Policy Research Working Paper 10470
A New Distribution Sensitive Index
for Measuring Welfare, Poverty, and Inequality
Aart Kraay
Christoph Lakner
Berk Özler
Benoit Decerf
Dean Jolliffe
Olivier Sterck
Nishant Yonzan
Development Economics
June 2023
Policy Research Working Paper 10470
Abstract
Simple welfare indices such as mean income are ubiqui- three main definitions of distribution sensitivity in the lit-
tous but not distribution sensitive. In contrast, existing erature. Variants on this index can be used as distribution
distribution sensitive welfare indices are rarely used, often sensitive poverty measures and as inequality measures, with
because they are difficult to explain and/or lack intuitive the same simple intuitive units. The properties of the new
units. This paper proposes a simple new distribution sen- index are illustrated using the global distribution of income
sitive welfare index with intuitive units: the average factor across individuals between 1990 and 2019, as well as with
by which individual incomes must be multiplied to attain a selected country comparisons. Finally, the index can be used
given reference level of income. This new index is subgroup to define the “prosperity gap” as a proposed new measure of
decomposable with population weights and satisfies the “shared prosperity,” one of the twin goals of the World Bank.
This paper is a product of the Development Economics. It is part of a larger effort by the World Bank to provide open access
to its research and make a contribution to development policy discussions around the world. Policy Research Working
Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at akraay@worldbank.
org, clakner@worldbank.org, bozler@worldbank.org, bdecerf@worldbank.org, djolliffe@worldbank.org, olivier.sterck@
uantwerpen.be, and nyonzan@worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
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A New Distribution Sensitive Index
for Measuring Welfare, Poverty, and Inequality
Aart Kraay1ⓡ Christoph Lakner1ⓡ Berk Özler1ⓡ Benoit Decerf1ⓡ
Dean Jolliffe1ⓡ Olivier Sterck2ⓡ and Nishant Yonzan1
Keywords: Welfare Index, Poverty Index, Inequality Index, Shared Prosperity
JEL Classifications: D63, I32
1 World Bank; 2 University of Oxford and University of Antwerp. We thank Matthew Adler, Benu
Bidani, Francisco Ferreira, Laurence Chandy, Gaurav Datt, Maya Eden, Garance Genicot, Peter Lanjouw,
Max Lawson, Luis-Felipe Lopez-Calvo, Daniel Mahler, Roy Van der Weide, and seminar participants at
the World Bank’s Poverty and Equity Global Practice and the Normative Economics and Economic
Policy e-seminar for helpful discussions and feedback. The names of the authors are in random order,
using the author randomization tool of the American Economic Association, indicated by “ⓡ” in the
author list and archived here. We acknowledge financial support from the UK government through the
Data and Evidence for Tackling Extreme Poverty (DEEP) Research Programme. The views expressed
here are the authors’, and do not necessarily reflect the official views of the World Bank, its Executive
Directors, or the countries they represent. A verified reproducibility package for this paper is available
here. Stata and R scripts for implementing the new welfare, poverty and inequality indexes introduced in
this paper are available at the end of this blog.
1 Introduction
Average income per capita and the poverty headcount are among the most common
measures used to summarize living standards and to assess progress towards national and
international development goals. The popularity of these welfare measures is in no small measure
due to their simplicity, which makes them easy to understand and to communicate to wide
audiences. However, these measures also have a fundamental limitation: they are silent on the
extent of inequality among all individuals (in the case of average income) and inequality among
the poor (in the case of the headcount).
This limitation does not sit well with the widely held view that inequality matters:
specifically, that welfare measures should be more responsive to changes in income that take place
at the bottom of the distribution. This view has at least four justifications. First, with decreasing
marginal utility of income, the social preferences of a utilitarian policymaker give more weight to
those at the bottom of the distribution.1 Second, most major philosophical traditions and religions
express a special concern for the worse-off (World Bank, 2005). Third, the fact that several of the
Sustainable Development Goals have distributional elements reveals a preference among
international policymakers for lower inequality.2 Fourth, direct empirical estimates suggest a high
degree of inequality aversion, although the variance of estimates is large (Clark and D’Ambrosio
2015, Kot and Paradowski, 2022).
This limitation is well-understood, and a long literature has generated an impressive variety
of distribution sensitive welfare measures to address it.3 However, these measures are rarely used
in practice. To illustrate this, Table 1.1 displays the results of a simple keyword search for leading
poverty and welfare measures, in three bodies of documents: academic research papers in EconLit;
policy documents of major multilateral development banks as curated at nlp4dev.org; and general
writing in the Google N-gram Viewer. Across all three categories of documents, the frequency of
references to poverty headcount is two to three orders of magnitude larger than the frequency of
references to the most common distribution sensitive poverty measures, i.e., the squared poverty
gap and the Watts index. The pattern is similar for welfare indices in the bottom panel: references
to average income are two orders of magnitude more common than references to the best-known
distribution sensitive welfare measures like the Atkinson and Sen indexes.
1
This is related to prioritarianism, an ethical theory that gives extra weight to the well-being of the worse-off (Adler
and Norheim, 2022).
2
Such preferences are not just revealed but explicitly stated in international efforts to move beyond per capita GDP
as a welfare measure. For example, the Stiglitz-Sen-Fitoussi Commission on the Measurement of Economic
Performance and Social Progress criticized measures of per-capita income and wealth because they give no indication
of how the available resources are distributed across people (Stiglitz, Sen and Fitoussi 2009).
3
For example, early critiques of the headcount index for its lack of distribution sensitivity go back to Watts (1969)
and Sen (1976). Distribution sensitive poverty and welfare measures have been introduced by numerous scholars,
including Watts (1969), Atkinson (1970), Sen (1976), Thon (1979), Takayama (1979), Kakwani (1980), Clark,
Hemming, and Ulph (1981), Donaldson and Weymark (1980), Chakravarty (1983), Foster, Greer, and Thorbecke
(1984), Hagenaars (1987), Chakravarty (2009).
1
Table 1.1. Use of Poverty and Welfare Measures in Academic, Policy, and Public Discourse
Academic Use Policy Use Public Discourse
Multilateral Development
EconLit* Banks at nlp4dev.org* Google Ngram**
Poverty Measures
Poverty Rate OR Poverty Headcount *** 40832 26730 40.49
Poverty Gap 4438 4609 6.84
Squared Poverty Gap 473 681 0.32
Watts Index 47 52 0.09
Welfare Measures
Average income 61493 15492 36.11
Atkinson Index 744 122 0.49
Sen Index 258 70 0.12
Notes: * Number of documents. ** 1 million times fraction of corpus in 2019. *** Search results in last column for
“poverty rate” only as Ngram viewer does not support Boolean search. Search results retrieved March 22 -24, 2023.
One plausible reason for the limited uptake of distribution sensitive measures is that they
are difficult to explain in non-technical terms. For example, it is challenging to define the Atkinson
index as anything other than a constant elasticity of substitution aggregate of individual incomes.
Compounding the problem, distribution sensitive measures also often have non-intuitive units. For
example, the squared poverty gap is measured in squared percent deviations from the poverty line,
while the Watts poverty index is measured in log-point differences from the poverty line that
cannot easily be interpreted as percent differences unless they are small. While these complications
of interpretation and units are of little concern to mathematically minded users, they represent only
a small proportion of the audience for welfare measures. For the wider audience of policymakers
and the public, “average income in dollars” or “the number of poor people” are far more
understandable, and therefore far more effective in influencing policy and public opinion despite
their shortcomings. This dilemma is well-summarized by the succinct remark in Watts (1969) that
the headcount has “little but its simplicity to recommend it.”
To address this problem, we propose a new distribution sensitive welfare index with a
simple mathematical formulation and intuitive units. Our basic measure, described more formally
in Section 2.1, is the average across all individuals of / , where is the income of individual
and is a reference income level. As an average, its formulation is simple to describe. Moreover,
the average has very intuitive units. Each term in the average is the factor by which the income of
the corresponding individual needs to be multiplied to attain the reference level of income, and the
index simply is the average factor by which individual incomes need to be multiplied to reach the
reference level of income.
In addition to being extremely simple to explain, this measure has several desirable
properties. In Section 2.3, we show that it satisfies all three definitions of distribution sensitivity
in the literature: Pigou-Dalton sensitivity, transfer sensitivity, and growth sensitivity. In addition,
the new index is subgroup decomposable with population weights, which means for example that
its value at the world level is simply a population-weighted average of its value in each country in
2
the world. In fact, we show that our measure is unique among all existing named welfare indices
we could identify in the literature in that it is both population-weighted subgroup decomposable
and satisfies all three concepts of distribution sensitivity.
We develop two variants on our basic measure. The first variant limits attention to
individuals with incomes below the reference income level by censoring incomes above it (Section
2.2). This results in a traditional poverty measure that is responsive only to changes in income
below the reference income level. The second one transforms our welfare index into an associated
relative inequality measure by replacing the absolute level of reference income with average
income (Section 2.5). In both cases, the variants retain the simple interpretation of the welfare
index: the average factor by which individual incomes must be multiplied to attain the reference
level of income.
We illustrate the empirical properties of our new welfare index in Section 3. We first
document levels and trends in the new measure calculated across the world income distribution
between 1990 and 2019. This was a period of strongly pro-poor growth, in the sense that growth
was on average much higher in lower percentiles of the world income distribution than in higher
percentiles. Consistent with its strong distribution sensitivity, we show our new welfare index
improves substantially faster than other welfare measures over this period. In addition, we focus
on three pairs of countries, a pair of low-income countries (Burkina Faso and Mali), a pair of
middle-income countries (Colombia and Peru) and a pair of high-income countries (France and
the United States), showing that our new measure can lead to reordering of welfare comparisons
relative to existing measures.
In Section 4, we conclude with a specific policy application of the new welfare index. Since
2013, the World Bank has articulated its twin goals as “ending extreme poverty and boosting
shared prosperity” (World Bank, 2015). Extreme poverty is measured as the share of the world’s
population living below the international poverty line, currently set at $2.15 per day in 2017 $PPP,
while shared prosperity is measured at the country level as the growth rate of average incomes in
the bottom 40 percent. We propose using our new index a new measure of shared prosperity, and
refer to it as the global “prosperity gap”. In this application, we set the reference income level to
$25 per day, a “prosperity standard” roughly corresponding to the median poverty line of high-
income countries. The global prosperity gap in 2019 is 5.0, indicating that on average incomes
need to be multiplied by a factor of 5.0 to reach the prosperity standard. We show how the
prosperity gap varies across countries and over time, and demonstrate that it has better theoretical,
empirical, and communications properties than growth in the bottom 40 percent. Readers who are
primarily interested in the prosperity gap measure can skip Sections 2 and 3 of the paper and jump
directly to Section 4.
3
2 A New Distribution Sensitive Index
2.1 Basic notation and intuition
The new index is given by:
1
(, ) = ∑ , (1)
=1
where = 1, … , indexes individuals; denotes individual ’s economic resources, which we
refer to as “income” throughout the paper; ≡ (1 , … ) denotes the distribution of incomes
ordered from smallest to largest; and > 0 is a parameter whose value is interpreted as a reference
income level. As discussed further below, can be a poverty line if it is set at some accepted fixed
minimum level; it can be a “prosperity standard” if it is set as some accepted fixed higher
aspirational level; or it can be replaced by a linear function of incomes () such as average
income to obtain a measure of relative inequality.
The key innovation of our index is to express the contribution of individuals to the welfare
index as a ratio of the reference level of income to their individual income . This ratio
represents the factor by which their income must be multiplied to reach . By contrast, existing
poverty indices typically express individual contributions as a difference or gap − . Expressing
the contribution of individuals as a ratio has two advantages. First, the ratio formulation is a
convex function which is distribution sensitive, as we discuss more formally in Section 2.3. By
contrast, the simple difference or gap − is a linear function and is therefore not distribution
sensitive. To make it distribution sensitive, a further convex transformation needs to be applied
(e.g., exponentiation or minus log), at the cost of complicating interpretation. Second, the ratio
formulation leads to two intuitive and appealing interpretations:
(1) The index is literally the average factor by which individual incomes must be multiplied to
reach . It is therefore intimately related to economic growth, capturing the average (gross)
growth rate needed to reach .4 This property also means that the index responds intuitively
to distribution-neutral growth: when all incomes are doubled, the value of the index is
divided by two.
(2) The new index captures the intuition that if person A has half the income of person B,
person A is twice as “poor” as person B. The index is a simple average of individual
contributions, where the contribution of a person at the threshold is 1, the contribution of
a person with half the threshold is 2, the contribution of someone with one-third of the
threshold is 3, and so on (Figure 2.1). In this way, individual contributions to the overall
index are intuitively and transparently distribution sensitive. In Appendix B we
4
We note the parallel between this interpretation and Morduch (1998)’s interpretation of the index of Watts (1969) as
continuously compounded growth needed to get to in years.
4
demonstrate that our new welfare index is the only symmetric subgroup decomposable
welfare index that satisfies this proportionality property.
Finally, the value selected for the reference income affects the value taken by the index: higher
values of reference income naturally imply a greater average multiple needed to reach that
reference income level. However, it does not affect the ranking of distributions associated with
index : rather, it only scales the value of the index by a fixed factor equal to .
Figure 2.1 - Individual contribution (ratio ) as a function of income .
2.2 Censored indices
The new index is inclusive in the sense that it is affected by the incomes of all individuals.
This is a typical feature of welfare indices. By contrast, poverty indices conventionally focus
exclusively on individuals who earn less than the reference income, i.e. the poverty line. We
propose two ways of adjusting the general index in Equation (1) to focus on individuals earning
less than the reference income. The first approach is simply to top-code incomes at the reference
income . This yields the following censored version of our measure:
1
(, ) = ∑ , (2)
=1
where censored income
is defined as ≡ if < and ≡ if ≥ . The interpretation
of the index is the same as that of the index , except that it applies to the censored distribution
of incomes. It retains the simple intuitive interpretation as the average factor by which incomes
5
need to be multiplied to attain the standard of living defined by the threshold , although with the
minor qualification that no increase, i.e., a factor of one, is needed for people with incomes above
the threshold.
The second approach is closer in spirit to the poverty measurement literature because it
forces individual contributions to be equal to zero (as opposed to one for above) for individuals
with income greater than or equal to . To do this, we define the index = − 1, which yields
the expression:
1 1 −
(, ) = ∑ ( − 1) = ∑ ( ), (3)
=1 =1
where corresponds to the number of individuals with incomes below .
By definition, both and yield the same ranking of income distributions. However,
can be interpreted as a growth rate instead of a factor: it is the average growth rate needed to
attain the standard of living defined by the threshold, again with the qualification that zero growth
is needed for people who are above the threshold.
2.3 Properties
The index is not only easy to interpret and communicate, but it also satisfies two key
properties that make it especially useful for policy purposes: it is distribution sensitive according
to all three main notions of distribution sensitivity in the literature, and it is subgroup
decomposable with population weights. Furthermore, it also satisfies several additional standard
properties that are expected of welfare indices.
The literature has advanced three main definitions of distribution sensitivity.5 First, a
measure is Pigou-Dalton sensitive if it improves when a progressive transfer is carried out between
two individuals.6 Formally, Pigou-Dalton sensitive measures should satisfy the monotonicity and
transfer axioms. Second, a Pigou-Dalton sensitive measure is transfer sensitive if it satisfies the
transfer sensitivity axiom, which goes one step further, requiring measures to give more weight to
equal-sized transfers taking place lower in the distribution. Third, a Pigou-Dalton sensitive
measure is growth sensitive if the elasticity of the measure with respect to income of the p-th
percentile is monotonically decreasing in p, i.e., a one percentage point growth in income of a
poorer person improves the welfare measure by a greater percentage amount than a one percentage
point growth for a richer person (Ray ⓡ and Genicot, 2022; Dollar, Kleineberg and Kraay, 2015).
This requirement is the “growth” analogue of the transfer axiom and is very much related to the
notion of pro-poor growth proposed by Ravallion and Chen (2003). Transfer- and growth-sensitive
5
For all the mathematical definitions and proofs required for this sub-section, please see Appendix A.
6
An index satisfying the monotonicity and transfer axioms is often referred to as “distribution sensitive” in the
literature. In contrast, we refer to such an index being Pigou-Dalton-sensitive to reserve the term “distribution
sensitive” for the larger set of properties encompassing all three notions that we use.
6
measures satisfy Pigou-Dalton sensitivity by definition, while the opposite is not necessarily true.
In contrast, growth sensitivity does not imply and is not implied by transfer sensitivity.
The new index satisfies these three definitions of distribution sensitivity. First, it is Pigou-
Dalton sensitive as it satisfies the monotonicity and transfer axioms. Second, it is transfer sensitive
because it also satisfies the transfer sensitivity axiom. Finally, it is growth sensitive, i.e., it satisfies
the growth progressivity axiom proposed by Ray ⓡ and Genicot (2022).7 The uncensored version
of the index ( ) satisfies these axioms globally, while the censored version ( , as well as )
satisfies them for incomes below the threshold.
Table 2.1 presents several poverty measures and welfare indices found in the literature,
providing their definitions, elasticities with respect to income, as well as showing whether they
satisfy the three definitions of distribution sensitivity discussed above. Among poverty measures,
the class of Foster-Greer-Thorbecke (FGT) indices is the most popular. FGT indices are Pigou-
Dalton sensitive when > 1 and transfer-sensitive when > 2. However, no FGT index is
growth sensitive. Table 2.1 shows that the same holds for many standard poverty measures. Among
the welfare indices listed by Dollar, Kleineberg and Kraay (2015), some of which are included in
Table 2.1, only the Atkinson index with parameter > 1 is growth sensitive.
The new index is also subgroup decomposable with population weights, a property shared
by many poverty indices but very few welfare indices. In the family of Atkinson welfare indices,
only mean income, with parameter = 0, is decomposable with population weights. To the best
of our knowledge, (, ) is the first welfare index proposed in the literature that is decomposable
with population weights while satisfying all three definitions of distribution sensitivity.8
Our index also enjoys other classical properties. The uncensored version of the index ( )
is continuous in all incomes, which formalizes the idea that poverty is not a discrete condition that
7
Ray ⓡ and Genicot (2022) propose the Growth-Progressivity axiom in the context of upward mobility measurement.
They do not study social welfare functions.
8
To be sure, our index is not the only population-weighted subgroup decomposable index that satisfies the three
definitions of distribution sensitivity. For instance, this class of indices contains any index obtained by raising an
1
Atkinson index with > 1 to the power 1 − . More generally, any additive welfare index ∑ ( ) is population-
=1
weighted subgroup decomposable, and will satisfy the three definitions of distribution sensitivity if the derivatives of
( ) satisfy the conditions in Appendix A. Among all the members of this class, we single out (, ) for two related
reasons. First, it has a very simple interpretation as the average factor by which incomes must increase to reach the
reference level of income. Second, (, ) is homogenous of degree minus one in income: if all incomes double, it
is halved. Since (, ) is decreasing in income, this is the natural analog of the homogeneity of degree one property
enjoyed by most welfare measures that are increasing in income. In Appendix B, we show that (, ) is the only
welfare measure that is homogenous of degree minus one and population-weighted subgroup decomposable.
Admittedly, both homogeneity of degree (minus) one and subgroup decomposability are cardinal properties, which
are less desirable than ordinal ones from a normative perspective because they discriminate among indices that are
ordinally equivalent. However, cardinal properties may still be desirable from a practical perspective – for instance,
when they simplify analysis for a policymaker.
7
is acquired or lost when one's income crosses a particular threshold (Watts, 1969).9 Our censored
index ( , as well as ) is also continuous because incomes are top-coded to the reference income
.10 We note that the main axiomatic difference between the uncensored and censored versions of
the new index is that the latter satisfy the focus axiom, which states that the income distribution of
people above the poverty line should not influence the measurement of poverty (Sen, 1976).
Policymakers wishing to only focus on incomes below the threshold can therefore use the
censored index as a legitimate measure of poverty.
9
It is straightforward that the new index also satisfies the symmetry and replication invariance axioms, which imply
that any two income distributions having the same cumulative distribution function must have the same poverty value,
as well as the scale invariance axiom, which requires the measurement of poverty to be unaffected by the unit and
currency against which income is measured.
10
If instead of top-coding, we drop individuals whose income is larger than from the summation, then a discontinuity
arises when the income of an individual crosses the reference income . Indeed, dropping individuals implies that their
ratio drops from 1 to 0 when they cross .
8
Table 2.1. Properties of Welfare and Poverty Measures
Definition Elasticity Distribution sensitivity
decomposability
Dalton
Pigou-
Subgroup
Transfer
/
Growth
=
/
Poverty
Foster, Greer, and Thorbecke (1984)
1
∗ ∗ (1 −
−1
)
If >1 If >2 ✓
≥ 0 = ∑ (1 − )
=−
∗ ∗
=1
Headcount ( = 0)
0 =
0
=0 ✓
Poverty Gap ( = 1) ✓
1
1
=−
1 = ∑ (1 − ) ∗ 1 ∗
=1
Squared poverty gap ( = 2) ✓ ✓
1 2 2 ∗ ∗ (1 − )
2 = ∑ (1 − ) 2
=−
∗ 2 ∗
=1
Sen poverty index
2
2 ∗ ∗ (1 −
) ✓ ✓
= ∑ (1 − ) (1 − ) + 1
+ 1 =−
=1
∗ ∗
Class of Chakravarty (1983)
1
−1
∗ ∗ ( ) ✓ ✓ ✓
0 < < 1 ℎ = ∑ (1 − ( ) ) ℎ
=−
∗ ℎ ∗
=1
9
Watts (1969) ( → 0)
1
=−
1 ✓ ✓ ✓
= ∑(() − ( )) ∗
=1
Welfare
Average income per capita ̅
✓
1 =
̅ = ∑ ∗ ̅
=1
1 1 1−
Atkinson welfare index If >0 If >0 If >1 If =0
≥ 0 1
1− = ( )
= ( ∑ 1− )
=1
Atkinson welfare index ( = 1)
1
1
=
1 ✓ ✓
1 = (∏ )
=1
Atkinson welfare index ( = 2)
1
−1
2
=
2 ✓ ✓ ✓
2 = ( ∑ −1 ) ∗
=1
Sen welfare index 2
=
2 ( + 1 − ) ∗ ✓
= 2 ∑( + 1 − ) ∗ 2
=1
Shared Prosperity
Share of income of Bottom 40%
1
40 ̅
40
=
for ≤ 40 ,
40 ̅ 40
∗
̅
= ∑ 40
40 ̅
= 0 for > 40
=1
New indices
(, ) ✓ ✓ ✓ ✓
1
= −
= ∑ ∗ ∗
=1
10
(, ) ✓ ✓ ✓ ✓
1
= −
= ∑ ∗ ∗
=1
✓ ✓ ✓ ✓
1
= −
(, ) = ∑ ( − 1) ∗ ∗
=1
Notes: In this table, a property is considered as violated when it is failed within the population deemed relevant by the index (Welfare indices: all individuals;
Poverty indices: all individuals below the reference income , and Share of income of Bottom 40%: all individuals in the bottom 40). The notation 40 corresponds
to the number of individuals in the bottom 40% of the distribution. For poverty measures, the elasticity is equal to zero when individual is non-poor. For the sake
of brevity, we do not discuss the welfare measures discussed by Bonferroni (1930) and Donaldson and Weymark (1980), which are not decomposable. For
conciseness, we also skip the measures proposed by Thon (1979), Kakwani (1980), and Takayama (1979) as they are directly related to the index of Sen (1976),
and we skip the class of poverty measures proposed by Clark, Hemming and Ulph (1981) and Hagenaars (1987) because of their direct relationship with the class
of Chakravarty (1983), which has better properties.
11
2.4 Underlying social welfare function
The new index is ordinally equivalent to a utilitarian social welfare function (SWF) based
on an iso-elastic utility function:
() = ∑ ( ) ℎ ( ) = −−1 , (4)
=1
where > 0 is an irrelevant constant which for simplicity of interpretation we set to = −.
To interpret this SWF, recall that under the veil of ignorance of Harsanyi (1953), the
social planner evaluates an income distribution by the expected utility they get when drawing at
random any income in the distribution. Therefore, the degree of inequality aversion in the SWF
corresponds to the coefficient of relative risk aversion associated with their utility function.11 In
our case, the degree of inequality aversion of the SWF corresponds to a coefficient of relative risk
aversion (CRRA) equal to 2. This is an appealing value as it is widely used in calibrations of
macroeconomic models. It is also consistent with direct empirical estimates of the degree of
inequality aversion in Kot and Paradowski (2022), who find a global mean estimate of 1.92 (their
Table 5). Together, this provides some additional justification for focusing on the specific
functional form we propose for (, ).12
The ordinal equivalence between our index and the SWF defined in Eq. (4) has two
implications. First, our index is ordinally equivalent to the Atkinson welfare function for parameter
value = 2 (2 ), as (, ) = (). Hence, (, ) shares the same degree of inequality
2
aversion as 2 . Also, given that 2 corresponds to the harmonic mean of the distribution, our index
is ordinally equivalent to the harmonic mean.13
Second, our index inherits a nice interpretation associated with Atkinson’s SWF. The
Atkinson welfare functions are such that () can be interpreted as the income level that, if
earned by all individuals, yields the same welfare as distribution . This income level is called the
equally distributed equivalent income (EDEI()), and is thus implicitly defined by:
11
This argument assumes that risk preferences are homogenous across individuals. See Eden (2020) for a
generalization to heterogenous risk preferences.
12
For further justification, note that if people have iso-elastic preferences, which is consistent with empirical evidence
(Chiappori and Paiella, 2011), a CRRA of 2 corresponds to the inverse of the elasticity of intertemporal substitution
(EIS), i.e., 0.5, which corresponds almost exactly to the average EIS obtained in the meta-analysis of Havranek et al.
(2015). That study collected and analyzed 2,735 estimates of the EIS in consumption from 169 published studies that
cover 104 countries. They estimate an average EIS of 0.492, with a standard deviation of 1.298. This provides some
additional empirical support for the degree of inequality aversion implied by the new index and its underlying SWF.
Finally, please note that (, ) can easily be generalized to vary the degree of inequality aversion, , using the ratio
formulation ( ) . However, this comes at the expense of simplicity and ease of interpretation.
13
The relationship between our index and Atkinson’s welfare function connects our index to a measure of pro -poor
growth proposed by Foster and Székely (2008). These authors suggest that growth be considered “pro -poor” when the
percentage change in mean income is smaller than the percentage change in some generalized mean, like the geometric
or the harmonic mean. Hence, when the percentage reduction in our index is larger than the percentage increase in
mean income, growth can be considered “pro-poor.”
12
() = ((), . . . , ()). (5)
Hence, we can straightforwardly compute the EDEI that our index associates with any
distribution as () = . For instance, the two-person income distribution ′ = (3 , )
(,)
is such that the ratio of individual 1 is = 3 and the ratio of individual 2 is = 1, which means
1 2
that (′; ) = 2. The EDEI of distribution ′ can thus be computed as (′) = = 2.
(′,)
By definition, the fact that its EDEI is equal to means that the equal distribution ′′ = (/2, /2)
2
yields the same welfare as distribution ′, i.e., (′′; )=( ′ ; ) = 2.14
2.5 Associated inequality measures
Atkinson (1970) pointed out that any social welfare function automatically defines an
associated inequality index that quantifies how much welfare is lost due to the inequality observed
in the distribution. Following this insight, we can construct an inequality index associated with our
welfare index. When income is equally distributed, all individuals earn an income equal to the
mean. Thus, our welfare index becomes an inequality index when is set equal to mean income,
i.e.,
1 ̅
y
̅) = ∑ ,
(, y (6)
=1
where y ̅) is a measure of inequality because
̅ ≥ 0 denotes mean income in the distribution . (, y
it is homogeneous of degree zero (i.e, it is unchanged when all incomes in a distribution are
multiplied by a common factor) and it satisfies the transfer axiom. One noteworthy feature of
̅ ) is that it is equal to one when all incomes are equal, as opposed to zero for most inequality
(, y
indices. This difference reflects the fact that our inequality index contrasts individual incomes to
mean income in terms of ratios rather than in terms of differences. When individual incomes are
equal to mean income, the ratios are equal to one, whereas the differences are equal to zero.15
14
With a small value for , our index admits another interesting interpretation, related to the headcount ratio. Consider
a “two-groups” society that is divided between a “poor” group, where everyone earns , and a “non-poor” group,
where everyone earns a very large income. Now, consider any distribution for which (, ) < 1 , which is always
possible when selecting a sufficiently small value for . Then, the value (, ) corresponds to the fraction of
individuals in the “poor” group that the two-group society should have in order to have the same welfare as distribution
, which is (, ) = ((, … , , ∞, … , ∞), ).
15
We note that (, y ̅ ) is related to the Generalized Entropy class of inequality measures () =
1 1
∑
̅ ) = 2(−1) + 1. We are grateful to
=1 (( ̅ ) − 1) of Shorrocks (1980). Specifically, we have (, y
(−1)
Gaurav Datt for pointing this out to us.
13
Unlike many existing inequality measures, the inequality index (, y ̅) has a simple
interpretation with intuitive units: it is the average factor by which incomes must be multiplied to
attain mean income y ̅ , recognizing that this factor is less than one for individuals above the mean.
̅ ) is ordinally equivalent to the inequality index 2 proposed by Atkinson
We note that (, y
(1970), which has rarely been used in practice, perhaps because it lacks a straightforward
interpretation.
It follows from Equation (6) that our welfare and inequality indices are closely related.
Specifically, we can write our welfare index as a product of inequality and (the inverse of) mean
̅) (̅). This in turn implies that we can decompose the growth in
income, i.e. (, ) = (, y
y
̅ ), and mean income, y
(, ) into the difference between the growth rates of inequality, (, y ̅.
This inequality index admits a multiplicative decomposition into within-group and
between-group inequality. Let denote the number of groups, indexed by . Let denote
inequality within group as measured by our index. Let denote between-group inequality,
which corresponds to the inequality measured by our index in a counterfactual distribution
obtained from the initial distribution by replacing the income of each individual by the mean
income in their group. Formally, we have = ( , ̅ ) where denotes the distribution of
income and ̅ ) where is defined for all
̅ denotes mean income in group g and = ( ,
groups and individual in as ̅ . Our index decomposes into:
≡
)
̅ ) = ( ,
(, y ̅ ̅ )),
(∑ ( , (7)
=1
with weights:
̅
/
= , (8)
∑′=1 ′ /
̅′
where denotes the number of individuals in group and the weights satisfy ∑
=1 = 1.
This decomposition has a simple interpretation.16 ( , ̅ ) represents the average factor
by which income in each group g must be multiplied to reach average income within the group.
The within-group contribution to overall inequality is simply a weighted average of these factors,
∑ ̅ ) , which assigns greater weight to groups that are larger and/or have lower mean
=1 ( ,
income. If incomes are equally distributed within all groups, this term is equal to one. The first
̅ ) captures between-group inequality in an obvious way – it simply is our inequality
term ( ,
measure applied to the counterfactual distribution where inequality within groups has been
16
Amartya Sen, referring to the Theil index, which is a member of the General Entropy Class of inequality with a
parameter value of one, says: “But the fact remains that [the Theil index] is an arbitrary formula, and the average of
the logarithms of the reciprocals of income shares weighted by income is not a measure that is exactly overflowing
with intuitive sense.” (Sen & Foster, 1973, page 36)
14
eliminated by setting everyone’s income to group average income. If there are no differences in
mean income across groups, this term also is equal to one. Overall inequality simply is the product
of these two terms.
̅ ) generalizes to a class of inequality indices, whose members only
Finally, the index (, y
differ by the definition given to . Each member in the class defines to be equal to the mean
income among the X% richest individuals:
1 ̅
̅ )= ∑
(, , (9)
=1
where ̅ ≥ 0 denotes mean income among the X% richest individuals. The index (, ̅ ) is a
measure of inequality because it is homogeneous of degree zero and it satisfies the transfer axiom.
The index (, y ̅) corresponds to the extreme case where X=100%, i.e., (, y ̅ 100% ).
̅ ) = (,
The other extreme member in the class defines to be the largest income in the distribution. This
latter measure can be interpreted as the factor by which incomes need to be multiplied to attain the
income of the richest person in the distribution. Any member of this class admits an interpretation
that is related to a Kuznets ratio (Kuznets, 1955). Indeed, a Kuznets ratio measures the ratio of
average income among the top X% to average income among the bottom 1-X% of the
̅ ) measures the average of income ratios, namely those of
distribution.17 In contrast, index (,
the average income among the top X% to each individual income. This illustrates that index
̅ ) is more distribution sensitive than the corresponding Kuznets ratio.
(,
3. Empirical Applications
3.1 Data
We use data from the World Bank’s Poverty and Inequality Platform (PIP) (World Bank,
2022a) to illustrate the empirical properties of the new welfare measures proposed in this paper
and contrast them with existing ones. The PIP is a large compendium of over 2000 household
surveys covering 168 countries and over 97 percent of the world’s population.18 For most of the
analysis in this section, we rely on the “lined-up” version of the PIP data which
interpolates/extrapolates between/beyond survey years assuming distribution-neutral growth to
create a continuous annual dataset covering 1990-2019.19 This is the same dataset used by the
17
A closely related measure is the Palma ratio, defined as the ratio of the top 10 percent income share to the bottom
40 percent share (Palma, 2011). Setting = 10% in (, ̅ ) so that it measures the average factor by which all
incomes must increase to reach the mean income of the top 10% of the distribution is similar in spirit to the Palma
ratio and moreover gives greater weight to the shortfall of the poorest.
18
For documentation on PIP, see pip.worldbank.org and for technical details on the data and estimates used in PIP,
see the Global Poverty Monitoring Technical Notes at https://pip.worldbank.org/publication.
19
For details on the methodology used for the distribution-neutral shifts to the survey distribution to line up surveys
to a common reporting year, see the PIP methodological handbook (https://datanalytics.worldbank.org/PIP-
15
World Bank for its headline global poverty monitoring. For ease of reproducibility, the calculations
reported here are performed on the “binned” version of the dataset created by Mahler ⓡ et. al.
(2022), where the income distribution in each country-year is represented by data grouped into
1000 bins of equal population size per country and year, rather than on the underlying microdata
itself.20
Due to differences in the underlying survey instruments, the PIP combines surveys in
which consumption is the main household-level measure of well-being (covering about three-
quarters of the world’s population) with surveys in which income is the measure of well-being
(covering the remaining quarter of the world’s population). This presents challenges, since
conceptually consumption and income are very different measures of well-being, and these
differences are likely to be more pronounced in the tails of the distribution. For example, it is
biologically impossible to survive with zero consumption, but it is possible to survive with zero
income if households can draw down stocks of wealth. This conceptual difference is apparent in
the data. For example, zero values for reported consumption are virtually nonexistent, while it is
common for income surveys to have some mass of observations with zero or negative incomes. 21
Beyond these extremes in the lower tail of the distribution, household saving introduces conceptual
and empirical differences between income and consumption throughout the income distribution.
We follow the practice of the World Bank’s global poverty monitoring by abstracting from the
difference between income- and consumption-based surveys when combining distributions across
countries. For terminological convenience we will refer to the measure of well-being from the
survey as “income” irrespective of whether it is drawn from an income or a consumption survey
and note that it is measured in constant 2017 $PPP, per person per day.
An additional challenge is how to handle the effect of very low or zero measured income
when constructing distribution-sensitive welfare measures. Specifically, our welfare measures are
based on the ratio / which is undefined when reported income is zero and can be very large
when reported income is close to zero, to the point of dominating the index calculated as the
average of these ratios across all individuals.22 To address this problem, we bottom-code all binned
observations at $0.50 per day. We arrive at this value by applying the consumption floor
methodology outlined in Ravallion (2016) to estimate minimum consumption levels. Applying
this method to consumption surveys in PIP, we find that essentially no country has a daily per
Methodology). For countries without any household survey data, it is assumed that their income distribution is equal
to the population-weighted average income distribution in their region.
20
The dataset is available at https://datacatalog.worldbank.org/search/dataset/0064304/1000-Binned-Global-
Distribution.
21
When zeros in consumption are reported in surveys, they typically are treated as unit non-response. Regarding
incomes, Hlasny, Ceriani and Verme (2022) note that more than 75 percent of the 354 surveys from the Luxembourg
Income Study that they examine, contain income values taking the value of zero.
22
The concern that distribution sensitive measures are highly sensitive to low incomes is not new. Cowell and Victoria-
Feser (1996) discuss this issue in the context of data contamination (e.g., measurement error), using influence
functions to demonstrate the extent to which a poverty or inequality measure is changed by small errors in the data.
16
person consumption floor below this level. This procedure results in bottom-coding 0.3% of the
population covered by both income and consumption surveys in 2019.
3.2 The evolution of global welfare, inequality, and poverty over time
We begin by illustrating the features of our new measures in the setting of the world
interpersonal income distribution. Table 3.1 reports our new welfare index , comparing it with
mean income in Panel A with the help of our new inequality measure (Panel B). We also compare
our new censored poverty measures and with the headcount ratio (Panel C). While we compare
our new measures to other distribution sensitive welfare indices and poverty measures later in this
section, we start with the mean and the headcount index as comparators, precisely because they
are simple, commonly used, and not distribution sensitive by any of the three definitions in the
literature (see Table 2.1). is calculated using = $6.85 as the reference income level,
corresponding to the World Bank’s poverty line for upper-middle income countries, while the
censored measures in Panel C are presented for = $6.85 as well as = $2.15, the World Bank’s
international poverty line capturing extreme poverty.
Table 3.1: Global Welfare, Poverty, and Inequality Measures 1990-2019
Avg Annual Avg Annual
1990 2019 1990 2019
Change, % Change, %
Panel A: Welfare Measures (z=$6.85 per day) Panel B: Inequality Measures
W 2.95 1.36 -2.67 I 5.02 3.58 -1.16
Mean 11.6 18.0 1.51
Sen 3.53 6.88 2.30 Gini 0.70 0.62 -0.41
A1 4.29 8.76 2.46 AI1 0.63 0.51 -0.71
A2 2.32 5.03 2.67 AI2 0.80 0.72 -0.36
Panel C: Poverty Measures
(i) z = $6.85 per day (ii) z = $2.15 per day
C 3.15 1.66 -2.20 C 1.31 1.05 -0.74
P 2.15 0.66 -4.06 P 0.31 0.05 -5.95
FGT0 0.69 0.47 -1.34 FGT0 0.38 0.08 -5.20
FGT1 0.45 0.21 -2.56 FGT1 0.14 0.03 -5.84
FGT2 0.32 0.12 -3.36 FGT2 0.07 0.01 -6.06
Watts 0.85 0.34 -3.18 Watts 0.20 0.04 -5.91
̅), (, ), and (, ) and selected comparators. The
Notes: This table reports the global estimates of (, ), (,
welfare measure, , is reported using the $6.85 per person per day threshold in 2017 $PPP (Panel A). and are
reported for the $6.85 and $2.15 thresholds (per person per day in 2017 $PPP). The Avg Annual Change column
reports the average annual log differences. Abbreviations: – Atkinson Social Welfare Index with = 1 2; FGT
– Foster-Greer-Thorbecke poverty measure with = 0, 1, 2; – Atkinson Inequality Index with = 1 2.
17
Welfare and inequality indices
In 1990, our new welfare measure was equal to 2.95, indicating that on average, incomes
of people around the world needed to be multiplied by a factor of about three to attain the reference
income level of $6.85 (Table 3.1, Panel A). Our proposed inequality measure, (Table 3.1, Panel
B) was equal to 5.02, meaning that on average, incomes needed to be multiplied by a factor of
about five to reach mean income, which was equal to $11.6 per person per day for the world in
1990 (Table 3.1, Panel A). By 2019, had fallen by over half from its value in 1990, to 1.4, i.e.,
on average, individual incomes needed to be multiplied only by a factor of 1.4 to reach $6.85 in
2019. had declined to 3.6, while mean income had increased to $18.0 over those three decades.
Column 3 in Table 3.1 indicates an average annual decline (i.e., improvement) of -2.7
̅ increased from $11.6 to $18.0, an
percent per year for . In contrast, average per capita income
average improvement of 1.5 percent per year. To understand the difference in the time trends of
these two measures between 1990 and 2019, recall from Section 2.5 that the growth rate of is
̅ . Panel B of
the growth rate of our inequality measure, , minus the growth rate of mean income,
Table 3.1 indicates that our inequality measure declined at a rate of 1.2 percent per year, which
accounts for the faster rate of improvement in as compared with mean income.
The concept of equally distributed equivalent income (EDEI) discussed earlier in Section
2.4 is helpful to interpret how the numerical value of reflects its distribution sensitivity. Recall
that EDEI is the level of income that, if earned equally by all individuals, would generate the same
level of welfare as the actual distribution of income. For our measure, = / , which means
that our welfare measure would value the actual world income distribution in 1990 the same as a
$6.85
counterfactual one in which every individual had an income equal to = $2.32. This is roughly
2.95
one-fifth of actual world average income in 1990, indicating a substantial degree of inequality
$6.85
aversion. By 2019, the had increased to = $5.04. This represents 28 percent of world
1.36
average income in 2019, again reflecting the same considerable degree of inequality aversion, but
now applied to the lower level of inequality prevailing in 2019.23
To better understand the sources of this reduction in inequality that contributes to the
improvement in relative to the mean, a simple decomposition is helpful. Define ≡ ∆ as
̅ as the growth rate of average income.
the growth rate of income of individual , and ≡ ∆
The growth rate of can be decomposed as follows:
−∆ ≈ + ∑ ( − ) (10)
=1
23
In Appendix C, we further quantify the distribution sensitivity of our indices by considering the fraction of their
value that is contributed by individuals below each percentile and by contrasting this fraction with that obtained for
other subgroup decomposable indices.
18
where is the elasticity of the welfare measure with respect to the income of individual given
in Table 2.1.24 The first term in Equation (10) is the growth rate of average income, while the
second term is a first-order approximation to the growth rate of inequality, decomposing it into a
weighted average of relative growth rates across the income distribution, − , with weights
reflecting the sensitivity of the welfare measure to growth at each point in the income distribution,
. For , these weights are proportional to −1 , assigning a higher weight to growth that occurs
lower in the income distribution, which is what makes a growth sensitive measure.
Panel A of Figure 3.1 reports the distribution of growth rates across the world income
distribution, plotting the average annual growth rate over the period 1990-2019 at each percentile
of the initial income distribution in 1990, i.e., the global growth incidence curve.25 World average
growth is shown as a horizontal line at 1.5 percent, and the relative growth rates − are the
deviations of the growth incidence curve from this global average growth rate. Virtually every
percentile in the bottom 80 percent of the world income distribution experienced higher-than-
average growth over this period. Since our measure assigns uniformly higher weights to growth in
lower percentiles of the income distribution (Panel B of Figure 3.1), it is not surprising that
improves faster than average income over this period. Conversely, since average income assigns
higher weights to lower growth rates observed near the top of the income distribution, it follows
that it improves more slowly than over this period.
24
Please see Appendix D for the derivation of this decomposition.
25
Note that this is an “anonymous” growth incidence curve in that it tracks the growth rate of percentiles, not the
growth rate of individuals (which would require true panel data for all individuals in the world and does not exist).
As is well-known, the anonymous growth incidence curve understates (overstates) growth at the bottom (top) of the
income distribution relative to non-anonymous growth rate that tracks the growth rate of individuals starting at
different points in the income distribution (Jenkins and Van Kerm (2006, 2016), Grimm (2007), Van Kerm (2009),
Bourguignon (2010), and Kraay and Van der Weide (2022)). For terminological convenience, in the main text we
refer to “individuals” and “percentiles” interchangeably even though this is not strictly correct.
19
Figure 3.1: Growth Incidence and the Sensitivity of Welfare Measures to Growth
Notes: Panel A reports the average annual growth for each global income percentile for the period 1990 to 2019.
The horizontal line at 1.5% shows the average annual growth in the global mean over the same period. Panel B
shows the elasticity of various welfare measures with respect to income at each percentile of the global income
distribution in 1990. See Table 2.1 for the expressions for these elasticities.
Equation (10) can be used to decompose growth in the other welfare functions in Table
3.1, which are also distribution sensitive. All we need to do is replace with the corresponding
elasticity, , for ∈ {, 1 , 2 }, as reported in Table 2.1. This clarifies that the only source of
difference in the growth rates of these welfare indices compared with that of is the fact that the
relative weights they assign to different points along the growth incidence curve are different
from . Table 3.1 also shows that the average annual changes in the Sen, 1 , and 2 indices are
2.30, 2.46, and 2.67, respectively (Panel A, Rows 3-5, Column 3). This indicates that these other
measures on average place less weight on the faster growth observed in lower percentiles of the
world income distribution over this period. This can be seen clearly in Panel B of Figure 3.1, which
plots the elasticities of all five measures shown in Panel A of Table 3.1. As noted above, the
elasticity for mean income is increasing with income, while leading distribution sensitive measures
such as the Sen index and the Atkinson index with = 1 also do not place uniformly higher weight
on growth rates in poorer percentiles. Only the Atkinson index with = 2 (2 ) assigns the same
20
downward-sloping weights to growth across the income distribution. This is because, as discussed
in Section 2.4, = /2 . For this reason, also we see in Panel A of Table 3.1 that its growth rate
is the same (in absolute value) as the growth rate of .
Finally, recall from Section 2.5 that our inequality measure can be expressed as a product
of between-group and within-group inequality. Applying this decomposition to the world income
distribution in 1990, we find that between-country inequality is 3.4 (indicating that the average
factor by which country average incomes must be multiplied to reach world average income is
3.4), while within-country inequality is 1.5 (indicating that the average factor by which individual
incomes in each country in the world need to increase to reach their corresponding country average
is 1.5). The product of these two, 3.4 x 1.5, gives overall inequality of = 5 as shown in Table
3.1. By 2019, the between-country component of global inequality had declined by a third to 2.3,
while the within-country component had hardly changed, at 1.6. Consistent with the evidence for
the decomposition of global inequality using the generalized entropy index with = 0 in Mahler
ⓡ et. al. (2022), the decomposition above indicates that most of the decline in global inequality
between 1990 and 2019 is due to a narrowing of gaps between countries.
Poverty indices
Panel C of Table 3.1 reports the censored measures, and , comparing them to the Foster-
Greer-Thorbecke poverty measures (headcount ratio, poverty gap, and squared poverty gap), as
well as the Watts index. Recall that is the average factor by which incomes need to be multiplied
to attain the reference income level (equal to $6.85 or $2.15 in Panel C), setting this factor equal
to one for individuals at or above the poverty line. At the threshold of = $6.85, was 3.15 in
1990 and improved to 1.66 in 2019.26 We also report = − 1, the average percent increase in
individual incomes needed to bring everyone below the poverty line to the poverty line, which was
2.15 (or 215%) in 1990. We use for comparisons of changes over time with other poverty
measures because, like those measures, it is also bounded by zero from below, while the minimum
value taken by is one when poverty is eliminated. 27
Between 1990 and 2019, the average annual decrease in the headcount index relative to a
$6.85 per day poverty line was 1.3 percent per year (dropping from 0.69 to 0.47), while the same
figure was 4.0 percent per year for (dropping from 2.15 to 0.66). The rates of decline of the
other poverty measures are between these two extremes. As with the welfare measures discussed
above, these differences reflect the interaction between the elasticities of these measures with the
distribution of growth rates across percentiles.
Panel B of Figure 3.2 graphs the elasticities of the various poverty measures with respect
to income, shown here for the $6.85 poverty line, while Panel A repeats the global growth
26
= 3.15 is only slightly higher than = 2.95 in 1990, because less than a third of incomes are censored and
attaches a relatively low weight to these incomes.
27 −1
The growth rate of mechanically is lower than the growth rate of by a factor of < 1 , as = + 1, which
makes its growth rate less comparable with that of the other poverty measures.
21
incidence curve for reference purposes. The elasticities are all negative, since all the poverty
measures decline when incomes increase. However, only our proposed measure gives a
consistently higher weight (in absolute terms) to income growth in poorer percentiles. The
elasticity of the Watts index does not vary with income (thus failing the growth progressivity
axiom defined in Section 2). The elasticity for the poverty gap is increasing throughout (in absolute
terms) as the poverty gap measures mean income below the poverty line, while for the squared
poverty gap it starts to decline only as incomes approach the poverty line.
22
Figure 3.2: Growth Incidence and the Sensitivity of Poverty Measures to Growth
Notes: Panel A reports the average annual growth for each global income percentile for the period 1990 to 2019. The
horizontal line at 1.5% shows the average annual growth in the global mean over the same period. Panel B shows the
elasticity of various poverty measure with respect to income at each percentile of the global income distribution in
1990 using the $6.85 per person per day threshold in 2017 $PPP. See Table 2.1 for the expressions for these elasticities.
The interaction between global growth incidence curve (Figure 3.2, Panel A) and these
elasticities (Figure 3.2, Panel B) accounts for the different rates of reduction across the different
poverty measures. Taking advantage of the fact that poverty measures are additively sub-group
decomposable, Figure 3.3 plots the contribution of growth at each percentile of the income
distribution to the percent change in the poverty measure for the $6.85 poverty line. The
contribution of each percentile is expressed in percent changes, and the average of all points along
each line represents the overall percent change in each poverty measure.28
28
See Appendix E for the precise formulation of the measures plotted in this graph.
23
Changes in poverty measures reflect both individuals crossing the poverty line and growth
in incomes below the poverty line (except for the headcount, which reflects only the former). The
downwards step pattern in the line for the headcount ratio in Figure 3.3 corresponds to the change
in the poverty rate over this period. The growth of percentiles between the 47th and 69th (the poverty
rate in 1990 and 2019, respectively, at the poverty line of = $6.85) contributes to the reduction
in the headcount ratio, and each percentile contributes equally, since the headcount measures only
whether people cross the poverty line. The growth of percentiles below the 47th percentile that did
not cross the poverty line contributes nothing to the change in the headcount.
Compared to the headcount ratio, all other poverty measures are less responsive to people
who cross the poverty line. This is because they are continuous while the head-count ratio features
a discontinuous jump at the poverty line. Instead of merely recording poverty incidence, the other
poverty measures all capture the distance from the poverty line in some way. For our measure,
the contribution of each percentile is − 1, which approaches 0 as approaches the poverty line.
The main difference between the growth rate of and the growth rate of the headcount can be
seen in the region below the 47th percentile, i.e., all people who do not cross the poverty line and
therefore do not contribute to a reduction in the headcount, but who still contribute to the reduction
in the distribution sensitive poverty measure . Overall, this explains why the percent change in
these measures is bigger (in absolute terms) than the percent change in the headcount ratio. The
contribution of growth to the reduction in poverty measures differs most starkly in the bottom tail
of the distribution, where growth has been relatively low. Because of the high weight that assigns
to the poorest, they make a large contribution to the overall reduction in the measure, despite their
relatively low growth.
The Watts index is an intermediate case. Below the 47th percentile, the Watts mimics the
growth incidence curve (with a negative sign) since it assigns equal weight to everyone (Panel B
of Figure 3.2). Comparing with the Watts, growth in the bottom 30 contributes more to reducing
than the Watts, while the opposite is true for percentiles 30 to 70. As a result, and the Watts
both fall proportionately by similar amounts (decline of 4.0 percent and 3.1 percent, respectively).
Finally, it is interesting to note that the change in the headcount ratio is much more like the
change in the distribution sensitive measures when using the $2.15 line (Table 3.1, Panel C). The
poverty rate at $2.15 declined from 38 percent to 8 percent. As before, the headcount is the most
responsive measure between the 8th and 38th percentile, which is counteracted by the distribution-
sensitive measures considering the income growth of the population that is always poor.29
However, the latter corresponds to only the bottom 8 percent of the distribution that remains below
the $2.15 line, rather than 47 percent with the $6.85 line. Since growth in this region of the
distribution contributes most to the difference between the growth rates of the poverty measures,
these differences largely disappear when this region is smaller. As a result, the growth rates of all
the poverty measures are quite similar in Panel C of Table 3.1.
29
Figure E.1 replicates Panel B of Figures 3.1 and Figure E.2 replicates Figure 3.3 for the $2.15 line.
24
Figure 3.3: Contributions of Growth at Different Percentiles to Changes in Poverty
Notes: This figure reports the contributions of each global percentile to the percent change in overall poverty from
1990 to 2019 for various poverty measures reported in Table 3.1. Poverty measures are reported using the $6.85 per
person per day threshold in 2017 $PPP. See Appendix E for the expressions of the various contributions.
3.3 Cross-country comparisons of welfare, inequality, and poverty
We conclude this section with cross-country comparisons of three pairs of countries around
2019 – one low-income pair (Burkina Faso and Mali); one middle-income pair (Colombia and
Peru); and one high-income pair (France and the U.S.) – using the measures that were presented
in Table 3.1. , , , and 0 are estimated at the $2.15, $6.85, and $25 per person per day
thresholds, respectively. The U.S. and Colombia, even though they have higher mean incomes
than their pairwise comparators, France and Peru, respectively, also have higher values of ,
because they have higher inequality, as measured by . In the case of United States, incomes on
average need to be multiplied by more than three, indicating the high level of inequality there, in
contrast with countries like France (1.4) with much lower inequality. Within each of the three
country pairs, the difference in is substantially larger than that indicated by the headcount ratio,
0 . The table serves to demonstrate that welfare judgments across countries (or regions or
groups) at a given point in time can also vary meaningfully if our proposed measure is used instead
of some of the other commonly used measures of welfare.
25
Table 3.2: Cross-country Comparison of the Simple Welfare, Poverty, and Inequality
Measures
Low-income, consumption Middle-income, income High-income, income
Burkina Faso Mali Colombia Peru France United States
2018 2018 2019 2019 2019 2019
Panel A z = $2.15 per day z = $6.85 per day z = $25 per day
W 0.82 0.66 1.14 0.92 0.58 0.93
Mean 4.85 4.77 15.6 13.9 58.9 83.6
I 1.85 1.47 2.59 1.86 1.36 3.09
Panel B
C 1.15 1.05 1.47 1.26 1.04 1.46
P 0.15 0.05 0.47 0.26 0.04 0.46
FGT0 0.31 0.17 0.35 0.29 0.10 0.11
Notes: This table reports the estimates of (, ), mean, and (, ̅) in Panel A and (, ), ( , ), and the poverty
headcount ratio (0 ) in Panel B. , , , and 0 are estimated at the $2.15, $6.85, and $25 per person per day
thresholds in 2017 $PPP for the low-, middle-, and high-income groups respectively. Mean is expressed in 2017 $PPP
per person per day. Consumption is used to measure welfare in Burkina Faso and Mali while income is used in other
countries.
4 The Prosperity Gap: a New Measure of Shared Prosperity
In this section of the paper, we present a policy application of our new welfare measures.
We show how our new welfare measure can be used as a natural measure of “shared prosperity”,
the promotion of which is one of the two main institutional goals of the World Bank. We first
provide background on the concept of shared prosperity as it is understood at the World Bank. We
then interpret as a global “prosperity gap” in incomes relative to a prosperity standard set at
$25 per day, roughly equal to the median poverty line among high-income countries. We show
how the prosperity gap varies across countries and over time, highlighting the benefits of it being
both distribution sensitive and subgroup decomposable. For the benefit of readers who are
primarily interested in the prosperity gap and are less interested in the technical properties of our
new welfare measures, this section is designed to be self-contained and may be read separately
from the rest of the paper.
4.1 Background on Shared Prosperity
Since 2013, the World Bank has articulated its twin goals as “ending extreme poverty and
boosting shared prosperity” (World Bank, 2015). Extreme poverty is measured as the share of the
world’s population living below the international poverty line. This line is currently set at $2.15
per day in 2017 $PPP, corresponding to the median national poverty line across low-income
countries. The concept of “shared prosperity” lacks an accepted precise definition but is clearly
intended to capture improvements in a distribution-sensitive welfare measure. This can be seen in
the following explanation of the concept:
26
“One way to think about the World Bank’s new shared prosperity goal is as an
alternative to average income as the benchmark of development progress. Instead
of assessing and measuring economic development in terms of the overall average
growth in a country, the shared prosperity goal places emphasis on the bottom 40
percent of the population. In other words, good progress is judged to occur not
merely when an economy is growing, but, more specifically, when that growth is
reaching the least well-off in society. Thus, the shared prosperity goal seeks to
increase sensitivity to distributional issues, shifting the common understanding of
development progress away from average per capita income and emphasizing that
good growth should benefit the least well-off in society.” (World Bank (2015), p.10)
As indicated in the quote above, shared prosperity is measured as the growth rate of average
incomes in the bottom 40 percent of the population in each country. This in turn implies that the
underlying distribution-sensitive welfare measure currently used by the World Bank to measure
“shared prosperity” is mean income in the bottom 40 percent of the income distribution in each
country.
Average income in the bottom 40 percent is a welfare measure that is simple to explain.
However, it has two key shortcomings. First, it is not subgroup decomposable with population
weights. This implies, for example, that average income in the bottom 40 percent of the world
cannot be expressed as a population-weighted average of average income in the bottom 40 percent
of each country in the world. This is simply because not every person in the bottom 40 percent of
the world income distribution is in the bottom 40 percent of the income distribution of the country
where they live, and vice versa.30 This makes the welfare measure less interpretable, because it is
not possible to have a global measure that corresponds to a simple aggregation of the same measure
at the country level, and similarly, it is not possible to simply decompose the measure across groups
of interest within a country.
Second and more importantly, average income in the bottom 40 percent only partially
satisfies one of the three definitions of distribution-sensitivity in the literature and fails to satisfy
the other two. It is Pigou-Dalton sensitive only with respect to transfers from a person in the top
60 percent to a person in the bottom 40 percent, but not within the bottom 40 percent since the
measure is simply mean income within this group. It does not satisfy transfer sensitivity or growth
sensitivity. In fact, within the bottom 40 percent, average incomes of the bottom 40 percent inherit
the upward-sloping pattern of growth elasticities of mean income, making it a “pro-rich growth”
rather than a “pro-poor growth” measure within the bottom 40 percent.
4.2 A New Measure of Shared Prosperity
To address these shortcomings, we propose a new “prosperity gap” to measure shared
prosperity. The prosperity gap is defined as the global average shortfall in income from a standard
30
In fact, 39 percent of people in the bottom 40 percent of the world income distribution in 2019 are in the top 60
percent of the income distribution of their country, and conversely, 26 percent of the people in the top 60 percent of
the world income distribution are in the bottom 40 percent of the income distribution of their country.
27
of prosperity set at $25 per day (adjusted for differences in purchasing power across countries).
Formally, it is an application of the censored version of the index, (, ) defined in Section 2.2:
1
≡ (, ) = ∑ (11)
=1
where the “prosperity standard” set at $25 per day for reasons discussed further below.
The prosperity gap has the same simple interpretation discussed earlier in the paper: it is
the average factor by which incomes must be multiplied to reach the prosperity standard. For a
person below the prosperity standard, multiplying their income by a factor will raise income
to the prosperity standard . For a person above (or at) the prosperity standard, the factor is less
than (or equal to) one, indicating that they have already attained the prosperity standard. The
prosperity gap is the average across all people of these factors. A value greater than one indicates
a prosperity gap or shortfall in prosperity, since it means that incomes need to increase by an
average multiple equal to the prosperity gap to achieve the prosperity standard.
The prosperity gap is subgroup decomposable with population weights. This means, for
example, that the global prosperity gap can be written as a population-weighted average of the
prosperity gaps at the country level. This makes it a useful tool for measuring shared prosperity at
the country level while having a global measure of the prosperity gap measured in the same terms.
Similarly, it means that the prosperity gap can readily be decomposed across groups of interest
within a given country.
̅, and the inequality
The prosperity gap also is readily decomposable into average income,
̅ ) (y
measure discussed in Section 2.5 of the paper, i.e., (, ) = (, y ̅
). The inequality measure
̅ ) in turn has a simple interpretation as the average factor by which incomes must increase to
(, y
attain mean income (recognizing that this factor is below one for incomes above the mean). This
clarifies how reductions in inequality and/or increase in average income will lead to improvements
in the prosperity gap. Specifically, the growth rate of the prosperity gap is simply the difference
between the growth rate of inequality (, y ̅) and the growth rate of mean income y
̅.
4.3 Empirical Implementation
We empirically implement the prosperity gap using the same data discussed in Section 3.1.
In 1990, the prosperity gap for the world was 10.8, meaning that incomes would have to increase
nearly 11-fold on average to reach the prosperity standard of $25 per day. By 2019 the prosperity
gap had fallen to 5.0, meaning that incomes would on average have to increase 5-fold to reach the
prosperity standard (Figure 4.1). As discussed in Section 3.2, the prosperity gap improves (i.e.,
falls) faster than average income during this period, reflecting the fact that growth was
concentrated in lower percentiles of the income distribution. Specifically, the prosperity gap
improved at an average annual rate of 2.7 percent, while average incomes increased by 1.5 percent
28
per year. The difference between these represents the contribution of a 1.2 percent per year
̅ )) to the improvement
reduction in global interpersonal income inequality (as measured by (, y
in the prosperity gap.
Figure 4.1: The Global Prosperity Gap
Notes: This figure reports the global estimate of (, ) using the prosperity threshold of $25 per person per day in
2017 $PPP for the period 1990 to 2019.
Globally, 81 percent of the world’s population, or about 6.2 billion people, falls short of
the $25 per day prosperity standard in 2019. In low-income countries, 99 percent of people fall
short of this standard, with a prosperity gap of 12.4 – meaning that incomes would have to increase
more than twelve-fold on average to attain the $25 per day standard. In lower-middle-income
(upper-middle-income) countries, 98 (86) percent of people also fall short of this standard, with a
prosperity gap of 6.8 (3.1). Even in high-income countries, 19 percent of people fall below the
prosperity standard, although the prosperity gap for them is much smaller at 0.9 (Table 4.1). This
reflects the fact that most people in high-income countries have incomes that exceed the prosperity
standard, so that their contribution to the prosperity gap is less than one.
Figure 4.2 illustrates the subgroup decomposability property of the prosperity group,
decomposing the global prosperity gap into regional contributions. The largest contributors to the
prosperity gap are the 5 billion people in East Asia, South Asia, and Sub-Saharan Africa
contributing 88% of the global multiple in 1990 and 84% in 2019. It is discouraging to note that,
in contrast with East Asia & Pacific and South Asia, the absolute contribution of Sub-Saharan
Africa to the global prosperity gap has not declined since 1990, with its share increasing from less
than 15% to more than 32%.
29
Table 4.1: The Prosperity Gap in 2019 – By Regions, by Income Groups, and for Selected
Countries
Prosperity Gap, Share of Population Millions of People
Multiple by Which below Prosperity below Prosperity
Income Must Increase Threshold, % Threshold
Low income 12.4 99.2 663
Income
group
Lower middle income 6.8 98.3 2,864
Upper middle income 3.1 85.9 2,463
High income 0.9 18.9 234
East Asia & Pacific 3.2 90.0 1,893
Europe & Central Asia 2.5 75.5 374
Latin America & Caribbean 3.5 81.1 521
Region
Middle East & North Africa 5.0 93.8 370
South Asia 6.7 99.1 1,820
Sub-Saharan Africa 11.4 98.8 1,094
Rest of the world 0.8 13.9 153
Burkina Faso 9.3 98.1 20
Bangladesh 6.5 99.4 162
Bolivia 2.5 76.3 9
Brazil 3.6 77.7 164
China 2.8 88.5 1,246
Colombia 4.2 85.8 43
Germany 0.5 7.7 6
Select countries
Ethiopia 8.2 99.6 112
France 0.6 9.8 7
Indonesia 5.2 98.1 265
India 6.9 99.1 1,354
Japan 0.8 16.5 21
Mali 7.3 99.5 20
Nigeria 10.1 99.9 201
Peru 3.4 88.7 29
United States 0.9 11.0 36
South Africa 7.8 88.1 52
Global 5.0 81.0 6,224
Notes: This table reports the global estimate of (, ) using the prosperity threshold of $25 per person per day in
2017 $PPP, the share of the population living below the prosperity threshold, and the number of people living below
the prosperity threshold in 2019 for various income groups, regions, and select countries. Income groups are classified
according to World Bank FY21 definitions, which uses data from 2019 (see
https://datahelpdesk.worldbank.org/knowledgebase/articles/906519-world-bank-country-and-lending-groups).
Regions are classified according to definitions used in the Poverty and Inequality Platform (see
https://datanalytics.worldbank.org/PIP-Methodology/lineupestimates.html#regionsandcountries).
30
Figure 4.2: Regional Contributions to the Global Prosperity Gap
Notes: This figure reports the decomposition of the global estimate of (, ) into contributions by regions using
the prosperity threshold of $25 per person per day in 2017 $PPP. See also Table 4.1.
Although they measure different concepts, empirically changes over time in the prosperity
gap and growth in the bottom 40 percent are similar at the country level. This can be seen in Figure
4.3, which compares average annual growth in the bottom 40 percent (on the vertical axis) against
growth in the prosperity gap (on the horizontal axis and multiplied by -1 so that positive growth
corresponds to an improvement in the prosperity gap). The left panel of Figure 4.3 considers the
full period 1990-2019 using the lined-up data discussed in Section 3.1, while the right panel
focuses on a smaller set of 53 “spells” of various lengths calculated between actual survey years
around 2014 and around 2019.31 Each panel includes the 45-degree line, with points above (below)
the line corresponding to episodes where average incomes in the bottom 40 percent improved
faster (slower) than the prosperity gap. Overall, the correlation between the two measures is high,
with most countries close to the 45-degree line.
31
The comparable spells can be found in the Global Dataset for Shared Prosperity here:
https://www.worldbank.org/en/topic/poverty/brief/global-database -of-shared-prosperity. Fifty-three countries have
growth spells between -5% and 5%.
31
Figure 4.3: Growth in the Prosperity Gap and Mean Income of the Bottom 40%
Notes: The left panel of this figure reports annualized growth between 1990 and 2019 for all countries with data in
PIP. The right panel reports the same information using the latest comparable spells for each country from the 10th
edition of the Global Database of Shared Prosperity (World Bank, 2022b) covering circa 2014-2019. For visual
clarity only cases of annual growth between -5% and 5% are reported.
4.4 Rationale for the Prosperity Threshold
We conclude with a brief discussion of the rationale for the prosperity threshold, which we
set at $25 per day at 2017 $PPP. There are two complementary rationales for this figure. First, it
is approximately equal to the median poverty line among high-income countries, which is close to
$24.4 per day (Jolliffe ⓡ et. al. 2022, Table B7). Second, it is close to the average level of
household per capita income in countries that cross the per capita GDP threshold that the World
Bank uses to distinguish middle- and high-income countries. This can be seen in Figure 4.4, which
plots average household daily per capita income (on the vertical axis) against annual GDP per
capita (on the horizontal axis), with both measured in constant 2017 $PPP and each point
corresponding to a survey-year observation for all surveys in the Poverty and Inequality Platform.
The solid upward-sloping line shows the line of best fit, with the vertical line at $21,600 ($15,000)
in 2017 $PPP indicating the high-income (IBRD graduation) threshold.32 The intersection of the
high-income threshold with the line of best fit occurs at $23 per day, with the intersection of the
IBRD graduation threshold occurring at $16 per day.
32
The high-income threshold and the IBRD graduation threshold in the Bank’s income classification for operational
purposes is defined in terms of US dollars at market exchange rates and is currently $13,205 and $7,445 respectively.
Hamadeh et. al. (forthcoming) propose a conversion of the income classification system to PPP terms. Preserving the
same number of countries in each group of the operational classification results in a high-income threshold of
approximately $23,745 and an IBRD graduation threshold of $16,500 in PPP-adjusted terms using 2021 dollars, which
is equivalent to $21,599 and $14,977 in 2017 dollars.
32
Figure 4.4: High Income Threshold as Justification for $25 per day Prosperity Standard
Notes: The solid upward-sloping line shows the line of best fit between average household daily per capita income
(on the vertical axis) and annual GDP per capita (on the horizontal axis). Each observation represents a survey
available in the Poverty and Inequality Platform. The vertical line at $21,600 indicates the high income threshold
and the vertical line at $15,000 indicates the IBRD graduation threshold in PPP-adjusted terms (updated using
Hamadeh et al. forthcoming).
Naturally, the prosperity threshold could be set at other levels as appropriate to the application, in
the same way that different poverty lines can be deployed when measuring poverty in different
contexts. For example, a higher threshold such as $50 per day could be justified based on being
approximately equal to median income in high-income countries – an unambiguously high global
standard of prosperity. On the other hand, a lower threshold around $15 per day could be justified
because it is close to the IBRD graduation threshold. Figure 4.5 illustrates how the prosperity gap
would change with different values of the prosperity threshold.
Finally, we note that a key feature of our measure is that changing the prosperity threshold only
changes the units of the prosperity gap, and otherwise has no effect on cross-country or over-
time comparisons of the measure. In fact, converting the prosperity gap from a measure based
on a prosperity threshold of = $25 to any other threshold ∗ requires nothing more than
multiplying the prosperity gap by ∗ /25. This greatly adds to the simplicity and interpretability
of the proposed new prosperity gap measure of shared prosperity.
33
Figure 4.5: The Global Prosperity Gap with Different Prosperity Thresholds
Notes: This figure reports the global estimate of the prosperity gap using $15, $20, $25, and $50 per person per day
prosperity thresholds for the period 1990 to 2019. The thresholds are expressed in 2017 $PPP.
34
5 Conclusions
The most used measures of welfare, mean income and the poverty headcount, are not
distribution sensitive. Despite widespread agreement that inequality is important, the large menu
of existing distribution sensitive welfare measures is rarely used in academic, policy, or public
discourse. We argue that a main reason for this is their complexity and lack of intuitive units. To
remedy this shortcoming, we have introduced a new family of distribution sensitive welfare,
poverty and inequality measures that has a simple mathematical formulation and simple units. The
key ingredient of these measures is the ratio of reference income to individual individual
incomes, / , which captures the factor by which individual incomes must be multiplied to attain
the reference level of income. This ratio formulation ensures that our measures satisfy all three
definitions of distribution sensitivity in the literature. Our measures are based on averages of this
ratio, giving a simple “average factor by which incomes must increase” interpretation to our
measures. This simple average property also ensures that our measures are sub-group
decomposable with population weights, further enhancing its ease of communication.
We demonstrate the empirical properties of our measures in the context of the world
interpersonal income distribution over the period 1990-2019, as well as in selected country
comparisons. Consistent with their strong distribution sensitivity, we find that our welfare and
poverty measures decline more strongly than their non-distribution sensitive counterparts in a
setting that has featured a substantial decline in global inequality. Finally, we exploit the simple
structure and units of our new measure to propose a new “prosperity gap” measure that the World
Bank could use to track progress towards its goal of promoting shared prosperity.
One area that requires further work to facilitate the implementation of these new measures
is a more careful consideration of the treatment of low incomes in household surveys. As discussed
in Section 3, by virtue of being based on / , which makes them distribution sensitive, our
measures also are sensitive to mismeasurement of very low or, at the extreme, zero incomes. In
this paper, we bottom-coded income at $0.50 per day, at which we arrived by applying the
consumption floor methodology outlined in Ravallion (2016) to estimate minimum consumption
levels. Further work is needed on different approaches to address the issue of mismeasurement of
income (or consumption) at the bottom of the distribution.
******************************
Contributors: Sterck had the original idea for the welfare index, which aggregates individual
contributions to the index in the form of / . Decerf, Kraay, Özler, and Sterck drafted Sections
1 and 2, which were revised by all authors. Jolliffe, Kraay, Lakner, and Yonzan drafted Sections
3 and 4, which were revised by all authors. All authors interpreted the data, contributed to
writing the paper, and approved the final version.
35
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Appendix
A Definitions and Axioms
In this section, we provide formal definitions for three notions of distribution sensitivity
that have been proposed in the literature and present the mathematical conditions that they impose
on welfare indices.
Basic notation
The notation is as follows. Let = (1 , … , ) denote an ordered income distribution
whose incomes are sorted in increasing order, i.e., +1 ≥ for all ∈ {1, . . . , − 1}. Let ℵ
denote the set of natural numbers and let + denote the set of strictly positive reals. Let =
∪∈ℵ ( + ) denote the set of all income distributions whose individual incomes are all strictly
positive, where the dependence of on distribution is omitted in the notation (i.e., we simply
+
write instead of ()). Let a measure be any function : → 0 be such that () > 0 for
33
some ∈ .
Definitions of three notions of distribution sensitivity
We present the formal definitions of three central notions of distribution sensitivity. These
notions are defined based on axioms. Before presenting these axiomatic definitions, we make two
remarks.
First, welfare indices can either increase in incomes, like social welfare functions, or
decrease in incomes, like poverty measures. As an improvement is marked by an increase in the
index value in the former case and by a reduction in the latter case, the exact definition for their
axioms is slightly different. As our index belongs to the latter category, we consider here the case
of a “decreasing” welfare index, whose value is decreasing in incomes. This is without loss of
generality.
Second, some decreasing welfare indices, like poverty measures, are only affected by
incomes that are smaller than some reference income threshold, i.e., they satisfy the Focus axiom
proposed by Sen (1976).34 For the sake of presentation, we present axioms so that they apply to all
individuals in the distribution. It is straightforward to adapt the version of the axioms we present
in such a way that they only apply to poor individuals. When discussing distribution sensitivity of
indices that satisfy the Focus axiom, we consider only their application to poor individuals.
DEFINITION 1: A welfare index is Pigou-Dalton sensitive if it satisfies both the Monotonicity
and the Transfer axioms.35
33
To avoid trivialities, we rule out any measure that takes value 0 for all income distributions.
34
The Focus axiom could be defined as follows: For all ∈ , > 0 and for whom ≥ we have
() = (1 , … , −1 , + , +1 , . . . , ).
35
This definition is standard in the literature, see for instance Zengh (1997).
39
Monotonicity requires that the measure is reduced by an increment in the income of an individual.
(A1) Monotonicity: For all ∈ , > 0 and ∈ {1, . . . , }, we have:
() > (1 , … , −1 , + , +1 , . . . , ).
Transfer requires that the measure is reduced by a progressive equal-sized transfer.
(A2) Transfer: For all ∈ , > 0 and , ∈ {1, . . . , } with − ≥ + ,
we have: () > (1 , … , −1 , + , +1 , . . . , −1 , − , +1 , . . . , ).
DEFINITION 2: A welfare index is transfer sensitive if it is a Pigou-Dalton-sensitive index that
satisfies the Transfer Sensitivity axiom.
Transfer sensitivity requires that a progressive equal-sized transfer reduces the measure more when
it takes place further down the distribution.
(A3) Transfer Sensitivity: For all ∈ , > 0 and , , , ∈ {1, . . . , } with > , − ≥
+ and − ≥ + , we have:
(1 , … , −1 , + , +1 , . . . , −1 , − , +1 , . . . , ) > (1 , … , −1 , +
, +1 , . . . , −1 , − , +1 , . . . , ).
DEFINITION 3: A welfare index is growth sensitive if it is a Pigou-Dalton-sensitive index that
satisfies the Growth Progressivity axiom.
The Growth Progressivity axiom requires that a transfer of growth rate from a richer to a poorer
individual reduces the measure (Rayⓡ and Genicot, 2022).
(A4) Growth progressivity: For all ∈ , > 1, and , ∈ {1, . . . , } with > , +1 >
and +1 > we have:
(1 , … , −1 , , +1 , . . . , ) > (1 , … , −1 , , +1 , . . . ; ).
Mathematical relationships between the three notions of distribution sensitivity
We now discuss the conditions under which a welfare index that is both symmetric and
decomposable in population weights satisfies these three notions of distribution sensitivity. Both
symmetry and decomposability are highly desirable for policy work. Moreover, concentrating on
symmetric and decomposable indices yields simpler conditions.
A welfare index is symmetric if its value is unaffected when we permute the names given
to individuals. This is the standard anonymity requirement.
(A5) Symmetry: For all , ′ ∈ such that ′ is obtained from through a permutation of incomes
across individuals, we have (′) = ().
40
A welfare index is decomposable in population weights if its value corresponds to the
weighted sum of the value that it takes on sub-groups, where weights correspond to the population
shares of these subgroups.
(A6) Decomposability (in population weights): For all = ( , ) ∈ we have:
() = ( ) + ( ).
A decreasing welfare index that is symmetric and decomposable in population weights
must have its expression given by:36
1
() = ∑
=1 ( )
+
where function : + → 0 is decreasing in .
The mathematical conditions for the first two notions are well-known. As observed by Zengh
(1997), a symmetric and decomposable welfare index M is:
- Pigou-Dalton sensitive if and only if:
2
<0 and >0 for all > 0, (condition 1)
2
- Transfer sensitive if and only if:
2 3
<0, 2
>0 and 3 <0 for all > 0. (condition 2)
We next present the mathematical conditions for the third notion. By definition, a growth-
sensitive measure must satisfy the requirements of a Pigou-Dalton-sensitive index (condition 1).
Moreover, the growth progressivity axiom requires that for all 0< < , we have that a given
growth rate in income reduces function more than the same growth rate in income , which
is < . This inequality is equivalent to requiring that the first derivative of with
2
respect to be strictly positive, which yields + > 0 for all > 0. For a Pigou-
2
2
Dalton-sensitive decomposable measure, it is only if function is sufficiently convex ( 2>>0)
that this condition can hold (because < 0).
Hence, we have that a symmetric and decomposable welfare index M is:
- Growth sensitive if and only if:
36
See the proof of Proposition 1 for details.
41
2 2
<0, 2 >0 and 2 + > 0 for all > 0. (condition 3)
As can easily be seen from their respective conditions, growth-sensitive measures and
transfer-sensitive measures are necessarily Pigou-Dalton sensitive. The contrary is not necessarily
true. We also note that some growth-sensitive measures are not transfer sensitive, while some
transfer-sensitive measures are not growth sensitive. The two notions are not logically related.
Finally, we observe here that the growth progressivity axiom is equivalent to another
property proposed by Dollar, Kleineberg, and Kraay (2015), namely that the measure’s elasticity
to growth is strictly decreasing in income. By definition, the elasticity to growth is = ,
where < 0 for any measure satisfying monotonicity. The property proposed by these authors
2
thus requires that becomes less negative with income, i.e., > 0, which yields +
2
1
> 0 for all > 0. This last inequality yields for a symmetric and decomposable measure
2
that 2 + > 0 for all > 0, as desired. As a result, we can deduct from the sign of the
derivate of their elasticity to growth (given in Table 2.1) that the poverty indices of FGT, Sen and
Chakravarty, the welfare indices of Sen and Atkinson ( ≤ 1) and the shared prosperity indicator
mean of bottom 40 are not growth sensitive because for some we have ≤ 0.
In contrast, we show that our index W satisfies the three notions of distribution sensitivity.
By the above discussion, we need to show that it satisfies conditions 1, 2 and 3. By definition, our
decomposable welfare index is such that ( ) = / , which yields:
1 2 2 1 3 3 1 4
= − ( ) , 2
= 2 ( ) , and 3
= −6 ( ) .
The respective signs of these three mathematical expressions directly yield that our index
satisfies Pigou-Dalton sensitivity (condition 1) and transfer sensitivity (condition 2). Our index is
also growth sensitive as:
2
+ =
2 2
which has the appropriate sign for condition 3, as desired.37
B Axiomatic derivation
In this section, we show that our uncensored index is fully jointly characterized by a
37
From these reasonings, it is straightforward to see that our two indices and also satisfy conditions 1, 2 and 3
for the population that they deem relevant (i.e., for individuals whose income is smaller than ).
42
proportionality axiom together with symmetry and decomposability. The proportionality property
of a welfare index () can be formally defined by the following axiom:
1
(A7) Proportionality: For all ∈ and all > 0, we have () = ().
Equivalently, A7 is the requirement that the welfare index is homogenous of degree -1.
Axioms A5, A6 and A7 directly lead to the following result.
Proposition 1: A welfare index satisfies Decomposability, Symmetry and Proportionality if
and only if = .
Proof
“if part”. The proof that satisfies all three axioms is straightforward and thus omitted.
“only if part”. We must show that only measure satisfies these three axioms.
Take any measure that satisfies these three axioms and any ∈ . By the iterative application
of Decomposability, we get:
1
() = ∑ ( ),
=1
where function : + → could be specific to individual . By Symmetry, function must be the
+ +
same for all individuals. As the image set of is 0 , we must have : + → 0 . By
Proportionality, function must be homogeneous of degree -1, which implies that for all > 0
we have:
( ) =
for some factor > 0. We cannot have = 0, otherwise we would have (y)=0 for all
distributions, a contradiction to the definition of a measure . This shows that = , which
concludes the proof of the Proposition ∎
C Quantifying the distribution sensitivity of our indices
In Section 3.2 we used the concept of equally distributed equivalent income to interpret the
distribution sensitivity of our index . However, some readers may still fear that our index is
so distribution sensitive that it basically only accounts for low incomes in the first few percentiles.
If that is the case, then would not be highly robust to measurement errors of incomes at the
bottom of the distribution, and even in the absence of measurement error, might de facto be focused
on only the very poorest to the exclusion of everyone else. In this section, we take advantage of
43
the subgroup decomposability of to quantify the reliance of on incomes in the bottom of the
distribution.
Figure C.1 shows the fraction of the value of for the world in 2019 that is contributed
by individuals below percentile . Clearly, this fraction is one when considering percentile =
100 and zero when considering percentile = 0. The graph shows that individuals below
percentile = 20 contribute slightly more than 50% of the value of . This implies that almost
half of the value of is contributed by individuals above percentile = 20. Yet, individuals
below percentile = 20 contribute more than twice their population share, which provides some
quantified sense of ’s distribution-sensitivity. To fix ideas, we compare ’s sensitivity to the
bottom of the distribution to mean income’s sensitivity to the top of the distribution. This is a
meaningful comparison given that is decreasing with individual incomes while mean income
is increasing with individual incomes. As shown in the figure, individuals above percentile =
80 contribute more than 60% of the value of mean income. This suggests that is not necessarily
more sensitive to the bottom of the distribution than mean income is sensitive to the top of the
distribution.
Clearly, the global distribution is particularly unequal. When repeating the same exercise
for all countries in the world in 2019, we find that for the average country, individuals below
percentile = 20 contribute 43% of the value of . In contrast, in the average country,
individuals above percentile = 80 contribute 44% of the value of mean income.
44
Figure C.1: Contributions to (, ), and the Mean by Individuals Below each Percentile
Notes: This figure reports the cumulative contribution of each global percentile to the overall (, ) and to the
overall mean in the 2019 global income distribution. Vertical lines show the various poverty thresholds for low, lower-
middle, upper-middle, and high-income countries in per person per day using 2017 $PPP.
Figure C.2 provides the same information for our censored measure . The difference is
that the contribution to reaches one when considering the percentile = 47 at which incomes
become equal to the reference income $6.85, which serves as the poverty threshold. The graph
also shows the same information for the headcount ratio (0), poverty gap (1 ), the poverty
gap squared (2 ) and the . These poverty measures are interesting comparison points
because they are not distribution-sensitive enough to satisfy one definition of distribution-
sensitivity: 1 is at the frontier of being Pigou-Dalton sensitive; 2 is at the frontier of being
transfer sensitive; and is at the frontier of being growth sensitive. Thus, by contrasting the
graph for (i.e., censored ) to the graphs for these poverty measures, we get a sense of how
much each of the three definitions of distribution-sensitivity matter. The fraction contributed by
individuals below percentile = 20 is equal to 62%, 71%, 77% and 81% respectively for 1,
, 2 and .
45
Figure C.2: Contribution to (, ), and Various Poverty Measures by Individuals Below
each Percentile
Notes: This figure reports the cumulative contribution of each global percentile to the overall (, ), headcount
ratio, poverty gap, poverty gap squared and Watts index in the 2019 global income distribution. Vertical line shows
the $6.85 per person per day poverty threshold in 2017 $PPP used to estimate the various poverty measures.
D Decomposing growth in welfare
We provide in this appendix the derivation for Eq. (10) in Section 3.2, which decomposes the
growth in uncensored welfare.
Let () denote a generic uncensored existing welfare function from among those listed in Table
2.1, as well as our new index . The first-order Taylor series approximation to the change over
time in logarithm of is:
∆ 1 1
∆ = ≈ ∑ ∆ = ∑ ∆ = ∑ ∆
=1 =1 =1
= ∑ ( + − ) = + ∑ ( − )
=1 =1
where the third-last equality uses the definition of the elasticity ≡ (reported in Table 2.1
for the various welfare functions); the second-last equality uses the definitions of individual and
46
̅
average income growth ≡ ∆ and = ∆; and the last equality uses the fact that
∑
=1 = 1 for the uncensored welfare functions since they are homogenous of degree one in
incomes.
E Unpacking growth in the censored indices
In this Appendix, we explain Figure 3.4 in more detail and replicate Figures 3.3 and 3.4 for the
$2.15 poverty line.
Figure 3.4 has been derived as follows. Each of the poverty measures can be written as ( ) =
1
∑ ( ) where < is the indicator function and ( ) is the appropriate function
=1 <
for each poverty measure. The functions for the four poverty measures we consider are simply
− − 2
( ) = − 1, (0) ( ) = 1, (1) ( ) = , (2) ( ) = ( ) , and
( ) = ln(z) − ln (yit ). The percent change in poverty is given by:
( ) − (−1 ) 1 (< ( ) − −1 < (−1 ))
= ∑
(−1 ) (−1 )
=1
Figure 3.4 graphs each term in this average across all percentiles of the income distribution. The
average of the points along each line gives the total percent change in the poverty measure.
Figure E.1 shows the elasticities of the different poverty measures using $2.15 as the
poverty line. This graph is very similar to Figure 3.2, Panel B in the main text which is drawn at
$6.85, except that now the non-zero elasticities are compressed into a smaller region corresponding
to the lower poverty rate. Figure E.2 replicates Figure 3.3 using $2.15 as the poverty line. Again,
qualitatively the graph is very similar to Figure 3.3 in the main text, but is compressed into a
smaller region below the lower poverty line.
47
Figure E.1: Elasticities of Poverty Measures using the $2.15 per Person per Day Threshold
Notes: This figure shows the elasticity of various poverty measures for each percentile of the global income
distribution in 1990 using the $2.15 per person per day threshold in 2017 $PPP. See also Table 2.1.
Figure E.2: Contribution to Percent Change in Poverty for Various Poverty Measures
Notes: This figure reports the contributions of each global percentile to percent change in poverty overall poverty
from 1990 to 2019 for various poverty measures reported in Table 3.1. Poverty measures are reported using the $2.15
per person per day threshold in 2017 $PPP
48