a nation’s

welfare could be negatively affected by overall and intergroup inequalities.
Consider a case in which income inequality is moderate between two groups
in a region. Decomposition reveals low within-group inequality and very
high between-group inequality, indicating a polarized society and the poten-
tial for social conflict. Policy makers may overlook this critical situation if
they focus only on overall inequality data.
281
Dynamic Decomposition of Inequality Using the Second

Theil Measure
Among all relative inequality measures, the generalized entropy measures

are additively decomposable so that overall inequality is the sum of overall
within-group inequality and between-group inequality. Overall within-group
inequality is the weighted average of within-group inequalities of population
subgroups. Weights attached to within-subgroup inequalities do not necessarily
sum to 1. It turns out there are only two generalized entropy measures for which
the weights sum to 1: the first Theil measure and the second Theil measure.
For the first Theil measure, weight attached to each subgroup is the share
of overall income held by that subgroup. For the second Theil measure,
weight attached to each subgroup is that subgroup’s population share. For
dynamic decomposition of inequality, it is more interesting to understand
the change in within-group and between-group inequality and also the
change in subgroup population share.
Following Mookherjee and Shorrocks (1982), we use the second Theil
measure and decompose the change in overall inequality into four compo-
nents: (a) change in within-group inequality, (b) change in between-group
inequality, (c) shift in subgroup population shares, and (d) relative varia-
tion in subgroup mean incomes. Let us examine the process mathematically
before interpreting the empirical results. Recall from chapter 2 that the
second Theil measure is
N
WA (x ) 1 x
I T 2 (x ) = ln =
WG (x ) N
∑ ln x
n =1
, (A.1)
i
where x– is the mean of the income vector x and N is the total population
size.
Suppose the overall population is divided into K > 1 population
subgroups. These population subgroups may be different geographic regions,
ethnic groups, or rural and urban regions. For rural and urban decom-
position, K = 2. We denote the income vector of subgroup k by xk, the
population size of subgroup k by Nk, and the mean income of subgroup k by
x– k. Let us denote the population share of subgroup k by vk = Nk/N and the
income share of subgroup k by m k = x– k / x–. The second Theil measure can
then be decomposed as
IT 2 (x) = ∑ k =1 v k IT 2 (x k ) + ∑ k =1 v k ln 1 .
K k
mk (A.2)
282
Appendix
The first component is the population-share weighted average of within-

group inequalities, and the second term is the between-group inequality.
Now, suppose we are interested in the dynamic decomposition of the
second Theil measure between periods t0 and t1. The decomposition of
changes in inequality between these two periods is
ΔIT 2 (x) = IT 2 (x; t1) − IT 2 (x; t 0 )
K K K K
1 1
= ∑ v k (t 0 )ΔIT 2 (x k ) + ∑ IT 2 (x k ; t1)Δv k + ∑ ln Δv k
+ ∑ v k (t 0 )Δ ln k ,
m (t1) m
k
k =1 k =1 k =1 k =1
(A.3)
where Δ represents the change in the variables from time t0 to t1. The four
components can be interpreted as (a) the intertemporal change in within-
group inequality, (b) the change in the population shares of the groups in
the within-group component, (c) the change in population shares of the
groups in the between-group component, and (d) the change in the relative
incomes of the subgroups.
Table A.7 provides a dynamic decomposition of the overall Georgian
income inequality using the second Theil measure. Results in the table cor-
respond to changes across years 2003 and 2006. The variable for our analysis
is consumption expenditure in lari per month. Row 1 reports the change in
overall inequality. Rows 2 through 5 decompose this change into four factors,
as explained in the previous paragraph. Row 2 reports the change in overall
within-group inequality. Rows 3 and 4 report the effect of changes in popula-
tion shares on the within-group inequality and the between-group inequality,
respectively. Row 5 reports the change in relative subgroup incomes.
Table A.7: Dynamic Decomposition of Inequality Using

the Second Theil Measure
GE(0)
A
1 Change in aggregate inequality −0.011
2 Within-group inequality −0.015
3 Population shares of within-group inequality 0.000
4 Population shares of between-group inequality 0.000
5 Mean group incomes 0.004
Source: Based on ADePT Poverty and Inequality modules using Integrated

Household Survey of Georgia 2003 and 2006.
283
The decrease in the overall inequality between 2003 and 2006 is −0.011
[1,A]. Row 2 indicates that this decline is mostly attributed to the decrease
in the within-group inequality because it is evident from row 5 that the
relative income share does not change in the same direction. The effect of
change in population share on the within-group inequality [row 3] and the
between-group inequality [row 4] is negligible.
Decomposition of Generalized Entropy Measure by

Income Source
In table A.8, we first break down the single variable into several compo-
nents, then we decompose the overall inequality across that variable into
the inequality of its components. For example, the total disposable income
of a household has several components such as male earnings, female earn-
ings, benefits, and income taxes. Analyzing inequality across disposable
income may not reveal inequality across these various components. This
type of inequality decomposition into factor components was studied in
detail by Shorrocks (1982), but only for a single period. Jenkins (1995)
conducted a dynamic intertemporal decomposition analysis across the popu-
lation. Following Jenkins, we use the generalized entropy measure of order
Table A.8: Decomposition of Generalized Entropy Measure by Income Source
Relative Proportionate
Mean Correlation Absolute factor
mean GE(2) factor contribution
(GEL) with total contribution
(%) (%)
A B C D E F
2003
1 Food consumption 76.9 61.0 80.8 27.2 12.7 52.3
2 Expenditures on nonfood goods 15.2 12.0 62.5 57.2 3.0 12.4
3 Utilities 8.4 6.7 35.5 140.0 1.4 5.9
4 Expenditures on services 17.4 13.8 55.4 140.5 4.8 19.6
5 Other expenditures 8.2 6.5 48.5 179.6 2.4 9.8
6 Per capita consumption expenditure 126.1 100.0 24.2 24.2 100.0
2006
7 Food consumption 72.8 57.8 72.3 26.2 11.3 40.5
8 Expenditures on nonfood goods 13.2 10.5 56.3 74.8 2.9 10.5
9 Utilities 10.4 8.3 40.2 161.3 2.3 8.2
10 Expenditures on services 20.2 16.1 62.9 221.9 8.4 30.2
11 Other expenditures 9.3 7.4 50.0 186.7 2.9 10.6
12 Per capita consumption expenditure 126.0 100.0 27.8 27.8 100.0
284
Appendix
two for our analysis in this table, mainly because some components may be
zero and the measure is additively decomposable, as discussed in chapter 2.
Before discussing the results, let us provide a brief theoretical back-
ground. Interested readers can refer to Shorrocks (1982) for a further theo-
retical discussion. The following theoretical brief was heavily drawn from
Shorrocks (1982) and Jenkins (1995). Suppose the variable for our analysis
is income and is denoted by vector x. Income has K components, and the
distribution of the kth component across the population is denoted by xk.
The mean of incomes is denoted by x–, and the mean of the kth component
is denoted by x– k. Inequality across incomes is denoted by IGE(x; 2), and
inequality across the kth component is denoted by IGE(xk; 2). The overall
inequality can be expressed as
IGE (x; 2) = ∑ k = 1 Sk with Sk = rk c k IGE (x k ; 2)IGE (x; 2),

K
(A.4)
where rk is the correlation between x and xk, and Xk is the share of that
component in the overall income. Thus, Sk is the absolute contribution of
component k to the overall income. It turns out that the relative contribu-
tion of component k is
∑
K
sk = Sk / IGE (x; 2) = rk c k IGE (x k ; 2)/ IGE (x; 2) and S = 1. (A.5)
k =1 k
Jenkins shows that the absolute change in IGE(x; 2) between time

periods t and t + 1 can be decomposed as
k
t +1
ΔIGE (x; 2) = IGE (x; 2) − IGE
t
(x; 2) = ∑ ΔSk
k =1
k
= ∑ Δrk c k IGE (x k ; 2)/ IGE (x; 2). (A.6)
k =1
Similarly, the proportionate change in inequality can be expressed as
t +1
IGE (x; 2) − IGE
t
(x; 2) K ΔSk K
ΔS K
d IGE (x; 2) = t
=∑ = ∑ sk k = ∑ skd Sk . (A.7)
IGE (x; 2) k =1 Sk / sk k =1 Sk k =1
Table A.8 presents the results using the Georgian dataset for 2003 and
2006. Rows denote different categories of consumption expenditure on food
285
items, nonfood items, utilities, services, and other expenditures for two
years. Column A reports the mean consumption expenditure and the mean
expenditure in each category. Georgia’s mean per capita expenditure in
2003 is GEL 126.1 [6,A], which changes marginally to GEL 126.0 in 2006
[12,A]. The mean per capita expenditure on food in 2003 is GEL 76.9 [1,A],
which decreases to GEL 72.8 in 2006 [7,A]. Mean expenditure on nonfood
also decreases over three years. However, mean expenditures for the other
three categories increase.
Column B reports the mean expenditure of each category as a percent-
age of overall per capita expenditure. The food category accounts for 61.0
percent of per capita expenditure in 2003 [1,B], which falls to 57.8 percent
in 2006 [7,B]. Per capita expenditure on foods is highly correlated with the
overall per capita expenditure—the correlation in 2003 is 80.8 [1,C] (the
upper bound and the lower bound of correlation is 0), which falls to 72.3 in
2006 [7,C], while the correlation between per capita expenditure on utili-
ties and the overall expenditure increases. Inequality of GE(2) for Georgia
increases from 24.2 [6,D] to 27.8 [12,D]. Inequality in per capita food con-
sumption expenditure does not change much, but inequalities in utilities
and expenditures on services drastically increase.
Finally, we look at the contribution of each component to over-
all inequality. As expected, the food category contributes the most to over-
all inequality. This category’s contribution is more than half of the overall
inequality. Its proportionate contribution, however, falls to 40.5 percent in
2006. The proportionate contribution of expenditure on services increases
from 19.6 percent in 2003 [4,F] to 30.2 percent in 2006 [10,F].
Table A.8 is helpful for understanding the source of inequality. This table
can identify components responsible for changes in inequality across two
time periods and the contributory factor to the overall inequality in a single
period of time.
Quantile Function
Figure A.1 graphs the quantile function of per capita expenditure for urban
Georgia. The vertical axis reports per capita expenditure, and the horizontal
286
Appendix
Figure A.1: The Quantile Functions of Urban Per Capita Expenditure, Georgia
600
480
360
Quantile
240
120
0
0 0.2 0.4 0.6 0.8 1.0
Expenditure percentile
2003 2006
axis reports percentiles. A quantile function reports the level below which
per capita expenditure falls for a given population percentage, when the
population is ranked by per capita expenditure. The solid line represents
the quantile function for 2003, and the dotted line corresponds to the urban
distribution of consumption expenditure for 2006. The horizontal lines are
poverty lines for 2003 and 2006.
If a distribution’s quantile function lies completely above that of
another distribution, then the situation is called first-order stochastic
dominance. When a distribution first-order stochastically dominates
another distribution, then every income standard reported ranks the former
distribution better than the latter distribution. If two quantile functions
cross each other, then a dominance relationship may not hold and rank-
ing distributions would depend on the particular per capita expenditure
standards used.
The curve with the solid line represents Georgia’s urban quantile func-
tion in 2003, and the quantile function with the dotted line corresponds
to Georgia in 2006. If a quantile function lies completely above another
287
quantile function, then every lower partial mean of the former distribution
is larger than the corresponding lower partial mean of the latter distribu-
tion. However, in the case of urban Georgia, the two quantile functions
cross each other, which prevents an unambiguous ranking. As evident from
the figure, the 90th percentile in 2006 is larger than the 90th percentile
in 2003, whereas the 40th percentile in 2006 is smaller than that in 2003.
Given that a quantile function is an inverse of the cumulative distribution
function, the example implies that first-order stochastic dominance does not
hold between these two time periods.
Generalized Lorenz Curve
Figure A.2 graphs the generalized Lorenz curve of Georgia’s urban per capita
expenditure for 2003 and 2006. The vertical axis reports the cumulative
mean per capita expenditure and the horizontal axis reports the percentile
of per capita expenditure. A generalized Lorenz curve graphs the share of
mean per capita consumption expenditure spent by each percentile of the
Figure A.2: Generalized Lorenz Curve of Urban Per Capita Expenditure,

Georgia
150
Cumulative mean per capita
120
expenditure (lari)
90
60
30
0
0 0.2 0.4 0.6 0.8 1.0
Percentile of per capita expenditure
2003, Gini=33.49 2006, Gini=35.65
288
Appendix
population. The curve graphs the area under the quantile function up to
each percentile of population, or the height of the Lorenz curve times the
mean per capita expenditure. Thus, the height of the generalized Lorenz
curve is equal to the mean consumption expenditure when the percentile is
one. In other words, the share of the total consumption expenditure spent
by the entire population is 100 percent.
The curve with the solid line represents the generalized Lorenz curve
for urban Georgia in 2003. The generalized Lorenz curve with the dotted
line corresponds to urban Georgia in 2006. If a generalized Lorenz curve
lies completely above another generalized Lorenz curve, then every lower
partial mean of the former distribution is larger than the corresponding
lower partial mean of the latter distribution, and the former distribution
has a larger Sen mean than the latter distribution. Also, when one gener-
alized Lorenz curve lies above another, the distribution corresponding to
the former generalized Lorenz curve is said to second-order stochastically
dominate the distribution corresponding to the latter. In this particular
example, the distribution of per capita expenditure in 2003 second-
order stochastically dominates the distribution of per capita expenditure
in 2006.
General Mean Curve
Figure A.3 graphs the general mean curve of urban Georgia’s per capita
expenditure for two years. The vertical axis reports per capita expenditure,
and the horizontal axis reports parameter α, also known as a society’s degree
of aversion toward inequality. A general mean curve plots the value of
general means of a distribution corresponding to parameter α. The general
mean of a distribution tends toward the maximum and the minimum per
capita expenditures in the distribution when α tends to ∞ and – ∞, respec-
tively.
Given that the largest per capita expenditure in any distribution is usu-
ally several times larger than the minimum per capita expenditure, allowing
α to be very large would prevent any meaningful graphic analysis. So we
restrict α = 1 to be between –5 and 5, which we consider large enough.
The height of the curve at α = 1 denotes the arithmetic mean. Similarly,
the heights at α = 0, α = –1, and α = 2 denote the geometric mean, the
harmonic mean, and the Euclidean mean, respectively.
289
Figure A.3: Generalized Mean Curve of Urban Per Capita Expenditure,

Georgia
400
320
Per capita expenditure (lari)

240
160
80
0
–5 –4 –3 –2 –1 0 1 2 3 4 5
Parameter α
2003 2006
The solid line represents urban Georgia’s general mean curve for 2003.
The general mean curve with the dotted line corresponds to urban Georgia
for 2006. If a general mean curve of a distribution lies completely above
the general mean curve of another distribution, then every general mean
of the former distribution is larger than the corresponding general mean of
the latter. Then, for example, the former distribution would have a higher
arithmetic mean, higher geometric mean, higher harmonic mean, and
higher Euclidean mean than the latter distribution. Note that the standard-
ized general mean curve can be obtained from the general mean curve by
dividing the curve throughout by the arithmetic mean.
Generalized Lorenz Growth Curve
Figure A.4 graphs the generalized Lorenz growth curve for Georgia’s per
capita expenditure. The vertical axis reports the annual growth rate of
290
Appendix
Figure A.4: Generalized Lorenz Growth Curve for Urban Per Capita
Expenditure, Georgia
0.2
means consumption expenditure (%)
Annual growth rate of lower partial
0.15
0.1
0.05
–0.05
0 0.2 0.4 0.6 0.8 1.0
Percentile of per capita expenditure
the lower partial mean consumption expenditures and the horizontal axis
reports the cumulative population share. A generalized Lorenz growth curve
graphs the growth of lower partial mean per capita consumption expendi-
ture for each population percentile. Thus, a generalized Lorenz growth curve
indicates how every lower partial mean is changing over time.
General Mean Growth Curve
Figure A.5 graphs the general mean growth curve for Georgia’s per capita
expenditure. The vertical axis reports the annual growth rate of the general
mean consumption expenditures and the horizontal axis reports parameter
α, also known as a society’s degree of aversion toward inequality. A general
mean growth curve graphs the growth of different general means and thus
indicates how the general means are changing over time. The growth rate
in mean per capita expenditure is the same as the growth rate of general
mean at α = 1.
291
Figure A.5: General Mean Growth Curve of Urban Per Capita Expenditure,
Georgia
0.4
Annual growth rate of general mean

consumption expenditure (%)
0.2
–0.2
–5 –4 –3 –2 –1 0 1 2 3 4 5
Parameter α
References
Jenkins, S. P. 1995. “Accounting for Inequality Trends: Decomposition

Analyses for the UK, 1971–86.” Economica 62 (245): 29–63.
Mookherjee, D., and A. F. Shorrocks. 1982. “A Decomposition Analysis of
the Trend in UK Income Inequality.” The Economic Journal 92 (368):
886–902.
Shorrocks, A. F. 1982. “Inequality Decomposition by Factor Components.”
292
Index
Boxes, figures, notes, and tables are indicated by b, f, n, and t, respectively.
A setting parameters, 264

absolute poverty lines, 27–28, 140, 227–28 specifying datasets, 249–52
additive decomposability properties standard errors, 265–66
definition of, 10 tables, 254–56, 268–69
geographic targeting and, 132–33 variables, 258–63
inequality measures, 81 adjusted headcount ratio, 233
outlined, 86–87 adult equivalence (AE) scales, 48–49
poverty measures, 112–13 age-gender pyramid, headcount ratio and,
additively decomposable properties, 21, 197–99, 198f
37–38, 81 age groups, headcount ratio by, 196–97, 196t
ADePT analysis program aggregate welfare indicator, 232
analysis procedure overview, 248–49 aggregation method data, poverty measures,
calculation assumptions, 156 2–3, 44n1, 45, 46, 106
conventions used, 246 Alkire and Foster, methodology of, 234
examining output, 259 analysis of variance (ANOVA), 21
frequencies calculations, 266 analysis procedure overview, ADePT,
generating report, 257–58 248–49
graphs, 254–56 anonymity standards, 54
if-conditions, 266–67 ANOVA (analysis of variance), 21
installation, 246–47 applications
launching, 247–48 income standards, 9
map variables, 252–54 inequality measures, 18–21
projects, 264–65 poverty measures, 35–36
293
Index
arithmetic means, income standards, 63 components approach, Jalan and Ravallion,

Atkinson, Anthony 229–30
class of inequality measures, 16–17, 22, compositional properties, inequality
91–93 measures, 81
general class of welfare functions, 39 consistency properties subgroup, 37–38
inequality measures by geographic consumption expenditures
regions, 204–7, 205t currency in ADePT, 156
theorem, 24 elasticity of FGT poverty indices to per
capita, 199–201, 199t
B elasticity of poverty to, 275–76, 275t
between-group inequality measures, 20, 21, mean and median per capita, 158–59, 158t
239 mean and median per capita growth and,
bimodal density, 51 183–84, 184t
quantile PCEs and Quantile ratios of per
C capita, 165–67, 166t, 176–78, 177t
calibration properties, income standards, 54 survey data, 46–47
cardinal welfare indicator, 228 consumption regressions, 217–20, 217t
categorical variables, 228–29 counting measures, 29
cdf. See cumulative distribution function cumulative distribution function (cdf)
censored distribution income vector, 113 defined, 52–53, 52f
censored welfare function, 134 as income distribution, 5, 50
characteristics of household head curves. See also generalized Lorenz curves;
headcount ratio by, 184–86, 185t poverty curves
mean and median per capita general mean growth, 79–81, 80f
consumption expenditure, growth general mean of urban per capita
and Gini coefficient by, 184t expenditure, Georgia, 289–90, 290f
population distribution across quintiles growth incidence, 13, 76–77, 77f, 105
by, 187–88, 187t income standard growth, 12–13, 26, 75–81
standard of living and inequality by, Kuznets, 19
183–84 Lorenz, 23–25, 101–3, 102f, 213–15, 214f
chronic poverty measures, 229–30 custom tables, ADePT, 268–69
CHUC. See Clark-Hemming-Ulph- CV (coefficient of variation), 21, 97–98
Chakravarty (CHUC) family of
indices D
circumstances, identity variables, 239 dashboard
Clark-Hemming-Ulph-Chakravarty of dimension-specific deprivation
(CHUC) family of indices measures, 234
defined, 33 of dimensional indicators, 235–36
elasticity to per capita consumption datasets
expenditures, 275–76, 275t data and variable details, ADePT,
poverty measures, 125–26, 203–4, 203t 259–61
poverty orderings of, 40–41 specifying ADePT, 249–52
sensitivity to poverty line, 277–78, 277t variables, ADePT, 258–59
coefficient of variation (CV), 21, 97–98 decomposability property. See also additive
Commission on Growth and Development, decomposability properties;
The Growth Report: Strategies for additively decomposable properties
Sustained Growth and Inclusive geographic targeting and, 132–33
Development, The, 2008, 9 subgroup consistency and, 37–38
294
Index
decompositions rural and urban poor, 162–63, 162t

of generalized entropy measure by income sensitive measures, 236–37
source, 284–86, 284t sensitive poverty measures, 34–35,
of generalized entropy measures, 280–81, 129–30, 133–34
280t smoothed, 20
Gini coefficient, 278–80, 279t spread, 13
headcount ratio and subnational, 181–83, dominance properties
181t analysis, 207–16
of inequality measures, 21–22 conducting analysis of, 70–71
of inequality using second Theil measure, first-order stochastic. See first-order
282–84, 283t stochastic dominance
Oaxaca, 19 inequality measures, 81
poverty changes in growth and Lorenz, 23–25, 102
redistribution, 222–23, 222t poverty measures, 31, 107, 109
degenerate income distributions, 65 second-order stochastic, 11–12, 39
demographic composition, headcount ratio third-order stochastic, 39
by, 192–94, 192t types of, 54
density function, 50–52, 51f, 157–58, 157f unanimous relation and, 69–70, 101–3
deprivation cutoff, 232 doubly censored distributions, 34
deprivation measures, dimension-specific, dual cutoff approach to identification,
234 233
deprivation vector, 29
desirable properties E
income standards, 54–58 economies of scale, 47–48
inequality measures, 81–87 ede (equally distributed equivalent income),
poverty measures, 106–13 9
dimension-specific deprivation measures, 234 education levels, headcount ratios by,
dimensional indicators, dashboard of, 190–92, 191t
235–36 effect, distribution and growth, 43
diminishing marginal utility, 9, 57 efforts, identity variables, 239
distributions. See also income distributions elasticity
base, 105 consumption expenditures and, 275–76,
computing inequality of, 150n9 275t
density function, 157–58, 157f of FGT poverty indices to per capita
doubly censored, 34 consumption, 199–201, 199t
effect, 43 poverty line and, 42
FSD using quantile functions and cdf, 71f employment categories, headcount ratio by,
headcount ratio and poor, 172–73, 172t 188–90, 189t
income types of, 4–6 entropy measures, generalized. See
joint, 237–38 generalized entropy measures
mean of, 149n5 equal-weighted sample, 4–5
of population across quintiles, 169–70, equally distributed equivalent income
169t (ede), 9
of population across quintiles by equivalence scale, 47–48
household head’s characteristics, Euclidean means, income standards, 8, 17,
187–88, 187t 64
of population across quintiles by European Union’s country-level poverty
subnational region, 180–81, 180t lines, 28
295
Index
expenditures. See also consumption growth in, 143

expenditures Sen mean and, 272t
Georgia per capita, 286–92, 287f, 288f, standardized curves of Georgia, 215–16,
290f, 291–92, 292f 216f
per capita, 47 subgroup consistency, 10
total, 47 as welfare measures, 66
generalized entropy measures
F decomposition by geography, 280–81,
FGT. See Foster-Greer-Thorbecke measure 280t
index decomposition by income source,
first-order polarization, 241 284–86, 284t
first order-stochastic dominance (FSD) defined, 17
definition of, 11 by geographic regions, 204–7, 205t
for dominance analysis, 70–71, 70b inequality measures, 96–100
poverty ordering, 38–39 generalized Lorenz curves
using quantile functions and cumulative defined, 11–13
distribution functions, 71f for dominance analysis, 71
focus axiom property poverty measures, 30, growth, 77–79
108–9 income distribution and, 72–73, 73f
formulas. See mathematical formulas poverty deficit curve and, 151n19
Foster and Alkire, methodology of, 234 quantile function and, 72f
Foster-Greer-Thorbecke (FGT) measure index as second-order stochastic dominance,
family of poverty indices, 30, 33–35, 74
38–39, 163–65, 164t for urban per capita expenditure,
per capita consumer expenditure and Georgia, 290–91, 291f
elasticity of, 199–201, 199t of urban per capita expenditure, Georgia,
poverty measures, 30, 123–24 288–89, 288f
squared gap measures, 38–39, 127 geometric means
Foster-Wolfson polarization measure, HDI approach, 236
240–41 income standards, 7–8, 63
four-person income vector, 106, 134–35 Georgia
frequencies calculations, ADePT, 266 general mean curve of urban per capita
FSD. See first-order stochastic dominance expenditure, 290–91, 290f
functions. See cumulative distribution general mean growth curve of urban per
function; quantile functions capita expenditure, 291–92, 292f
generalized Lorenz curve of urban per
G capita expenditure, 288–89, 288f,
gap standard, 32–33 290–91, 291f
gender-age pyramid, headcount ratio and, growth incidence curve of, 212f
197–99, 198f Lorenz curves of urban, 214f
general means, income standards poverty deficit curves in urban areas,
curve, 66f, 75 209f
curve of urban per capita expenditure, poverty incidence curves in urban areas,
Georgia, 289–90, 290f 208f
definition of, 7 poverty severity curves in rural areas,
family of, 62–66 211f
growth curves, 13, 79–81, 80f, 291–92, quantile function of urban per capita
292f expenditure, 286–88, 287f
296
Index
standardized general mean curves of, head of household characteristics. See

216f characteristics of household head
Gini coefficient. See also inequality headcount ratio
measures adjusted, 233
calculating the, 16–17 and age-gender pyramid, 197–99, 198f
decomposition by geography, 278–80, by age groups, 196–97, 196t
279t definition of, 3
definition of, 13 by demographic composition, 192–94,
distribution measures, 93–96 192t
given-sized transfer, 21 distribution of poor and, 172–73, 172t
mean and median per capita by education levels, 190–92, 191t
consumption expenditure, growth by employment category, 188–90, 189t
using, 158–59 in FGT family, 38–39
mean and median per capita income, by household head’s characteristics,
growth using, 171–72, 171t 184–86, 185t
subgroup consistency, 22 by landownership, 194–95, 194t
twin-standard view, 25, 104–5 poverty incidence curve and, 135–36,
goods, private and public, 47 136f
graphs and tables, ADePT, 254–56 poverty measures, 29, 114–15, 130–31,
group-based inequality measures, 19 132–33
growth sensitivity to chosen poverty line, 201–3,
curve types, 12–13, 75–81 202t
curves, 26 subnational decomposition of, 181–83,
effect, 43 181t
elasticity of poverty line, 42 HOI (Human Opportunity Index), 239–40
incidence curve, 105, 212–13, 212f homogeneity
incidence curves, 13 of degree zero properties, 17
inequality and, 103–5 linear, 6, 54
mean and median per capita horizontal inequalities, 239
consumption expenditure and household, definition of, 47
Gini coefficient, 158–59, 158t, household head’s characteristics. See
183–84, 184t characteristics of household head
mean and median per capita income and Human Development Index (HDI),
Gini coefficient, 171–72, 171t 235–36
poverty and, 41–43, 141–44 Human Development Report 2010, 236
of quantile incomes, 76–77, 77f Human Opportunity Index (HOI),
rate of lower partial mean income, 239–40
78–79, 78f hybrid poverty lines, 140, 226–28
redistribution decomposition of poverty
changes and, 222–23, 222t I
The Growth Report: Strategies for Sustained IA-HDI (Inequality-Adjusted Human
Growth and Inclusive Development Development Index), 237
(Commission on Growth and identification
Development 2008), 9 dual cutoff approach to, 233
poverty measures, 26–27, 45, 46
H union approach to, 232–33
harmonic means, income standards, 8, 63 identity variables, 239
HDI (Human Development Index), 235–36 if-conditions, ADePT, 266–67
297
Index
income linear homogeneity, 6, 54, 57

elasticity of poverty line, 42 monotonicity, 55–56
four-person vector, 106 normalization property, 54
gap ratio, 164–65 partial means, 60–62, 62f
individual’s, 46–47 population invariance property, 54, 55
mean and median per capita growth and, poverty measures and, 32–34, 113
171–72 progressive transfer, 56
permutation of, 55 quantile function, 62f
share of the top 1 percent, 16 quantile incomes, 58–60, 59f
source decomposition of generalized regressive transfer, 56
entropy measure, 284–86, 284t robustness of, 10–12
variable, 4 Sen means, 8, 66–69
vectors, 4–5, 113, 128–30, 134–35 subgroup consistency property, 57–58
income distributions. See also Gini symmetry properties, 54–55
coefficient transfer principle property, 56–57
base of, 26–27 types of, 7–8
cdf, 5, 50, 52–53, 52f unanimous relation, 69–70
data collection, 4–5 weak monotonicity, 54, 55–56
degenerate, 65 weak transfer principle, 54
density function, 50–52, 51f welfare functions as, 35
generalized Lorenz curve and, 72–73, 73f independent, path, 237
nonpoverty censored, 106 inequality
poverty measures, 134–35 aversion parameter, 91
poverty measures and, 26–27 decomposition using second Theil
size of, 5–6 measure, 282–84, 283t
skewness of, 51–52 income standards and, 271–73
vector of incomes, 50, 113 origins of, 19
income standards. See also Sen means, standard of living and, 158–59, 171–72,
income standards 171t, 183–84, 184t
anonymity standards, 54 Inequality-Adjusted Human Development
applications for, 9 Index (IA-HDI), 237
arithmetic means, 63 inequality-based approach, 43, 142–43
calibration property, 54 inequality measures. See also Gini
censored, 273–75, 273t coefficient; measures
comparisons of, 10–12 applications of, 18–21
definition of, 6, 54 Atkinson by geographic regions, 204–7,
desirable properties, 54–58 205t
dominance relation, 69–70 Atkinson’s class of, 16–17, 91–93
Euclidean means, 17, 64 between-group, 20, 21
general means, 66f, 143 decomposition of, 21–22
general means as normative family of, desirable properties, 81–87
62–64 dominance and unanimity, 101–3
geometric means, 63 dominance properties, 81
growth curves, 12–13, 26 examples of, 15–18
harmonic means, 63 generalized entropy measures, 96–100
inequality and, 271–73 group, 238–40
inequality measures and, 87 group-based, 19
invariance properties, 54 growth and, 103–5
298
Index
income standards and, 87 linear homogeneity, income standards, 6,

interpreting, 81 54, 57
invariance properties, 81 Living Standard Measurement Study
Lorenz curve, 101–3, 102f (LSMS), 48–49
normalization properties, 81 logarithmic deviation measure, mean, 17,
normalization property, 83 22, 97
overall, 19 Lorenz curves, 23–25, 101–3, 102f, 213–15,
partial means ratio, 89–91 214f. See also generalized Lorenz curves
population invariance property, 82–83 Lorenz dominance, 23–25, 102
poverty measures and, 113 lower end quantile ratio, 89
properties of, 13–14, 20–21 lower partial means, income standards, 7,
quantile ratio, 87–89 60–61, 72–73, 78–79, 78f
scale invariance property, 83 LSMS (Living Standard Measurement
smoothed distributions, 20 Study), 48–49
subgroup consistency property, 85–86
subgroup levels, 21–22 M
symmetry properties, 81–82 map variables, ADePT, 252–54
transfer principle property, 83–85 marginal utility, diminishing, 9
twin-standard view of, 25 mathematical formulas
weak transfer property, 83 AE scales, 48–49
welfare function and, 18, 100 fundamental assumptions, 49
within-group, 21 mean logarithmic deviation measure, 17,
inequality of opportunity, 20, 238–40 22, 97
installation of ADePT, 246–47 means. See also income standards
intermediate poverty lines, 29 of distribution, 149n5
intersection approach, 232 gap measure, poverty measures, 124–25
invariance properties income distribution size, 5–6
income standards, 54 subgroup consistency of, 10
inequality measures, 81, 82–83 measures. See also inequality measures;
poverty measures, 30–31, 107, 108 poverty measures
Foster-Wolfson polarization, 240–41
J generalized entropy, 17
Jalan and Ravallion components approach, mean log deviation, 17
229–30 Theil’s first, 18, 24
joint distributions, 237–38 Theil’s second, 17, 22
of welfare, 8–9
K median income, distribution size, 6
Kuznets, Simon mixed quantile ratio, 89
curves, 19 money-metric wealth indicator, 47
hypothesis, 19 monotonic transformation, 21
ratio, 13, 16, 90–91 monotonicity, income standards, 55–56.
See also weak monotonicity
L Multidimensional Poverty Index (MPI),
landownership, headcount ratio by, 194–95, 233–34
194t multidimensional poverty measures, 230–34
launching ADePT, 247–48 multidimensional standards, 234–38,
lessons for policy makers. See policy maker 236–37
lessons multimodal density, 51
299
Index
N consumption regressions, 219–20

national level, analysis at, 157 decomposition of generalized entropy
90/10 ratio, 88–89 measures by geography, 281
nonpoverty censored distribution of decompositions of generalized entropy
income, 106, 113 measure by income source, 286
normalization properties distribution of population across
income standards, 6, 54, 57, 83 quintiles, 170
inequality measures, 81 distribution of population across quintiles
poverty measures, 107 by subnational region, 181
normalized gap vector, 30 elasticity of FGT poverty indices to per
capita consumption expenditures, 201
O elasticity of poverty to per capita
Oaxaca decompositions, 19 consumption expenditures, 276
OECD (Organisation for Economic Gini coefficient decomposition by
Co-operation and Development), 48 geography, 279–80
opportunity, inequality of, 20, 238–40 headcount ratio by age groups, 197
ordinal variables, 228 headcount ratio by demographic
Organisation for Economic Co-operation composition, 193–94
and Development (OECD), 48 headcount ratio by employment
output, examining ADePT, 259 categories, 190
overall poverty, 174–76, 174t, 175t overall poverty, 161–62
partial means and partial means ratios,
P 168–69
parameters for ADePT, setting, 264 partial means and partial means ratios in
partial means subnational regions, 178–79
income standards, 7, 60–62, 62f poverty deficit curves in urban Georgia,
partial means ratios and, 167–69, 168t, 209f
178–79, 178t poverty incidence curves in urban
ratio, 16, 89–91 Georgia, 208–9
path independent, 237 poverty measures, 203–4, 203t
PCEs (per capita expenditures), 47, 165–66, poverty severity curves in rural Georgia,
166t, 176–78, 177t 211–12
Pen’s Parade, 5 quantile PCEs and Quantile ratios of per
per capita expenditures (PCEs), 47, 165–66, capita consumption expenditures,
166t, 176–78, 177t 167, 178
permutation of income, 55 rural and urban poor population
persistent poverty measures, 229–30 distributions, 163
Pigou-Dalton transfer principle. See transfer sensitivity of measures to poverty line,
principle properties 278
polarization measures, 240–41 sensitivity of poverty measures to choice
policies, public, poverty measures of poverty line, 203
influencing, 130–31 squared gap measures, 165
policy maker lessons squared gap measures and subnational
Atkinson measures and generalized contribution to overall poverty, 176
entropy measures by geographic standard of living and inequality across
regions, 206–7 population, 159
changes in probability of being in standardized general mean curves of
poverty, 221–22 Georgia, 216
300
Index
subnational decomposition of headcount sensitivity of poverty measures to choice

ratio, 182–83 of, 201–3, 202t
poor income standard, 3, 32 types of, 140–41
poor population distribution, rural and U.S., 27
urban, 162–63, 162t poverty measures
population invariance properties advanced, 118–26
assumption, 4–5 aggregate data, 2–3, 44n1
income standards, 54, 55 analysis of, 106
inequality measures, 14, 82–83 applications of, 35–36
poverty measures, 30, 108 censored income standards and, 273–75
population subgroup consistency, 10 chronic, 229–30
poverty CHUC family of indices, 203–4, 203t
analysis across other population counting, 29
subgroups, 183 deficit curve, 39–40
changes in probability of being in, 220– definition of, 29–30
22, 220t desirable properties, 106–13
description of, 1 distribution-sensitive, 34–35, 129–30,
evaluating within society, 45–46 133–34
growth and, 41–43, 141–44 dominance properties, 31, 109
growth and redistribution decomposition elasticity of FGT indices to per capita
of changes in, 222–23, 222t consumption expenditures, 199–201,
growth curves, 41–42 199t
growth elasticity of, 42 elasticity to per capita consumption
incidence curve, 135–36, 136f expenditures, 275–76, 275t
overall, 160–62, 160t, 174–76, 174t, 175t FGT family of indices, 30, 33–34
poor income standard, 32 FGT index, 123–24
profile, 36 focus axiom properties, 30, 108–9
subgroup’s contribution to, 38 gap, 30, 38–39, 114–18, 164–65
ultra, 225–26 gap and contribution to overall poverty
World Development Report 2000/2001: in subnational regions, 174–75
Attacking Poverty, 231 gap and deficit curve, 137, 137f
poverty curves gap in subnational regions, 174t
deficit, 39–40, 136–38, 137f, 151n19, headcount ratio, 29, 114–15
209–10, 209f identification step, 26–27, 45
growth, 41–42 identifying, 2–3
incidence, 39, 135–36, 136f, 207–9, 208f incidence curve, 39
severity, 39, 137, 139–40, 139f, 210–12, income distribution and, 26–27
211f income standards and, 32–34, 113
value, 38 inequality measures and, 113
poverty lines invariance properties, 107
absolute, 27–28, 140, 227–28 mean gap measure, 124–25
definition of, 26–27 multidimensional, 230–34
hybrid, 140, 226–28 normalization properties, 107
identifying, 3 numerical, 105
income elasticity of, 42 ordering, 38, 40–41, 140
intermediate, 29 persistent, 229–30
relative, 28–29, 42, 227–28 policy relevance of, 128–31
sensitivity of measures to, 277–78, 277t poor income standard, 3
301
Index
poverty measures (continued) quintile population, distribution of, 169–70,

population invariance properties, 30 169t, 187–88, 187t
process of, 2–3
properties of, 30–32 R
pros and cons of, 126–28 Rawls’s welfare function, 151n17
public policy influenced by, 130–31 regressions, consumption, 217–20, 217t
scale invariance properties, 31 regressive transfer, definition of, 56
sensitivity of headcount ratio to chosen relative poverty lines, 28–29, 42, 140,
poverty line, 201–3, 202t 227–28
severity curve, 39 relative slope, 24
squared gap measures, 38–39, 121–23 replication variance, 5
SST index, 119–21, 203––204, 203t report generation, ADePT, 257–58
subgroup consistency properties, 112 residual term, geographical interpretation
symmetry properties, 30, 107 of, 150n9
transfer principle properties, 110–11 results, ADePT output, 259
transfer sensitivity properties, 111 robustness
Watts index, 118–19, 203–4, 203t of income standards, 10–12
weak monotonicity, 31, 109 of poverty comparisons, 41–42
weak transfer properties, 31, 110–11 rural/urban decomposition, 157
World Bank’s main standard, 27
private goods, 47 S
pro-poor growth, 77, 141–44 sample, equal-weighted, 4–5
probability density function, 50–51, 51f scale invariance properties
progressive transfer, definition of, 56 inequality measures, 14
projects, working with ADePT, 264–65 poverty measures, 31
public goods, 47 scales
public policy, poverty measures influencing, AE, 48–49
130–31 economies of, 47–48
equivalence, 47–48
Q invariance property, 82–83, 108
quantile functions second-order polarization, 241
definition of, 5–6, 53–54, 53f second-order stochastic dominance, 11–12,
FSD using cdf and, 71f 39, 74
generalized Lorenz curves, 11–12, 72f second Theil measure, 22
partial means and, 61–62, 62f Sen means, income standards
Pen’s Parade, 5 definition of, 8
quantile income and, 59–60, 59f general means and, 272t
of urban per capita expenditure, Georgia, naming of, 149n6
286–88, 287f as two incomes, 66–69
quantile incomes Sen-Shorrocks-Thon (SST) index
definition of, 7 definition of, 33
income standard, 58–60, 59f elasticity to per capita consumption
quantile ratios and, 165–67, 166t, expenditures, 275–76, 275t
176–78, 177t poverty measures, 119–21, 203–4, 203t
quantile ratios pros and cons of, 126–27
inequality measures, 87–89 sensitivity to poverty line, 277–78,
quantile incomes and, 165–67, 166t, 277t
176–78, 177t Sen’s capability approach, 231
302
Index
sensitivity. See also transfer sensitivity subnational regions

properties analysis at level of, 170
analysis of, 199–207 decomposition of headcount ratio,
distributions, sensitive measures, 236–37 181–83, 181t
distributions, sensitive poverty measures, distribution of population across quintiles
34–35, 129–30, 133–34 by, 180–81, 180t
headcount ratio and sensitivity to chosen headcount ratio in, 172–73, 172t
poverty line, 201–3, 202t mean and media per capita income,
to poverty line, 277–78, 277t growth and Gini coefficient in,
SES (socio-economic status), 20 171–72, 171t
single welfare indicator, 238–39 partial means and partial means ratios in,
skewness 178–79, 178t
density function and, 158 poverty gap measure and contribution to
of income distributions, 51–52 overall poverty, 174–75, 174t
slope, relative, 24 squared gap measures and contribution to
smoothed distributions, 20 overall poverty, 175–76, 175t
socio-economic status (SES), 20 survey data, consumer expenditure, 46–47
space, poverty assessment of, 2 symmetry properties
space selection, evaluating poverty with, 46 income standards, 54–55
spells method, 230 inequality measures, 14, 81–82
squared coefficient of variation, half, 17 poverty measures, 30, 107
squared gap measures
poverty measures, 38–39, 121–23 T
poverty severity curve and, 139, 139f table cells, ADePT, 156
pros and cons of, 127 tables and graphs, ADePT, 254–56, 268–69
squared gap vector, 30 targeting, additive decomposability and
SST. See Sen-Shorrocks-Thon (SST) index geographic, 132–33
standard errors, ADePT, 265–66 targeting exercise, poverty measures
standard of living across population, influencing, 128–30
inequality and, 158–59, 171–72, Theil’s first measure, 18, 24
171t, 183–84, 184t Theil’s second measure, 17, 22, 97, 282–84,
standardized general mean curves of 283t
Georgia, 215–16, 216f third-order stochastic dominance, 39
standards, multidimensional, 234–38, 236–37 total expenditure, 47
stochastic dominance. See dominance transfer neutral, 34
properties transfer principle properties
strong transfer, 150n13 income standards, 54, 56–57
studies, income standards, 9 inequality measures, 8–9, 14, 81, 83–85
subgroups poverty measures, 110–11
consistency properties, 37–38, 57–58, 81, regressive and progressive, 56
85–86, 112 strong transfer as, 150n13
contribution to overall poverty, 38 transfer sensitivity properties
decomposability properties, 37–38 inequality measures, 14, 20–21, 81,
inequality levels, 21–22 84–85
population consistency, 10, 22 poverty measures, 31, 111
poverty analysis across other population, transformation, monotonic, 21
183 twin income standards, 15–16
poverty measures, 132–33 twin-standard view of inequality, 25, 103–4
303
Index
U poverty orderings of, 40–41

ultra-poverty, 225–26 pros and cons of, 127–28
unanimous relation and dominance, 69–70, sensitivity to poverty line, 277–78, 277t
100 weak monotonicity
union approach to identification, 232–33 income standards, 6, 54, 55–56
unit consistency property, 149n7 poverty measures, 31, 109
upper end quantile ratio, 89 weak relativity axiom, 228
upper partial means, income standards, 7, weak transfer properties. See also transfer
60–61 principle properties; transfer
urban/rural decomposition, 157 sensitivity properties
U.S. poverty line, 27 defined, 150n14
utility, diminishing marginal, 9 income standards, 54
inequality measures, 14, 83
V poverty measures, 31, 110–11
variable income, 4 wealth indicator, money-metric, 47
variable line poverty ordering, 38 weighted sample, equal, 4–5
variable measure poverty ordering, 41–42, weighting, method of, 121–22
140 welfare
variables, ADePT aggregate indicator, 232
dataset’s data and details of, 259–61 Atkinson’s general class of functions, 39
deleting, 263 cardinal indicator, 228
expressions, 263 censored function, 134
generating, 261–62 function and inequality, 18
replacing, 262 functions as income standards, 35
variance, analysis of (ANOVA), 21 general means as measures of, 8–9, 66
variance, replication, 5 indicator, 45–46
variation, coefficient of, 21 inequality measures and, 100
vector of incomes, 4–5, 50 Rawls’s function, 151n17
single indicator, 238–39
W within-group inequality measures, 21
Watts index World Bank
definition of, 33 HOI (Human Opportunity Index),
elasticity to per capita consumption 239–40
expenditures, 275–76, 275t main poverty standard, 27
poverty measures, 118–19, 203––204, World Development Report 2000/2001:
203t Attacking Poverty, 231
304
ECO-AUDIT
Environmental Benefits Statement
The World Bank is committed to preserving Saved:
endangered forests and natural resources. The • 13 trees
Office of the Publisher has chosen to print A • 6 million BTUs
Unified Approach to Measuring Poverty and of total energy
Inequality: Theory and Practice on recycled • 1,111 pounds of net
paper with 50 percent postconsumer fiber in greenhouse gases
accordance with the recommended stand- • 6,030 gallons of waste
ards for paper usage set by the Green Press water
Initiative, a nonprofit program supporting
• 404 pounds of solid
publishers in using fiber that is not sourced
waste
from endangered forests. For more informa-
tion, visit www.greenpressinitiative.org.

A Unified Approach to Measuring Poverty and Inequality

Written by

Document Information

Copyright

ISBN

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

ISBN:

A Unified Approach to Measuring Poverty and Inequality

Written by

Copyright:

ISBN:

A Unified Approach to

Measuring Poverty and

Health Equity and Financial Protection: Streamlined Analysis with ADePT

Assessing Sector Performance and Inequality in Education: Streamlined

ADePT User Guide (forthcoming) by Michael Lokshin, Zurab Sajaia, and

For more information about Streamlined Analysis with ADePT software

Some rights reserved

Rights and Permissions

ISBN (paper): 978-0-8213-8461-9

Cover photo: Scott Wallace/World Bank (girl and child)

Library of Congress Cataloging-in-Publication Data

Applying If-Conditions to Outputs ................................................................266

Appendix ................................................................................................. 271

Index ........................................................................................................ 293

2.1: Probability Density Function ..................................................................51

3.1: Mean and Median Per Capita Consumption Expenditure,

3.13: Partial Means and Partial Mean Ratios for Subnational

This book is an introduction to the theory and practice of measuring

These two components—a unifying framework for measurement and the

We are grateful to the Oxford Poverty and Human Development

• Inadequate resources to buy the basic necessities of life

The presence of poverty commonly leads groups to undertake activities

At the United Nations Millennium Forum in 2000, 193 countries agreed

1. Choose the space in which poverty will be assessed. The traditional

2. Identify the poor. This step involves selecting a poverty line

Applying and interpreting poverty measures require understanding the

The Income Variable

Income distribution data can be represented in a variety of ways. The

invariant to the population size, in that a replication of the vector (having,

Income Standards and Size

What Is an Income Standard?

To understand what a measure or index means, explicitly stating a set of

• Normalization states that if all incomes happen to be the same, then

These basic requirements ensure that the income standard measures

Four types of income standards are in common use:

that everyone’s utility function is identical and strictly increasing. In this

Income standards are used to assess a population’s prosperity, the way it

In certain empirical applications, there is a natural concern for certain iden-

Dominance and Unanimity

However, if generalized Lorenz curves cross, then the final judgment is

Some analyses go beyond the question of which distribution is larger to con-

Inequality Measures and Spread

The second aspect of the distribution—spread—is evaluated using a numeri-

What Is an Inequality Measure?

There are four basic properties for inequality measures:

the distribution and b is the income at a higher percentile q of the distribu-

1. Create an N × N matrix having a cell for every possible pair of

Inequality and Welfare

The techniques for evaluating between-group inequality involve smoothing

Subgroup Consistency and Decomposability

defined with reference to an underlying variable such as schooling, the anal-

Dominance and Unanimity

One alternative to numerical inequality measures for making inequality

general class of welfare functions considered by Atkinson. This is a very use-

A similar discussion applies to all the generalized entropy measures, apart

Growth and Inequality

The twin-standard view of inequality offers fresh insights on the relation-

The growth curves described above can be useful in understanding

Poverty Measures and the Base of the Distribution

Absolute Poverty Line

• The $1.25-per-day standard of the World Bank that is used to com-

An absolute poverty line is frequently used for evaluating poverty within

• Several competing methods are available for deriving an absolute

Relative Poverty Line