Policy Research Working Paper 10133
Poverty-Adjusted Life Expectancy
A Consistent Index of the Quantity
and the Quality of Life
Jean-Marie Baland
Guilhem Cassan
Benoit Decerf
Development Economics
Development Research Group
July 2022
Policy Research Working Paper 10133
Abstract
Poverty and mortality are arguably the two major sources expected life-cycle utility approach a la Harsanyi. The paper
of loss of well-being. Most mainstream measures of human then proceeds to empirical comparisons between countries
development capturing these two dimensions aggregate and across time and focuses on situations in which poverty
them in an ad-hoc and controversial way. This paper devel- and mortality provide conflicting evaluations. Once it is
ops a new index aggregating the poverty and the mortality assumed that being poor is (at least weakly) preferable to
observed in a given period in a consistent way. It is called the being dead, the analysis finds that about a third of these
poverty-adjusted life expectancy index. This index is based conflicting comparisons can be unambiguously ranked
on a single normative parameter that transparently captures by the poverty-adjusted life expectancy index. Finally, the
the trade-off between well-being losses from being poor paper shows that this index naturally defines a new and
or from being dead. The paper first shows that the pover- simple index of multidimensional poverty, the expected
ty-adjusted life expectancy index follows naturally from an deprivation index.
This paper is a product of the Development Research Group, 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 bdecerf@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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names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
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Produced by the Research Support Team
Poverty-Adjusted Life Expectancy: A Consistent
∗
Index of the Quantity and the Quality of Life
† ‡
Jean-Marie Baland, Guilhem Cassan, Benoit Decerf§
JEL: D63, I32, O15.
Keyworks: Well-being index, Human development index, Multidimensional
poverty, Poverty, Mortality.
∗ Acknowledgments : We express all our gratitude to Kristof Bosmans, James Foster, Dilip
Mookherjee and Jacques Silber for helpful discussions and suggestions. This work was supported
by the Fonds de la Recherche Scientiﬁque - FNRS under Grant n◦ 33665820 and Excellence of
Science (EOS) Research project of FNRS n◦ O020918F. We are grateful to the audience at the
World Bank seminar for providing insightful comments. All errors remain our own. The ﬁndings,
interpretations, and conclusions expressed in this paper are entirely those of the authors and should
not be attributed in any manner to the World Bank, to its aﬃliated organizations, or to members
of its Board of Executive Directors or the countries they represent. The World Bank does not
guarantee the accuracy of the data included in this paper and accepts no responsibility for any
consequence of their use.
† CRED, DEFIPP, University of Namur.
‡ CEPR, CRED, DEFIPP, University of Namur.
§ World Bank, bdecerf@worldbank.org.
1 Introduction
Comparing well-being across societies in a simple, meaningful and unambiguous man-
ner is a diﬃcult task. The reason is that well-being is multidimensional. Looking at
a dashboard of dimension-speciﬁc indicators is complex and typically yields a very
partial ranking of societies. A summary index avoids these issues, but it is often
meaningless and its comparisons are non-robust and therefore remain ambiguous.
In this paper, we develop a new index of human well-being which focuses on two
main dimensions, poverty and mortality. This index, called the poverty-adjusted
life expectancy index, cumulates the following advantages which other competing
indices typically fail to satisfy. First, it accounts for the multi-dimensional aspect of
well-being in a straightforward manner by combining the two major ingredients of
well-being, which are the quantity of life (through mortality) and its quality (through
poverty). Being derived from a lifecycle utility approach, it is based on sound micro-
foundations and is easily interpretable. Moreover, it takes into account distributional
concerns, by focusing on individuals with low outcomes in each dimension. It also
provides a ranking of societies in which mortality and poverty evolve in opposite di-
rections. For a non trivial share of these comparisons, this ranking does not depend
on the normative weight one gives to each dimension. Finally, our index does not
require eliciting preferences and applies straightforwardly using available data.
There is a long-standing tradition looking for an indicator able to track the level
of human development in a society (Hicks and Streeten, 1979; Stiglitz et al., 2009;
Fleurbaey, 2009). Measuring well-being in a given period allows comparing human
development across countries and across time. For this purpose, simple monetary
measures, such as GDP per head, have been heavily criticized, essentially on two
accounts.1 First, income aggregates such as GDP are insensitive to the distribution
of consumption across the population. This concern led to the design and adoption
of income poverty measures (see, e.g., World Bank (2015)). Second, key aspects of
human well-being, such as health, are extremely hard to meaningfully translate into
monetary values. As a result, monetary measures do not provide a suﬃcient infor-
mational basis to account for the multi-dimensional nature of human development.
They are therefore unﬁt to assess a country’s performance at promoting well-being
or to evaluate policies that imply trade-oﬀs between diﬀerent dimensions, e.g. envi-
ronmental regulations or health policies.
Given these limitations, one strategy is to adopt instead a dashboard of indica-
tors, such as the 17 Sustainable Development Goals (SDG) adopted in 2015 by the
UN. One can also attempt to aggregate diﬀerent dimensions of well-being into a sin-
gle indicator of human development. Among these indicators, one ﬁnds the Human
Development Index (HDI) (UNDP, 1990), the Level of Living Index (Drewnowski
and Scott, 1966) or the Physical Quality of Life Index (Morris, 1978). Echoing the
distributional concern, some of them focus on deprivations, like the Global Multidi-
mensional Poverty Index (Alkire et al., 2015) or the Human Poverty Index (Watkins,
2006). These summary measures provide a rough yardstick of human development,
which is arguably easier to communicate than a full list of various indicators. More
importantly, they have the potential to solve the partial ranking of societies yielded
1 Another important critique relates to sustainability of the well-being achieved in a particular
period.
2
by a menu of indicators, when one society performs better along one dimension but
not along another. A dashboard cannot indeed compare two societies when two or
more dimensions are conﬂicting.
All these composite indices are subject to the same fundamental criticisms (Raval-
lion, 2011a,b; Ghislandi et al., 2019). First, the selection of the appropriate indicator
in each dimension and the choice of the aggregation function are often arbitrary and
do not follow from a defensible notion of individual well-being. This is particu-
larly true for some dimensions, such as sanitation, which are essentially “inputs” into
well-being, rather than “outcomes”. Second, the system of weights embedded in the
aggregation function is often arbitrary, for instance by giving an equal weight to
each dimension. Such weights are typically not related to the choices individuals
would make when facing a trade-oﬀ between these dimensions, and cannot therefore
be taken as representative of human well-being. More fundamentally, diﬀerent indi-
viduals may make diﬀerent choices, which implies that no system of weights can be
completely consensual or universal.
Taken together, these criticisms are devastating. Indeed, the full ranking of so-
cieties yielded by composite indices is of little value if the trade-oﬀs they imply
between “conﬂicting” dimensions are not meaningful. Moreover, the value of a sum-
mary indicator also depends on how quickly it can be grasped. Unfortunately, these
indicators, originally conceived as pragmatic ordinal indicators, do not usually oﬀer
a simple interpretation that can be easily communicated.
Given these weaknesses, some scholars even argue in favor of reducing the in-
formational basis to a unique dimension, such as health (Hicks and Streeten, 1979),
thereby avoiding the need to choose a particular aggregation process. In this respect,
a prominent and easily interpretable indicator is life expectancy at birth, which can
also be adapted in order to account for distributional concerns (Silber, 1983; Ghis-
landi et al., 2019; Gisbert, 2020). According to these authors, the cost of focusing
on a single dimension may not be that high, as not all dimensions carry the same
importance for human well-being.
In this paper, we propose to measure human well-being using the poverty-adjusted
life expectancy, P ALEθ , a new summary index that aggregates well-being losses
resulting from the poverty and mortality observed in a given period. There are
good reasons to focus on poverty and mortality when measuring human develop-
ment. First, poverty and mortality are arguably the two major sources of welfare
losses: poverty entails welfare losses by reducing the quality of life while mortality
entails welfare losses by reducing the quantity of life. Prominent scholars in wel-
fare economics such as Deaton and Sen have dedicated a large part of their work to
the study of poverty and mortality (Deaton, 2013; Sen, 1998). Unsurprisingly, the
ﬁrst two Sustainable Development Goals of the UN are directly related to poverty
while the third one refers to mortality.2 Second, focusing on poverty and mortality
naturally reﬂects distributional concerns as they are the worst possible outcomes
associated with consumption and health.
This summary index makes substantial progress on the criticisms identiﬁed above.
In particular, the aggregation of poverty and mortality is normatively grounded on
the expected life-cycle utility, the measure of social welfare proposed by Harsanyi
2 The ﬁrst two SDGs are entitled “No Poverty” and “Zero Hunger”, while the majority of the
indicators in the third “Good Health and Well-being” section refer to some form of mortality.
3
(1953). According to Harsanyi, social welfare in a given period can be understood
as the life-cycle utility expected by a newborn when drawing at random a life that
reﬂects the outcomes observed in that particular period. Our main simpliﬁcation is
to consider a binary quality of life: in any period, an individual is either poor or
non-poor.3 Life-cycle utility is then the sum of period utilities over one’s lifetime,
where period utility takes two values, one high when non-poor and one low when
poor. Our index therefore normalizes the expected life-cycle utility when one expects,
throughout her lifetime, to be confronted by the poverty and mortality prevailing in
the current period. We call this index “poverty-adjusted life expectancy” since, in
a stationary society, this index simply counts the number of years that a newborn
expects to live but weighs down the periods that she expects to live in poverty. That
is, our index has similar hypothesis and interpretation as the extremely popular
life expectancy index. Mathematically, our index is obtained by multiplying life
expectancy at birth by a factor one minus the fraction of poor, where the fraction of
poor is weighed down. This (normative) weight θ, the value of which lies between zero
and one, corresponds to the fraction of the period utility lost when poor. When being
poor has no utility cost, this weight takes the value zero and P ALE0 corresponds to
life expectancy at birth. When being poor is as bad as losing one year of life, θ = 1
and our index P ALE1 then corresponds to the poverty-free life expectancy at birth
(Riumallo-Herl et al., 2018), i.e. the number of years of life a newborn expects to
live out of poverty.
As stressed above, some pairs of societies cannot be compared using a dashboard
considering poverty and mortality separately because the two dimensions are “in
conﬂict”, for instance if one society has less poverty but higher mortality than the
other. Even though our index relies in general on some weight given in the trade-
oﬀ between poverty and mortality, we show that it can sometimes improve on this
partial ranking for all plausible values of its weight, as long as one considers that
being poor is not worse than being dead. (As we show below, a necessary and
suﬃcient condition for an unambiguous ranking is that the index makes the same
comparison for the two extreme values for its weight.) For instance, consider two
societies A and B where B has a higher fraction of poor but a higher life expectancy
at birth. Suppose that the situation is such that one may expect to spend more
periods in poverty in B than in A but also more periods out of poverty in B than
in A, as people live longer in society B. It is easy to show that life-cycle utility is
larger in B, regardless of the weight given to periods of poverty, because individuals
on average live more periods of both types in B. Hence, provided that being poor
is not worse than being dead, our index unambiguously ranks A and B, which a
dashboard approach is unable to do. As a result, our index increases the set of pairs
of societies that can be unambiguously compared. As long as the larger number of
years spent in poverty is more than compensated by a longer life expectancy, P ALEθ
and, therefore, welfare can only increase.
As we make clear later, our index, being closely related to the concept of life
expectancy, is based on “expectations” whereby a newborn assumes to be exposed
throughout his lifespan to the poverty and mortality observed in the current period.
3 Clearly, we do not claim that our index is superior to Harsanyi’s approach, but it is a plausible
measure of expected life-cycle utility when considering poverty as the main factor reducing the
quality of life. Also, the poverty status we consider here could also be a two-statuses measure
resulting from some aggregation of diﬀerent dimensions of the quality of life.
4
It is therefore not a projection or a forecast of the average life-cycle utility of the
cohort born in a particular period, implying that it cannot in general be interpreted as
the expected life-cycle utility of a newborn, unless the society is stationary. However,
even when mortality is selective and aﬀects predominantly poor people, we also
show that our index still provides a meaningful way to aggregate the two sources of
welfare losses observed in a particular period. The reason why a risk-neutral social
welfare function a la Harsanyi does not require to account for selective mortality
is that mortality is a peculiar dimension as, once dead, all the other dimensions of
deprivation become irrelevant. As a result, the aggregation of mortality and poverty
is much simpler than the aggregation of other dimensions of deprivations aﬀecting
alive individuals.4
Empirically, we combine data sets provided by the World Bank data on income
poverty (PovCalNet) and an internationally comparable data set on mortality data
(the Global Burden of Disease) from 1990 to 2019. Again assuming that one year
spent in poverty is (weakly) preferred to one year of life lost, we show that P ALEθ is
able to solve a nontrivial number of ambiguous comparisons across time or between
countries for which the two dimensions are conﬂicting. For instance, when comparing
all possible pair of countries in each year, across all years, there are about 21 percent
of such comparisons for which mortality and poverty move in opposite directions.
Out of these ambiguous cases, P ALEθ is able to solve 35 percent of them. We also
investigate the evolution of each country in the data set, by comparing the situation
in a particular year to that prevailing ﬁve years earlier. We ﬁnd that, out of 27
percent of conﬂicting comparisons, P ALEθ is able to solve 38 percent of them.
Finally, we propose a generalization of our index that explicitly addresses dis-
tributional concerns about unequal lifespans. We deﬁne a new indicator of mul-
tidimensional poverty that also captures deprivation in the quantity of life, which
requires the introduction of a normative age threshold below which one is considered
as deprived, i.e. a deﬁnition of premature mortality. This new index, which we call
the expected deprivation index (EDθaˆ ), is a weighted sum of the number of years
that a newborn expects to lose prematurely and the number of years she expects to
spend in poverty, using the same weight as in P ALEθ . (Again, these expectations
imply that a newborn assumes to be exposed throughout her lifespan to the poverty
and mortality observed in the current period.) We show that this index enjoys the
same advantages as P ALEθ and can usefully complement P ALEθ if one is concerned
with unequal lifespans. In particular, it also increases the set of pairs that can be
unambiguously compared when considering each dimension separately. In its spirit,
EDθaˆ is similar to the Generated Deprivation index recently proposed by Baland
et al. (2021), and they are in fact equal in stationary societies. We show that EDθaˆ
is more reactive to contemporaneous policies (e.g. in the case of permanent mortality
shocks), simpler to interpret and less data demanding than Generated Deprivation.
The poverty-adjusted life expectancy is reminiscent of several indicators proposed
in health economics, like the quality-adjusted life expectancy (QALE) or the quality-
adjusted life year (QALY).5 Both account for the quality and quantity of life, by
4 The mutual exclusivity of mortality and poverty simpliﬁes their aggregation (Baland et al.,
2021). In this paper, we show that a risk-neutral social welfare function justiﬁes to ﬁrst aggregate
within each dimension and then aggregate across dimensions. In this sense, our index satisﬁes a
form of path independence (Foster and Shneyerov, 2000).
5 See for instance Whitehead and Ali (2010) for an economic interpretation of QALYs, or Heijink
5
weighting down the quantity of life for periods with low quality. They have been
developed following the method of Sullivan (1971) and we show that these approaches
directly follow from the expected life-cycle utility approach in stationary societies.
Our index however accounts for another important dimension of well-being than
health, which is poverty. Also, P ALEθ takes advantage of the existence of the well-
established concept of a poverty threshold, which splits the population into poor
and non-poor, thereby transforming the quality of life into a binary variable. This
transformation is key to the simple interpretation of our index. There is, to the best
of our knowledge, no immediate equivalent of such threshold in health economics.
There exist other indicators of a society’s well-being which are arguably much
superior to the one we propose. Yet, these indicators either rely on techniques that
are not mature yet, require many arbitrary assumptions or cannot be readily applied
on a large scale using existing data. For instance, Becker et al. (2005) and Jones
and Klenow (2016) follow more sophisticated versions of Harsanyi’s expected life-
cycle utility approach by imposing a speciﬁc structure on preferences. Alternatively,
Fleurbaey and Tadenuma (2014), in the case of well-being, or Decancq et al. (2019),
for poverty, propose to aggregate diﬀerent dimensions using individual preferences.6
Also, there is a large literature investigating the weights to be given to diﬀerent
dimensions of well-being (Benjamin et al., 2014; Decancq and Lugo, 2013). However,
this literature has not reached full maturity, or cannot be applied on a large scale
due to data constraints.
The remainder of the paper is organized as follows. In Section 2, we present the
theory supporting our P ALEθ index and provide some empirical implications. In
ˆ index, which we compare to P ALEθ and Generated
Section 3, we present the EDθa
Deprivation. Section 4 concludes.
2 A transparent index of welfare
Our objective is to propose a simple indicator to measure and compare the level
of human development of diﬀerent societies in a given period. In particular, we
would like this indicator to aggregate two major sources of welfare losses: mortality,
which reduces the quantity of life, and poverty, which reduces the quality of life.
This aggregation should follow from the way individuals aggregate these losses and
therefore be related to life-cycle preferences.
The rationality requirements of decision theory provide a structure on admissi-
ble life-cycle preferences. Rational preferences over streams of consumption have
been axiomatized by Koopmans (1960) and later generalized by Bleichrodt et al.
(2008). Such preferences must be represented by a discounted utility function, which
aggregates these streams as a discounted sum of period utilities
d
U= β a u(ca ) (1)
a=0
where d ∈ N is the age at death, β ∈ [0, 1] is the discount factor, ca is consumption
at age a and u is the period utility function.
et al. (2011); Jia et al. (2011) for applications of the QALE index to comparisons of health outcomes
across populations.
6 The limits of these diﬀerent approaches are reviewed in Fleurbaey (2009).
6
Building on this representation of preferences, Harsanyi (1953) proposes to mea-
sure the welfare of a society by aggregating life-cycle utilities over the whole society.
According to Harsanyi (1953), behind the veil of ignorance, each newborn faces a
lottery whereby she ignores whether and when she will be poor and for how long she
will live. When evaluating her life-cycle utility, she considers the life of a randomly
drawn individual in that society. Following the formulation of Jones and Klenow
(2016), her expected life-cycle utility is given by
a∗ − 1
EU = E β a u(ca )V (a), (2)
a=0
where V (a) is the (unconditional) probability that the newborn survives to age a,
a∗ is the maximal lifespan one can reach and the expectation operator E applies to
the uncertainty with respect to consumption ca . The period utility when being dead
is normalized to zero. As a result, mortality is valued through its opportunity cost:
death reduces the number of periods during which a newborn expects to consume.
Although this approach has solid theoretical foundations, it does not seem that
the indicator deﬁned by Eq. (2) could be directly used as a summary measure
of human development. Indeed, this indicator requires the choice of a particular
expression for the period utility function u(). Moreover, the trade-oﬀ between the
quantity and quality of life that underlies this indicator depends on the deﬁnition
of u and remains relatively obscure. And, ﬁnally, this indicator, being expressed in
utility-units, does not lend itself to a direct interpretation.
2.1 The P ALEθ index
In order to improve on these issues, we consider two assumptions that simplify Eq.
(2) into a simple index of human development. Our ﬁrst simplifying assumption is
to ignore discounting, i.e. β = 1. We argue that such assumption is necessary in
order to assign equal weights to all individuals, regardless of their age. Indeed, Eq.
(2) equates a society’s welfare in a given period to the expected life-cycle utility of
individuals born in that period. Clearly, the expected life-cycle utility of newborns
is related to the society’s welfare in a given period only when one assumes that their
expected lives reﬂect at each age the outcomes observed for individuals of that age
during the period considered. Discounting with a factor less than one would give less
weight to the outcomes of older individuals.
Our second simplifying assumption is to transform consumption into a binary
variable, i.e., ca can be either being non-poor (N P ) or being poor (P ). This is
obviously a strong assumption because we ignore the impact on period utility of
consumption diﬀerences within these two categories. However, we argue that this
assumption reﬂects the distributional concern, i.e. the desire to evaluate a society’s
development by focusing on the fate of its least well-oﬀ individuals. We believe this
assumption is the price to pay when one wishes to focus on poverty as the main
source of welfare losses, rather than other more general determinants of the quality
of life.7
7 This assumption is also used by Decerf et al. (2021) in a study of the poverty and mortality
eﬀects of the Covid-19 pandemic. These authors compare the relative sizes of poverty and mortality
shocks, whereas we derive here an indicator of well-being.
7
Taken jointly, these two assumptions require the use of a simple indicator, which
we call the poverty-adjusted life expectancy (P ALEθ ). Our second assumption im-
plies Eu(ca ) = π (a)uP + (1 − π (a))uN P where uN P = u(N P ), uP = u(P ) and π (a)
is the probability to be poor at age a conditional on being alive at age a. As, by
a∗ − 1
deﬁnition, life expectancy at birth is LE = a=0 V (a), we can rewrite Eq. (2) as
a∗ − 1
EU = uN P LE − (uN P − uP ) V (a)π (a). (3)
a=0
In Section 2.3, we show that these two assumptions are suﬃcient to deﬁne P ALEθ .
We now provide a simple illustration showing how these two assumptions naturally
lead to our index under a third assumption of “independence”. Under the latter
assumption, the conditional probability of being poor at each age a is a constant equal
to the fraction of poor in the population, i.e., π (a) = H for all a ∈ {0, . . . , a∗ − 1}
where H is the head-count ratio. (Clearly, this independence assumption does not
hold when mortality is selective, for instance when the poor die younger than the
non-poor. We discuss this limitation in more details in Section 2.3.) We can then
normalize Eq. (3) as
EU uN P − uP
= LE 1 − H .
uN P uN P
θ
This last expression deﬁnes the poverty-adjusted life expectancy index:
P ALEθ = LE (1 − θH ). (4)
The monotonicity of the period utility function implies that uN P ≥ uP . Moreover,
since being poor is not worse than being dead, we have uP ≥ 0. The parameter
θ, which captures the fraction of utility lost when a non-poor individual becomes
poor in a given period, is therefore such that θ ∈ [0, 1]. Importantly, this parameter
directly captures the trade-oﬀ between poverty and mortality. Indeed, as the period
uN P −uD
utility of being dead uD is normalized to zero, we have 1θ = uN P −uP . Hence, the
ratio 1θ measures, for a non-poor individual, the number of periods in poverty that
are equivalent to being dead for one period.
P ALEθ has a simple expression: its ﬁrst factor measures life expectancy, whereas
its second factor captures the fall in the quality of life due to poverty. This reduction
depends on the value assigned to the parameter θ. When θ = 0, becoming poor does
not aﬀect the quality of life and P ALE0 corresponds to life expectancy at birth.
When θ = 1, being poor is equivalent to being dead and P ALE1 corresponds to the
Poverty Free Life Expectancy (PFLE), an indicator proposed by Riumallo-Herl et al.
(2018),which measures the number of years that an individual expects to live free
from poverty.8 For other values for θ, P ALEθ corresponds to the number of years
of life free from poverty that provides the same life-cycle utility as that expected by
a newborn.
P ALEθ aggregates a measure of mortality, LE , with a measure of poverty, H , in
8 Riumallo-Herl et al. (2018) do not relate their PFLE index to a formal notion of social welfare.
As our theory makes clear, the PFLE index reﬂects an extreme view on the trade-oﬀ between
poverty and mortality, namely that being poor is as bad as being dead. One key diﬀerence between
our work and Riumallo-Herl et al. (2018) is that, through our formal framework, we provide a sound
theoretical basis for the aggregation process, even when mortality is selective, solve a number of
“conﬂicting” situations and derive a parallel index of multidimensional deprivation.
8
a way consistent with life-cycle preferences. This is a progress over most composite
indices, but P ALEθ also relies on a normative parameter, θ, that weights these
two dimensions. Thus, one may wonder whether aggregating the two component
indices is very useful given that there is a priori no consensus on the value that this
parameter should take. Indeed, the welfare comparison of two societies based on
P ALEθ may depend on the particular value assigned to the parameter θ. We show
that a nontrivial part of these comparisons does not depend on the parameter value
even for some pairs not related by domination. In other words, there exist pairs of
societies such that one is poorer but the other has a higher mortality rate, that can
be ranked by P ALEθ unambiguously, in the sense that this comparison holds for
all admissible values of θ. Hence, the structure of expected life-cycle utility allows
to extend comparisons beyond those associated to domination independently of the
particular value assigned to θ.
We illustrate this property in Figure 1. Without aggregation, domination alone
allows comparing society A with the northwest quadrant (where societies have more
poverty and more mortality) and the southeast quadrant (where societies have less
poverty and less mortality). For any value of θ, we can draw the iso-PALEθ curves
passing through A. The iso-PALE0 curve (associated to θ = 0) is a vertical line since
poverty has no welfare costs and life expectancy is the sole determinant of welfare.
Note however that the iso-PALE1 curve (associated to θ = 1) is not a horizontal
line. This deﬁnes two additional areas for which welfare can be unambiguously
compared with that of society A. The iso-PALEθ curves associated to intermediate
values of θ ∈ [0, 1] are indeed all located in the area between the iso-PALE0 curve
and the iso-PALE1 curve. The area in the NE quadrant below the iso-PALE1 curve
yields an unambiguously higher welfare than A, even though these societies have a
higher poverty than A. The area in the SW quadrant above the iso-PALE1 yields an
unambiguously lower welfare than A, even though these societies have a lower poverty
than A. The size of these new areas depends on the marginal rate of substitution of
P ALE1 at A. For society A and P ALE1 , this marginal rate of substitution is given
by LE ( A)(1−H (A))
(LE (A))2 . If LE (A) = 70 and H (A) = 20, this marginal rate of substitution
is equal to 0.011, meaning that one additional year of life is exactly compensated by
an increase in the head-count ratio H of 1.1% percentage points.
We now provide some intuition for these additional unambiguous comparisons.
They follow from (i) the fact that expected life-cycle utility sums period utilities
and (ii) the assumption that a period in poverty is not worse than a period lost (i.e.
uP ≥ uD ). For simplicity let us compare the life-cycle utility of two individuals iA
and iB , who respectively live in societies A and B depicted in Figure 1. Assume that
the larger poverty and smaller mortality of society B is such that the life of iB has
more periods in poverty than that of iA , and the life of iB also has more periods out of
poverty than that of iA . As both types of period are positively valued (ii), the value
selected for the weight does not matter anymore. Indeed, iB has a larger expected
life-cycle utility because her life has more periods in each consumption status than
the life of iA . In other words, the larger poverty rates in society B is more than
compensated by a longer life expectancy, so that an individual in society B always
expect to live more years out of poverty than in society A. Conversely, in the SW
quadrant above the iso-PALE1 , societies exhibit lower poverty rates but the fall in
life expectancy in these societies is so large compared to society A that, despite the
9
H
θ=0
1
0<θ<1
θ=1
Dominated by A
Larger EU
B than A
A
H(A)
Smaller EU
Dominates A
than A
LE
0
LE(A)
Figure 1: A simpliﬁed version of Harsanyi’s expected life-cycle utility approach
increases unambiguous comparisons.
redution in poverty rates, an individual expects to live fewer years out of poverty
and society A is unambiguously preferred.
As an illustration, Table 1 below reports the situation of Pakistan and Bangladesh
in 2019. Note that Life Expectancy can trivially be decomposed into Poverty Ex-
pectancy (LE*H) and PALE1 (LE*(1-H)) which corresponds to Poverty Fee Life
Expectancy. Pakistan has a lower headcount ratio than Bangladesh, but life ex-
pectancy is also lower in Pakistan. Therefore, it is a priori diﬃcult to rank those
two societies. Assuming that poverty and mortality remain unchanged, an individ-
ual born in Bangladesh can expect to spend 4.9 years of his life in poverty and 68.8
years out of poverty. In Pakistan, he can expect 2.8 years in poverty and 62.1 years
out of poverty. Hence, a newborn in Bangladesh can not only expect to spend more
years in poverty, but also more years out of poverty since the longer life expectancy
there more than compensates for the higher poverty rate. As a result, P ALEθ ranks
Bangladesh above Pakistan for all possible values of θ.
Table 1: An example of unambiguous comparison: Pakistan and Bangladesh in
2019.
Headcount Life Poverty Poverty Free
ratio Expectancy Expectancy Life Expectancy
(LE ∗ H ) LE ∗ (1 − H ) = P ALE1
Pakistan 4.3% 64.8 2.8 62.1
Bangladesh 6.7% 73.6 4.9 68.8
Ignoring mortality leads to correct welfare comparisons whenever there is dom-
ination, meaning that H and LE yield the same ranking. These cases correspond
to the NW and SE quadrants in Figure 1. (Clearly, when H yield the same ranking
as LE , P ALE0 automatically yields the same ranking as P ALE1 .) In the absence
of domination (NE and SW quadrants in Figure 1), ignoring mortality may lead to
erroneous welfare comparisons. First there are cases such that ignoring mortality
always lead to unambiguously wrong comparisons, independently of the value as-
10
signed to the normative parameter θ. In Figure 1, these cases correspond to the
areas in the NE and SW quadrants that are between the iso-PALE1 curve and the
dashed horizontal line. For other cases, disregarding mortality leads to correct or
incorrect comparisons depending on the value of θ. This occurs when P ALE0 and
P ALE1 yield opposite rankings, which corresponds in Figure 1 to the areas between
the iso-PALE0 and iso-PALE1 curves. Proposition 1 provides the conditions under
which ignoring mortality, i.e., comparing two societies based on H , always leads to
wrong welfare comparisons.
Proposition 1. (Unambiguous comparisons of welfare)
(i) For any two societies A and B, P ALEθ (A) < P ALEθ (B ) for all θ ∈ [0, 1] if and
only if
P ALE0 (A) < P ALE0 (B ) and P ALE1 (A) < P ALE1 (B ) (Condition C1)
(ii) There exist societies A and B for which P ALEθ (A) < P ALEθ (B ) for all θ ∈
[0, 1] even though H (A) < H (B ). These societies are such that H (A) < H (B ) and
LE (A) < LE (B ).
Proof. See Appendix 5.1.
2.2 Applications of P ALEθ
The data on population and mortality by country, age group and year comes from the
Global Burden of Disease database (2019). Comparable information across countries
and over time is available for the 1990-2019 period and is, to our knowledge, the
most comprehensive mortality data available for international comparison.9 Data on
alive deprivation come from the PovcalNet website which provides internationally
comparable estimates of income deprivation level. This data set is based on income
and consumption data from representative surveys carried out in low- and middle-
income countries between 1981 and 2019.10 In our empirical application, we follow
the World Bank’s deﬁnition of extreme income deprivation, corresponding to the $1.9
a day threshold (Ferreira et al., 2016). We merged the two databases at the year and
country level. Since the Global Burden of the Disease data are only available since
1990, we focus on the 1990-2019 period for a total of 120 low- and middle-income
countries.
We ﬁrst present in Figure 2 the evolution of life expectancy, the headcount ratio
and P ALEθ for these countries during the period 1990-2019. When θ = 1, life ex-
pectancy can be trivially decomposed into poverty expectancy and poverty adjusted
life expectancy: the diﬀerence between LE and P ALE1 is the number of years a
9 To construct this database, population and mortality data are systematically recorded across
countries and time from various data sources (oﬃcial vital statistics data, fertility history data as
well as data sources compiling deaths from catastrophic events). These primary data are then con-
verted into data in ﬁve years age groups, at year and country level using various interpolations and
inference methods (see Global Burden of Disease Collaborative Network (2020) for more informa-
tion on the GBD data construction). Following the literature, we only consider the point estimate
in the number of deaths (see also Hoyland et al. (2012) for a critique of this approach).
10 The website address is http://iresearch.worldbank.org/PovcalNet/povOnDemand.aspx. Each
country’s income deprivation level in PovCalNet is computed on a three year basis, and yearly data
are obtained by linear interpolation. In order to keep the panel balanced, we also extrapolate the
data and keep countries for which only 5 years or less of data have been extrapolated. A more
detailed description of the data source is given in Chen and Ravallion (2013).
11
newborn expects to live in poverty. (For θ < 1, the corresponding P ALEθ curves
all lie between life expectancy and the P ALE1 curve.) Throughout the period, life
expectancy increased from 62.3 in 1990 to 71.1 in 2019 but the decrease in poverty
expectancy is even more spectacular, from 27.9 years in 1990 down to 6.9 years in
2019. This decrease in poverty combined with an increase in life expectancy resulted
in a large increase in P ALE1 , from 34.4 in 1990 to 64.2 years in 2019.
Figure 2: Evolution of P ALE1 and Life Expectancy, 1990-2019
70 60
60 50
50
40
40
Years
30 %
30
20
20
10 10
0 0
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
2012
2014
2016
2018
2020
Year
Life Expectancy Headcount ratio (right axis)
PALE Poverty Expectancy
We now attempt to quantify the value added of the P ALEθ index as compared to
a menu of two separate indicators (LE and H). To do this, we quantify the number
of situations for which the two indicators are “in conﬂict” and the percentage of
these conﬂicting’ situations that are unambiguously ranked by P ALEθ . We again
assume that the weight θ is equal to one, which corresponds to the most conservative
approach consistent with the idea that being poor is weakly preferable to being dead.
(Choosing a lower maximal value for θ, by decreasing the maximal weight given to
the poverty component, would mechanically increase the number of situations that
we can unambiguously compare with P ALEθ .)
We ﬁrst provide an empirical version of Figure 1 above by comparing the situa-
tions of diﬀerent countries in 2019. The resulting diagram is presented in Figure 3
below. The point of reference (point A in Figure 1) chosen for this diagram is deﬁned
as a hypothetical reference country with a median head count ratio and a median life
expectancy at birth, which corresponds roughly to the situation of Nepal in 2019.
The iso-P ALE1 curve is represented by the dotted curve. All countries below this
iso-P ALE1 curve have a larger P ALE1 value than the reference country. Among
these, some countries, located in the southeast quadrant, are obviously better oﬀ,
with a larger life expectancy and lower poverty levels. Others, located in the north-
12
west quadrant, are unambiguously worse oﬀ. In the other two quadrants, there is a
signiﬁcant number of countries for which the evolution of life expectancy and poverty
are conﬂicting as they go in opposite directions. Among these, those represented by
shaded triangles correspond to situations in which the comparison by P ALEθ is un-
ambiguous. In the northeast quadrant, P ALEθ is always larger, as higher poverty is
more than compensated for by lower mortality. In the southwest quadrant, P ALEθ
is unambiguously smaller, as the fall in poverty is not large enough to compensate for
the higher mortality. Countries represented by a small dots are countries we cannot
rank unambiguously, as this ranking depends on the particular value assigned to θ.
Figure 3: Comparing countries in 2019
100
80
Headcount ratio
40 20
0 60
50 60 70 80
Life Expectancy
Unambiguous cases Unsolved ambiguous cases
Solved ambiguous cases iso−PALE
(a) 2019
Note: for the sake of presentation, we only report in the ﬁgure observations for which
life expectancy is larger than 40.
Figure 4 replicates this exercise by comparing all pairs of countries for each year
between 1990 and 2019, and reports, among all these comparisons, the proportion
of cases which are ambiguous, and the share of these ambiguous cases for which
P ALEθ provides a unambiguous answer. Out of 23 percent of ambiguous compar-
isons, P ALEθ is able to solve at least 37 percent of them. The share of ambiguous
comparisons that our index unambiguously solves strongly increases over time due
to the falling incidence of absolute poverty in many countries.11
In Figure 5, we provide P ALEθ comparisons within countries between present
and past situations. More precisely, for each year, we compare the situation in period
t to the situation prevailing in the same country ﬁve years earlier. Given that each
11 The falling incidence of absolute poverty implies that diﬀerences in H across countries in a given
year become, on average, smaller over time. This explains why the share of ambiguous comparisons
that our index unambiguously solves increases over time. This is easy to see when assuming that
the diﬀerences in LE across countries in a given year remain constant over time. Indeed, a smaller
diﬀerence in H can be “over-compensated” by a smaller diﬀerence in LE .
13
Figure 4: Evolution of the resolution of ambiguous inter-country comparisons, 1990-
2019
50
40
%
30
20
1990 2000 2010 2020
Year
Share of ambiguous comparisons
Minimal share of ambiguous comparisons solved with PALE
Reading: in 1990, countries had on average 23% of ambiguous comparisons, out of
which at least 26% were solved by the use of PALE.
country’s situation changed over time, we need to adapt our graphical presentation
to represent the set of situations for which P ALEθ stays constant over time. We
again conservatively assume θ equal to one.
By deﬁnition, P ALE1 = LE (1 − H ), and thus P ALE1 increases if and only if
dLE/LE > d(1 − H )/(1 − H ). This simple expression allows us to contruct a ﬁgure
in the (dLE/LE, d(1 − H )/(1 − H )) plan, in which the rate of growth of LE is
measured on the horizontal axis, and the rate of growth of (1 − H ), which we refer to
as the “Non-poverty Headcount”, on the vertical axis.12 We deﬁne the “zero-growth
P ALE1 ” curve, which represents all the combinations of the two growth rates such
that P ALE1 remains unchanged: dLE/LE = d(1 − H )/(1 − H ) . Above this curve,
P ALE1 increases and below this curve P ALE1 decreases.
The situations of interest are located in the northwest and in southeast quadrants
in which the two indicators move in opposite directions. In these quadrants, there
are two regions, one in the triangle below the curve in the northwest quadrant, and
one in the triangle above the curve in the southeast quadrant for which P ALEθ is
able to provide a clear welfare comparison. In these two areas, the shaded triangles
represent situations in which, in a particular country, the situation either strictly
improved (in the southeast quadrant) or deteriorated (in the northwest quadrant)
12 For the sake of the graphical presentation, we excluded from the graph measures that could
be considered as outliers (growth in nonpoverty headcount larger or smaller than 100 percent, and
growth rates in life expectancy larger than 90 percent or smaller than -40 percent). These are
however adequately accounted for in the following graph.
14
compared to the situation prevailing in the same country ﬁve years earlier.13
Figure 5: Resolution of ambiguous countries’ evolutions, 1990-2019
100
Non Poverty Headcount Growth
−50 0
−100 50
−20 −10 0 10 20 30
Life Expectancy Growth
Unambiguous cases Unsolved ambiguous cases
Solved ambiguous cases zero growth−PALE
Finally, Figure 6 reports, using the same comparisons, the evolution over time
of the share of ambiguous situations in which life expectancy and poverty moved
in opposite directions in one country between t and t+5, and the share of these
ambiguous situations for which the most conservative deﬁnition of P ALEθ provides
a clear ranking. Overall, the share of ambiguous comparisons declines from about 30
to 20 percent over the period considered (with an overall average of 27 percent). Out
of these, we can solve an average of 38 percent of welfare comparisons, from about
20 in 1990 to more than 50 percent in the last years considered.
2.3 P ALEθ beyond the independence case
We have shown in Section 2.1 that P ALEθ corresponds to a simpliﬁed version of
expected life-cycle utility (Eq. (3)) under the assumption of independence. How-
ever, independence is unlikely in practice: mortality is selective as the poor die
younger than the non-poor (Chetty et al., 2016). Canudas-Romo (2018) points to
this limitation when criticizing the PFLE index of Riumallo-Herl et al. (2018), which
corresponds to P ALE1 . This may cast some doubts on whether P ALEθ is a valid
measure of the welfare losses suﬀered in a given period. In this section, we show
that, in the absence of independence, P ALEθ still corresponds to the expected life-
cycle utility in any stationary society. We then argue that this result provides the
conceptual foundation for the use of P ALEθ even in societies that are not stationary.
13 Again, if being dead is strictly worse than being poor, so that θ is always strictly lower than
one, more situations can be strictly signed. They are located in the triangle above the “zero-growth
P ALE1 ” in the NW quadrant, and in the triangle below the “zero-growth P ALE1 ” in the SE
quadrant.
15
Figure 6: Evolution of the resolution of ambiguous countries’ evolutions, 1990-2019
80
60
% 40
20
0
1995 2000 2005 2010 2015 2020
Year
Share of ambiguous comparisons
Minimal share of ambiguous comparisons solved with PALE
The particularity of a stationary society is that all outcomes observed in one
period are replicated in the following period. In our setting, a society is stationary if
natality, mortality and poverty are constant over time. As a result, in a stationary
society, one can perfectly infer his expected life-cycle utility from the mortality and
poverty prevailing at the time of his birth. P ALEθ is then a simple normalization
of his expected life-cycle utility, even when the independence assumption does not
hold.14
Proposition 2 (Correspondence between Harsanyi and P ALEθ ).
EU
For any stationary society, P ALEθ = uN P .
Proof. A formal statement and proof is provided in Appendix 5.2.
Clearly, in practice, populations are not stationary and we cannot in general
interpret P ALEθ as the expected life-cycle utility of a newborn. Indeed, the poverty
and mortality observed at birth might not be good predictors for the future, in
particular as mortality and mortality decline over time with medical progress or
economic growth. Therefore, P ALEθ should not in general be interpreted as a
projection or a forecast of the average life-cycle utility that the cohort of individuals
born in the period will enjoy during their lives. This being said, the validity of
P ALEθ as an indicator of a society’s welfare in period t does not rely on whether
this indicator correctly forecasts the future. Our objective is to aggregate the welfare
losses observed in period t using a lifecycle utility approach. This aggregation should
14 We discuss in the conclusion how to adapt P ALEθ if the social planner is not indiﬀerent to
the fact that some individuals cumulate poverty and early mortality, which typically happens when
mortality is selective. Harsanyi’s social welfare as deﬁned in Eq. (2) is indiﬀerent to such cumulation.
16
not depend on the future evolutions of poverty and mortality.15 Rather, the way to
aggregate the welfare losses in period t that is consistent with a lifecycle utility
approach is to take the perspective of a newborn who assumes that she is born in
a stationary population, i.e. that the poverty and mortality observed at the time of
her birth remain unchanged during her whole life. Proposition 2 shows that P ALEθ
is a normalization of the expected life-cycle utility of a newborn who makes this
assumption. In other words, even if we had a perfect forecast of the future average
lifecycle utility of individuals born in a non-stationary society, P ALEθ provides a
much better picture of the welfare losses in the period of their birth.
It is worth noting that the same point can be made about life expectancy at birth
(LE ). In practice, this measure is derived from the mortality vector observed in a
given period. As a result, this index does not correspond to the average lifespan of a
cohort born in that period if the society is not stationary. However, life expectancy
corresponds to the number of years of life that a newborn expects to live when she
assumes she is born in a stationary society. The way it aggregates current mortality
rates is widely accepted as a meaningful measure of period mortality.
3 A transparent index of deprivation
The normative relevance of one’s death may depend on the age at which death
occurs. This judgment is implicit in several mainstream multidimensional indicators.
For instance, the global Multidimensional Poverty Indicator only accounts for deaths
below 18 years old (Alkire et al., 2015), or the Human Poverty Index only accounts for
deaths below 40 years old (Watkins, 2006). The widespread focus on child mortality
follows the same logic.
This shows that one normative limitation of P ALEθ is that it does not reﬂect the
distributional concern in both its dimensions. Although P ALEθ does focus on a low
quality of life due to poverty, P ALEθ does not particularly focus on a low quantity of
life. Indeed, an additional year of life given to an old individual has the same impact
on P ALEθ as an additional year of life given to a young one. It is true that, in
general, lifespans are distributed less unequally than consumption (Peltzman, 2009),
which slightly tunes down the need to capture unequal lifespans when monitoring
human development. Nevertheless, concerns around unequal lifespans justify the use
of an indicator that is sensitive to very low lifespans.
In this section, we extend our welfare index to measure deprivation in both the
quality and quantity of life. As stressed in the introduction, multidimensional poverty
indices capturing the quality and quantity of life are plagued by the same limitations
as welfare indices. They typically lack solid theoretical foundations and black box
opaque trade-oﬀs (Ravallion, 2011b).
Two properties of a measure of deprivation require us to adapt the P ALEθ in-
dex. First, we must deﬁne deprivation in the quantity of life. Borrowing from a long
tradition focusing on absolute poverty, we consider as deprived an individual who
ˆ. Follow-
dies prematurely, i.e. who dies before reaching a minimal age threshold a
15 For instance, a transitory mortality or poverty shock – due to war or to another disaster –
does reduce current welfare, even if the country fully recovers in the next period. In contrast, the
transitory nature of the shock implies that its consequences aﬀect essentially the current generations.
Its impact on the actual expected life-cycle utility of newborns can therefore be negligible, or nil if
the shock did not aﬀect the mortality rates of the newborns.
17
ing Baland et al. (2021), we call “lifespan deprived” an individual who dies before
reaching this age threshold. Second, deprivation is the opposite concept of welfare,
i.e. deprivation decreases when welfare increases. As we now show, under these two
properties, P ALEθ naturally leads to a particular index of deprivation.
3.1 Expected deprivation: The EDθa
ˆ index
ˆ ) the generalization of index P ALEθ in
We call expected deprivation at birth (EDθa
a deprivation context. The main diﬀerence is that EDθa ˆ is based on an indicator of
mortality diﬀerent from LE . Indeed, when focusing on deprivation in the quantity of
ˆ matter. We
life, only the years of life lost before reaching the minimal age threshold a
therefore deﬁne another indicator of mortality, the lifespan gap expectancy, which
measures the number of years that a newborn expects to lose prematurely.16 Letting
nt (a) denote the number of individuals born in period t who survive at least to age
a and nt = nt (0), we have:17
ˆ −1
a
nt (a) − nt (a + 1)
ˆ =
LGEa a − (a + 1)) ∗
(ˆ .
a=0
nt
We illustrate in Figure 7 the close connection between LGEa
ˆ and LE , where
a∗ −1 nt (a)
LE = a=0 nt . In the ﬁgure, we construct a counterfactual population pyramid
by reporting for each age a the number nt (a) of newborns who are still alive at age a.
As explained in Section 2.3, this counterfactual pyramid corresponds to the popula-
tion pyramid in period t if the society is stationary in period t.18 In the left panel of
Figure 7, LE is proportional to the area below the population pyramid. By contrast,
LGEa ˆ is proportional to the area between this pyramid and the age threshold. The
ˆ is
right panel illustrates the property that, for large enough age thresholds, LGEa
ˆ ≥ a∗ , where a∗ is the maximal lifespan,
the complement of LE . Formally, when a
ˆ = a
LGEa ˆ − LE .
ˆ , aggregates the poverty and lifespan de-
The expected deprivation index, EDθa
privation expected by a newborn, if she considers facing, throughout her life-cycle,
the poverty and mortality prevailing at the time of her birth. It combines a com-
ponent for deprivation in the quality of life and a component for deprivation in the
quantity of life:
LGEaˆ LE ∗ H
ˆ =
EDθa +θ , (5)
LE + LGEa
ˆ LE + LGEa
ˆ
quantity deprivation quality deprivation
where the parameter θ ∈ [0, 1] is deﬁned in exactly the same way as for P ALEθ and
ˆ ∈ N0 must respect a lower-bound a
the age threshold a ˆ ∈ N0 , such that a
ˆ≥aˆ ≥ 0.
The value for the lower bound a ˆ inﬂuences the set of comparisons that are robust to
ˆ (see below).
the values selected for θ and a
ˆ jointly deﬁne the respective importance
The two normative parameters θ and a
16 LGE is a particular version of the Years of Potential Life Lost, an indicator used in medical
ˆ
a
research in order to quantify and compare the burden on society due to diﬀerent causes of death
(Gardner and Sanborn, 1990).
17 See Proposition 5 for a mathematical expression for LGE that only depends on the mortality
ˆ
a
observed in period t. See also Appendix 5.2 for more details on the formal framework.
18 In a stationary society, the current population pyramid can be obtained by successively applying
the current age-speciﬁc mortality rates to each age group.
18
Figure 7: Life Expectancy and Lifespan Gap Expectancy
Number ˆ
a Number
indiv . indiv .
ˆ
a
nt nt
nt ∗ LGEa
ˆ
nt ∗ LGEa
ˆ
nt ∗ LE nt ∗ LE
Age Age
0 1 2 3 4 0 1 2 3 4 5
Note: in the Left panel, the light grey area below the counterfactual “stationary”
population pyramid is a multiple of LE and the dark grey area is a multiple of
LGEaˆ.
attributed to poverty and mortality. Parameter θ determines the relative weights of
being dead or being poor for one period. In contrast, parameter aˆ determines the
number of periods for which “being prematurely dead” is accounted for by the index.
ˆ also aﬀects the relative sizes of these two sources of welfare losses through
Hence, a
its impact on LGEa ˆ.
Both components have the same denominator, which measures a normative lifes-
pan corresponding to the sum of LE and LGEa ˆ . This normative lifespan can be
interpreted as the (counterfactual) life expectancy at birth that would prevail if all
premature deaths were postponed to the age threshold. It is at least as large as
LE , and corresponds to LE if the age threshold is equal to 0. The numerator of
each term measures the expected number of years characterized by one of the two
dimensions of deprivation. The numerator of the quantity deprivation component
measures the number of years that a newborn expects to lose prematurely (when
ˆ. The numerator of the
observing mortality in the period) given the age threshold, a
quality deprivation component measures the number of years that a newborn expects
to spend in poverty.
As said above, these expectations are correct in the case of independence (see
Section 2.1) or in stationary societies (see Section 2.3). As discussed above, in a
stationary population, the poverty and mortality rates prevailing at its birth perfectly
reﬂect the poverty and mortality a new cohort will be confronted to in the future.
ˆ as an indicator of
This restriction does not however invalidate the use of EDθa
deprivation in the current period (see Section 2.3): again, a widely used index such
as life expectancy suﬀers from exactly that same limitation but is nevertheless widely
interpreted as if the society was in a stationary state.
Finally, the deﬁnition of EDθa ˆ is such that each year prematurely lost is as bad
as 1/θ years spent in poverty. This trade-oﬀ between the relative costs of poverty
and mortality is the same as for P ALEθ .19 When θ = 1, EDθa ˆ has a transparent
interpretation, as it computes the expected proportion of the normative lifespan that
a newborn expects to lose prematurely or spend in poverty.
Unlike P ALEθ , EDθa ˆ accounts for the distributional concern in the mortality
ˆ, above which some deaths are normatively irrelevant,
dimension. The age threshold a
19 We assume here that the welfare cost of a year prematurely lost is equal to uNP .
19
is to mortality what the poverty line is to poverty. In Proposition 3, we show that
index EDθaˆ is a generalization of index P ALEθ : EDθ a ˆ ranks societies exactly in the
same way as P ALEθ as long as its age threshold a ˆ is at least as large as the maximal
age a∗ . For such values, the age threshold becomes not binding, and all deaths
become relevant in terms of deprivation because they all occur at younger ages than
the age threshold. When the age threshold is binding (smaller than the maximal age
a∗ ), the rankings obtained under EDθa
ˆ do not correspond to the rankings obtained
under P ALEθ .
ˆ generalizes P ALEθ ).
Proposition 3 (EDθa
For all aˆ ≥ a∗ we have P ALE θ = a ˆ ), which implies that, for any two
ˆ(1 − EDθa
societies A and B,
ˆ (A) ≤ EDθ a
P ALE θ (A) ≥ P ALE θ (B ) ⇔ EDθa ˆ (B ).
Proof. See Appendix 5.3.
Taken together, Propositions 2 and 3 show that EDθa ˆ aggregates two indices of
mortality, LE and LGEa
ˆ , with an index of poverty, H , in a way which is consistent
with life-cycle preferences. This improves on standard multidimensional poverty
indices. However, EDθa ˆ). Proposition 4
ˆ relies on two normative parameters: (θ, a
below provides the conditions under which the ranking by EDθa ˆ for some pairs of
societies A and B does not depend on the values selected for its normative parameters.
Proposition 4 (Unambiguous comparisons of deprivation).
(i) For any two societies A and B we have EDθa ˆ (B ) for all θ ∈ [0, 1]
ˆ (A) > EDθ a
and all a
ˆ≥a ˆ if and only if
a (A) > ED0ˆ
ED0ˆ a (B ) for all a ˆ, and
ˆ≥a
a (A) > ED1ˆ
ED1ˆ a (B ) for all a ˆ
ˆ≥a (generalized Condition C1)
ˆ > 1, there exist societies A and B for which EDθa
(ii) For any a ˆ (B ) for
ˆ (A) > EDθ a
all θ ∈ [0, 1] and all a ˆ even though H (A) < H (B ). These societies are such that
ˆ≥a
LE (A) < LE (B ).
Proof. See Appendix 5.4 for the straightforward proof.
ˆ provides unambiguous compar-
The ﬁrst part of Proposition 4 tells us that EDθa
a and ED1ˆ
isons if ED0ˆ a provide the same ranking for all age thresholds a ˆ.
ˆ above a
The intuition for this result is essentially the same as that provided in Proposition 1
ˆ ≥ a∗ , EDθa
above (since, when a ˆ is equivalent to P ALEθ ).
The second part of the Proposition indicates when ignoring mortality and focusing
exclusively on H leads to deprivation comparisons that are unambiguously correct
or wrong. When a ˆ < a∗ , it no longer suﬃces that H and LE yield separately the
same ranking for that ranking to be unambiguously correct. The reason is that, when
ˆ < a∗ , LE no longer contains all the relevant information on mortality: for instance,
a
two societies can share the same life expectancy at birth but one with several deaths
occurring below aˆ while the other has all deaths occurring above a ˆ. Note also that
the larger the lower-bound a ˆ, the
ˆ, i.e., the smaller the set of plausible values for a
larger the set of comparisons for which the generalized condition can be met.
20
We illustrate the above results in Figure 8.20 The vertical axis represents the
share of pairs of societies for which H and LE provide similar (at the top) or opposite
rankings (at the bottom). By deﬁnition, these rankings are insensitive to the age
ˆ considered. The horizontal axis represents all possible values of a
threshold a ˆ, the
lower bound on the age threshold.
The left panel describes the share of cases in which EDθa ˆ provides unambiguous
ˆ. Lower values of a
rankings as a function of a ˆ imply a fall in the share of cases that
EDθa ˆ has to
ˆ can rank unambiguously. Indeed, a larger age interval over which EDθ a
be computed implies a larger number of comparisons for ED. As a result, the number
of cases for which it can provide the same ranking for all age thresholds falls. Second,
if H and LE agree, EDθa ˆ provides the same ranking as H when a ˆ = a∗ . Finally, as
ˆ implies that the share
discussed above, when H and LE disagree, a larger value of a
of cases for which H provides a unambiguously wrong ranking gets larger.
ˆ, the share of pairs of societies for which
The right panel reports, for all values of a
P ALEθ and EDθa ˆ provide unambiguous rankings. Since P ALEθ does not depend
on the age threshold, it is able to rank a larger set of comparisons. As shown in
ˆ = a∗ , the two indices are equivalent.
Proposition 3, when a
Figure 8: H may make unambiguously wrong deprivation comparisons when a
ˆ > 1.
ˆ, the lower the share of societies pairs unambiguously
Reading: The smaller the lower-bound a
ranked by EDθa ˆ, the
ˆ even when H and LE agree with one another. The higher the lower-bound a
ˆ and P ALEθ .
higher the share of societies pairs unambiguously ranked by both EDθa
3.2 ˆ and P ALEθ
Mortality shocks and the evolution of EDθa
ˆ , as-
We now brieﬂy contrast the impact of mortality shocks on P ALEθ and EDθa
suming that these mortality shocks are independent of the poverty status. Consider
ˆ while
a mortality shock that equalizes individual lifespans across the age threshold a
keeping life expectancy LE constant. This lower dispersion in mortality does not
aﬀect P ALEθ , which only accounts for mortality through LE . By contrast, this
20 All graphs that follow are constructed using a lower bound on a ˆ equal to 1. Indeed, for θ = 0 and
ˆ = 0, EDθa
a ˆ is equal to zero for all societies and cannot therefore deliver unambiguous comparisons.
21
ˆ , since LGEa
shock reduces EDθa ˆ is thereby reduced. It is indeed easy to show that
∂EDθa
ˆ
∂LGEa
ˆ
> 0 (for θH < 1).
Consider instead a mortality shock that reduces mortality above the age thresh-
ˆ. Such shock increases LE but does not aﬀect LGEa
old a ˆ . As a result, P ALEθ
mechanically increases. It is also easy to show that deprivation, as measured by
∂EDθa
EDθa ˆ , decreases: ∂LE < 0, for θH < 1. Moreover, P ALEθ is more sensitive to
ˆ
ˆ , as the elasticity of P ALEθ to LE is equal to 1 while
this kind of shock than EDθa
ˆ to LE lies in (−1, 0). If the mortality shock is such that it
the elasticity of EDθa
ˆ, this shock simultaneously increases LE
reduces mortality below the age threshold a
and reduces LGEa ˆ . Again, P ALEθ improves and deprivation decreases since both
LE increases and LGE decreases.
3.3 ˆ and P ALEθ
Empirical relation between EDθa
Figure 9 reports the diagnostic delivered by P ALEθ and EDθa
ˆ over all pairwise com-
parisons of countries in 2019 for which H and LE yield opposite rankings, focusing
ˆ and P ALEθ independently
on the share of these cases that can be ranked by EDθa
of the value assigned to θ for a given age threshold.
Since P ALEθ does not depend on the age threshold, the discrepancies between
ˆ across ages can only come from variations in EDθ a
P ALEθ and EDθa ˆ . As is clear
from the ﬁgure, the share of cases solved by P ALEθ is constant but the share of cases
solved by both P ALEθ and EDθa ˆ increases with the age threshold. Since the age
threshold acts as a form of weight on the poverty component of EDθa ˆ , the relative
∗
importance of poverty in EDθa ˆ increases. When a
ˆ decreases as a ˆ ≥ a , mortality and
ˆ and P ALEθ and the two indices yield exactly
poverty have the same weight in EDθa
the same ranking (Proposition 3). For lower age thresholds, the number of periods
ˆ becomes more sensitive
of life considered as prematurely lost decreases, and EDθa
to its poverty component. In the extreme case in which aˆ = 0, EDθ0 can solve all
the cases for which H and LE yield opposite rankings since EDθ0 is unidimensional
and equal to θH .
Figure 10 presents the evolution of EDθa ˆ = 50) and the head-
ˆ (with θ = 1 and a
count ratio, H , for the world. As can be seen from the ﬁgure, the incidence of poverty
massively decreased over that period, while premature mortality (that is, before the
age 50) decreased at a much lower rate. As a result, at the world scale, ED1,50
follows closely the evolution of H and gets closer in the recent years. Overall, in
2019, a newborn can expect, under stationarity, to lose 15% of his normative lifespan
in poverty of through premature mortality. The corresponding ﬁgure in 1990 was as
high as 50%.
3.4 Relation with other indices of deprivation
We limit here our comparison to other indices in the literature to the index of gen-
erated deprivation (GDθaˆ ) we proposed in a companion paper (Baland et al., 2021).
ˆ , and Baland et al. (2021)
Generated deprivation is indeed the closest index to EDθa
discuss in details the relationships between GDθa ˆ and other indices of multidimen-
sional poverty. In short, EDθa ˆ and GD ˆ
θa are identical in stationary societies, but
ˆ has a simpler interpretation than GDθ a
EDθa ˆ , reacts faster to permanent mortality
22
ˆ , by age
Figure 9: Resolution of ambiguous comparisons of P ALEθ and EDθa
threshold
25
20
% 15
10
5
1 10 20 30 40 50 60 70 80 90 100
Age threshold
Remains Ambiguous Solved by ED only
Solved by PALE only Solved by PALE and ED
ˆ and H, 1990-2019 (where θ = 1 and a
Figure 10: Evolution of EDθa ˆ = 50).
60
50
40
% 30
20
10
0
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
2012
2014
2016
2018
2020
Year
Income Deprivation Lifespan Deprivation (50)
Headcount ratio ED (50)
ˆ is decomposable in
shocks and is less demanding in terms of data. However, GDθa
subgroups whereas EDθa ˆ is not.
23
21
ˆ index is deﬁned as follows:
In any year t, the GDθa
Y Lt N t ∗ Ht
ˆ =
GDθa +θ , (6)
Nt + Y L t Nt + Y L t
quantity deprivation quality deprivation
a∗ − 1
where θ ∈ [0, 1] and Nt = a=0 nt−a (a) is the population observed in t. Y Lt is the
total number of years of life prematurely lost due to mortality in year t, and can be
deﬁned as follows:
ˆ −2
a
Y Lt = nt−a (a) ∗ µt a − (a + 1)),
a ∗ (ˆ
a=0
where µt = (µt t
0 , . . . , µa∗ −1 ) stands for the vector of age-speciﬁc mortality rates.
22
The GDθaˆ index is also based on two components, one capturing quality depri-
vation, measured by the number of person-years spent in poverty in year t, and the
other quantity deprivation, measured by all the years prematurely lost in year t.
When compared to EDθa ˆ , the same normative weight is also used for these two com-
ˆ are however harder to interpret. This is because
ponents. The components of GDθa
GDθaˆ combines a number of poor with a number of years of life prematurely lost.
The rationale behind this aggregation is that, in a given year, the total number of
“poor individuals” in a given year also corresponds to the total number of “years”
lived in poverty in that year. This equivalence also explains why the denominator of
GDθa ˆ sums a number of individuals, Nt with a number of years Y Lt . By contrast,
ˆ are more easily interpretable: they are the
the numerators of the two terms in EDθa
number of years that a newborn expects to prematurely lose or spend in poverty (if
she expects mortality and poverty to stay at their currently observed levels).
ˆ and EDθ a
The following proposition establishes that GDθa ˆ are identical in sta-
tionary societies:
Proposition 5 (EDθa ˆ are identical in stationary societies).
ˆ and GDθ a
For any stationary society,
ˆ −1
a a− 1
ˆ =
LGEa a − (a + 1)) ∗ µt
(ˆ a∗ (1 − µt
k ), (7)
a=0 k=0
which yields GDθa ˆ.
ˆ = EDθ a
Proof. See Appendix 5.5.
ˆ and GDθ a
EDθa ˆ yield the same ranking for stationary societies. However, so-
cieties are typically not stationary so that EDθaˆ and GDθ aˆ may rank countries
diﬀerently. The main diﬀerence between EDθa ˆ comes from the way the
ˆ and GDθ a
two indices compute the number of years prematurely lost. EDθa ˆ takes the perspec-
tive of a newborn who faces throughout her life the mortality rates observed in t. In
contrast, GDθaˆ computes the number of years that are lost by the current population
due to the premature mortality observed in t. It records, over all premature deaths
in t, the number of years prematurely lost. Thus, if an individual dies at age 20 and
21 Strictly 1
speaking, the generated deprivation index proposed in Baland et al. (2021) is θ ˆ,
GDθa
ˆ since θ is a constant.
which is ordinally equivalent to GDθa
22 They are more formally deﬁned in Appendix 5.2.
24
the age threshold is 70, her premature death leads to a loss of 50 years of life in that
ˆ also counts the number of years prematurely lost, but instead of being
year. EDθa
computed on the actual population pyramid, EDθaˆ uses a counterfactual population
pyramid, which is the one that would prevail in a stationary society characterized by
the age-speciﬁc mortality rates observed in the period.
A major implication of this diﬀerence is that EDθa ˆ is more reactive to policy
changes than GDθaˆ . Consider a permanent mortality shock. The population dy-
namics is such that a transition phase sets in during which the population pyramid
slowly adjusts to the new mortality rates. This transition stops when a new station-
ary population pyramid is reached, typically after a∗ periods. GDθa ˆ records each
step of this transition and therefore exhibits inertia in its response to a permanent
mortality shock. By contrast, EDθaˆ immediately refers to the new stationary pop-
ulation pyramid and disregards the inertia caused by these transitory demographic
adjustments. We provide an illustration of this property in Appendix 6.
Finally, Baland et al. (2021) show that GDθa ˆ is essentially the only index de-
composable into subgroups to compare stationary societies in a way that satisﬁes
23
some basic properties. As a result, EDθa ˆ cannot be decomposable into subgroups.
This is no surprise given that EDθa ˆ is based on life expectancy, which cannot be
decomposed into subgroups. In Appendix 7, we also show that EDθa ˆ is essentially
the only index that is independent on the actual population pyramid and compares
stationary populations in a way that respects basic properties of deprivation. As a
ˆ , the only information
result, the actual population pyramid is irrelevant for EDθa
ˆ is age-speciﬁc mortality rates.
required for EDθa
4 Concluding remarks
ˆ,
An important limitation of the two indices proposed in this paper, P ALEθ and EDθa
is that they account for the distributional concern “dimension-by-dimension” instead
of accounting for them in terms of life-cycle utility. Indeed, our indices are insensitive
to the allocation of years of life prematurely lost between the poor and the non-poor.
This allocation may however have important implications for the distribution of life-
cycle utility. Indeed, when the poor die early, they cumulate low achievements in
the two dimensions and the diﬀerence between their life-cycle utility and that of the
non-poor increases.
Without denying the importance of this limitation, let us ﬁrst note that this
limitation is shared by most standard indices of human development.24 Second,
addressing this limitation requires data that are typically not available. One natural
way of accounting for such “concentration” of deprivations on the same individuals
would be to deﬁne as “life-cycle poor” individuals whose life-cycle utility is smaller
than that of a reference life, e.g., a life characterized by a lifespan of 40 years with
23 In other words, if decomposability into subgroups is seen as a key property, one should use GD .
ˆ
θa
Indeed, this index yields the same ranking as EDθa ˆ in stationary populations. In those populations,
GDθa ˆ thus yields the same ranking as P ALEθ when all deaths are normatively relevant (a ˆ ≥ a∗ ).
24 To the best of our knowledge, the global MPI index is the only one to account, in an indirect
way, for such concentration. In a nutshell, the deprivation-score of an individual is increased if she
lives in a household that has experienced the death of a less than 18 year child in the past ﬁve
years. Arguably, this aggregation of the quantity and quality of life is essentially practical. It is not
related to any concept of life-cycle utility. Moreover, it critically depends on the deﬁnition of the
household as it does not account for the occurrence of multiple deaths in the same households.
25
no period of poverty. One can then deﬁne an index of human development that
would, for instance, correspond to the expected fraction of newborns who will be
“life-cycle poor”. This type of index would not be ad-hoc, but would require better
data, combining poverty and mortality at the individual level, than what is currently
available in most countries. Moreover, this type of data, recording mortality up to
a given threshold, would necessarily be historical in nature, with little relevance
to the current situation. Alternatively, one may want to deﬁne indices that are
less demanding in terms of information, and are based on the observed mobility
between poverty and non-poverty, as well as on mortality ﬁgures for the poor and
the non-poor. Some additional assumptions would be required to then translate this
information into the lifecycle proﬁles for newborns.
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5 Appendix: Proofs
5.1 Proof of Proposition 1
Proof of (i). We start by the “only if” part. Assume to the contrary that P ALE0 (A) >
P ALE0 (B ) or P ALE1 (A) > P ALE1 (B ). This directly implies that P ALEθ (A) >
P ALEθ (B ) for some θ ∈ {0, 1} and therefore we cannot have P ALEθ (A) < P ALEθ (B )
for all θ ∈ [0, 1].
We now turn to the “if” part. By deﬁnition of the PALEθ index, we have to show
that
LE (B ) − LE (A) > θ ∗ (H (B ) − H (A)), (8)
for all θ ∈ [0, 1]. As P ALE0 (A) < P ALE0 (B ), we directly have that LE (B ) −
LE (A) > 0 because P ALE0 = LE . As P ALE1 (A) < P ALE1 (B ), we have LE (B ) −
LE (A) > H (B ) − H (A). It immediately follows that the inequality (8) is veriﬁed for
all values of θ smaller than 1.
Proof of (ii). From (i), proving (ii) only requires providing societies A and B with
H (A) < H (B ) such that P ALE0 (A) < P ALE0 (B ) and P ALE1 (A) < P ALE1 (B ).
If H (A) = 0.2, H (B ) = 0.4, LE (A) = 50 and LE (B ) = 75 we have P ALE1 (A) = 40
and P ALE1 (A) = 45, the desired result because P ALE0 = LE .
5.2 Stationary societies and PALEθ
We ﬁrst provide a formal deﬁnition of a stationary society. Consider a discrete set
of periods {. . . , t − 1, t, t + 1, . . . }. In each period, some individuals are born and
some individuals die (at the end of the period). All alive individuals are assigned a
consumption status for the period (P or N P ) . We deﬁne the life of an individual
i as the list of consumption statuses li = (li0 , . . . , lidi ) she enjoys between age 0 and
age di ∈ {0, . . . , a∗ − 1} at which she dies, where lia ∈ {N P, P }. The set of lives is
thus L = ∪d∈{0,...,a∗ −1} {N P, P }d+1 .
The number of newborns in period t is denoted by nt . The proﬁle of lives for
the cohort born in t is denoted by Ct = (li )i∈{1,...,nt } , where {1, . . . , nt } is the set of
newborns in t. Clearly, the proﬁle of lives Ct contains all the information necessary
to compute a newborn’s expected life-cycle utility (Eq. (3)). Let nt (a) denote the
number of individuals born in period t who are still alive when reaching age a. In
particular, we have nt (0) = nt . Let pt (a) denote the number of individuals born in
period t who are poor at age a, with pt (a) ≤ nt (a). By deﬁnition, the probability
( a)
that an individual born in t survives to age a is given by Vt (a) = ntnt , and the
conditional probability that an individual born in t will be poor when reaching age a
pt (a)
is πt (a) = nt ( a)
. To compute Eq. (3), it is suﬃcient to know the distribution of the
set of lives that Ct implicitly deﬁnes. We denote this distribution by Γt : L → [0, 1],
with l∈L Γt (l) = 1.
In period t, we cannot observe the proﬁle of lives for the cohort born in t. The only
elements of Ct that we observe in that period are nt (0), pt (0) and nt (1). However, we
also have information about the proﬁle of lives of the cohorts born before t. Formally,
let a society St be the list of proﬁles of lives for all cohorts born during the a∗ periods
29
in {t − (a∗ − 1), . . . , t}, i.e. St = (Ct−a∗ +1 , . . . , Ct ). In period t, we observe (i) the
number Nt of individuals who are alive in t:
a∗ − 1
Nt = nt−a (a),
a=0
(ii) the fraction Ht of alive individuals who are poor in t:
a∗ − 1
a=0 pt−a (a)
Ht = a∗ − 1 ,
a=0 nt−a (a)
and (iii) the age-speciﬁc mortality vector µt = (µt t
0 , . . . , µa∗−1 ) in period t where for
each a ∈ {0, . . . , a∗ − 1} we have
nt−a (a) − nt−a (a + 1)
µt
a = ,
nt−a (a)
with µt
a∗ − 1 = 1 .
We now show that, in stationary societies, the information available in period t
is suﬃcient to compute the value of the expected lifecycle utility using Eq. (3). The
particularity of stationary societies is to have their natality, mortality and poverty
constant over time, so that all (average) outcomes in a given period are replicated
over the next period. More formally, a society is stationary if both the distribution
of lives and the size of generations are constant over the last a∗ periods.
Deﬁnition 1 (Stationary Society).
A society St is stationary if, at any period t′ ∈ {t − a∗ + 1, . . . , t}, we have
• Γt′ = Γt (constant distribution of lives),
• nt′ = nt (constant size of cohorts).
It follows from this deﬁnition that nt (a) = nt−a (a) and pt (a) = pt−a (a) for all
a ∈ {1, . . . , a∗ − 1}.25 These equalities lead to the following Lemma, which allows us
to relate Eq. (3) to the information available in period t.26
Lemma 1. If St is stationary,
a− 1
Vt (a) = Πk t
=0 (1 − µk ) for all a ∈ {0, . . . , a∗ − 1}, (9)
Nt = nt ∗ LEt , (10)
a∗ − 1
N t ∗ Ht = n t ∗ V (a)π (a). (11)
a=0
Proof. We ﬁrst prove Eq (9). As St is stationary, we have nt (k ) = nt−k (k ) for all
k ∈ {1, . . . , a∗ − 1} and nt (k + 1) = nt−k (k + 1) for all k ∈ {0, . . . , a∗ − 2}. Therefore,
25 Clearly, a constant distribution of lives is not suﬃcient for these equalities, one also needs a
constant size of cohorts.
26 Lemma 1 also requires that n (a + 1) = n ∗
t t−a (a + 1) for all a ∈ {0, . . . , a − 2}, which follows
from the deﬁnition of a stationary society.
30
we have for all a ∈ {1, . . . , a∗ − 1} that
nt (a)
Vt (a) = ,
nt
a−1 nt (k + 1)
= Πk =0 ,
nt (k )
a−1 nt−k (k + 1)
= Πk =0 ,
nt−k (k )
a− 1 t
= Πk =0 (1 − µk ).
We then prove Eq (10). As St is stationary, we have nt (a) = nt−a (a) for all a ∈
nt (a)
{1, . . . , a∗ − 1}. Recalling that Vt (a) = nt , we can successively write
a∗ − 1
LEt = Vt (a),
a=0
a∗ − 1
a=0 nt (a)
= ,
nt
a∗ − 1
a=0 nt−a (a)
= ,
nt
= Nt /nt .
Finally, we prove Eq. (11). As St is stationary, we have pt (a) = pt−a (a) for all
pt (a) nt (a)
a ∈ {1, . . . , a∗ − 1}. Given that πt (a) = nt (a) and Vt (a) = nt , we can successively
write
a∗ − 1
a=0 pt−a (a)
Ht = a∗ − 1 ,
a=0 nt−a (a)
a∗ − 1
a=0 pt (a)
= ,
Nt
a∗ − 1
a=0 πt (a)Vt (a)nt
= .
Nt
The three equations in Lemma 1 imply that an individual born in a stationary
society can infer her expected life-cycle utility from the information available at the
year of her birth. (These direct relationships between current and future outcomes
in stationary societies are well-known to demographers (Preston et al., 2000).) We
illustrate this important insight using an example. Consider a stationary society
for which two individuals are born in each cohort, one living only for one period in
poverty and the other living for two periods out of poverty, i.e. n = 2, l1 = (P ) and
l2 = (N P, N P ). In period t, three individuals are alive: the poor born in t, the non-
poor born in t and the non-poor born in t − 1. Also, two individuals die at the end of
period t: the poor born in t and the non-poor born in t − 1. Eq. (9) states that the
mortality rates observed in period t (the right hand side of the equation) can be used
to infer the mortality rates that the newborn can expect to face during her life-cycle
(the left hand side). Thus, in our example, a newborn observes that, at the end of
period t, half of the individuals of age 0 die and all individuals of age 1 die. Eq. (9)
implies that he has a 50 percent chance to survive period t and a zero percent chance
31
to survive period t + 1. According to Eq. (10), the number of individuals who are
alive in period t, Nt , is equal to the number of person-periods in the proﬁle of lives
of the cohort born in period t. In our example, there are three individuals alive in
period t and there are three person-periods in Ct = (l1 , l2 ) = (P ; N P, N P ). Finally,
Eq. (11) states that the number of poor observed in period t, Nt ∗ Ht , is equal to
the number of person-periods of poverty in the proﬁle of lives of the cohort born in
period t. Indeed, there is one poor individual alive in period t and one person-period
P in Ct .
Lemma 1 shows that, in a stationarity society, the poverty and mortality ob-
served in a given period perfectly deﬁne the life proﬁle of newborns. Proposition 6
shows that P ALEθ is a normalization of the expected life-cycle utility of a newborn
in a stationary society even when mortality is selective, i.e. when the conditional
probability of being poor depends on age.
Proposition 6 (Equivalence between Harsanyi and P ALEθ ).
EUt
If society St is stationary, then P ALEθ = uNP
.
Proof. The result follows directly when substituting Eq. (10) and (11) into Eq.
(3).
5.3 Proof of Proposition 3
The proof builds on the complete framework presented in Appendix 5.2.
We ﬁrst show that LE + LGEa
ˆ =a ˆ ≥ a∗ .
ˆ when a
ˆ −1
a ˆ −1
a
nt (a) − nt (a + 1) nt (a) − nt (a + 1)
ˆ (Ct ) =
LGEa ˆ∗
a − (a + 1) ∗ ,
a=0
nt a=0
nt
ˆ −1
a
1
= ˆ ∗ (nt (0) − nt (ˆ
a a)) − ˆ ∗ nt (ˆ
nt (a) + a a) ,
nt a=0
ˆ −1
a
nt (a)
ˆ−
=a .
a=0
nt
ˆ ≥ a∗ , this implies
By deﬁnition of a∗ , we have nt (a) = 0 for all a ≥ a∗ . When a
ˆ −1 nt (a)
a a∗ −1 nt (a) a∗ −1 nt (a)
that a=0 nt = a=0 nt , where by deﬁnition LE = a=0 nt , the desired
result.
The fact that LE + LGEa ˆ implies that P ALE θ = a
ˆ = a ˆ ) because
ˆ(1 − EDθa
ˆ(1 − EDθa
a ˆ )(1 − ED θ a
ˆ ) = (LE + LGEa ˆ ) = LE (1 − θH ).
ˆ ≥ a∗ , P ALEθ is a linear function of EDθa
Thus, when a ˆ that depends negatively
ˆ . Therefore, these two indicators yields opposite ranking of any two societies
on EDθa
A and B , i.e. P ALE θ (A) ≥ P ALE θ (B ) ⇔ EDθa ˆ (B ).
ˆ (A) ≤ EDθ a
5.4 Proof of Proposition 4
We ﬁrst prove the following: for any a ˆ≥a ˆ and any two societies A and B, we have
EDθa ˆ (B ) for all θ ∈ [0, 1] if and only if
ˆ (A) > EDθ a
a (A) > ED0ˆ
ED0ˆ a (B ) and ED1ˆ a (B ).
a (A) > ED1ˆ
32
We start with the “only if” part. Assume on the contrary that ED0ˆ
a (A) <
ED0ˆa (B ) or ED1ˆa (A) < ED1ˆ a (B ). This implies that EDθ a
ˆ (A) < EDθ a ˆ (B ) for
some θ ∈ {0, 1} and therefore we cannot have EDθa ˆ (A) > ED ˆ
θa (B ) for all θ ∈ [0, 1].
ˆ index, we have to show that
We turn to the “if” part. By deﬁnition of the EDθa
LGEa ˆ (A) LGEa ˆ (B )
− >
LE (A) + LGEa ˆ (A) LE (B ) + LGE ˆ (B )
a
LE (B ) ∗ H (B ) LE (A) ∗ H (A)
θ − for all θ ∈ [0, 1].
LE (B ) + LGEa ˆ (B ) LE (A) + LGEaˆ (A)
(12)
As ED1ˆ a (A) > ED1ˆa (B ), Eq. (12) holds for θ = 1. As ED0ˆ a (A) > ED0ˆ a (B ),
the left hand side of Eq. (12) is strictly positive. As a result, inequality (12) holds
for all values of θ smaller than 1.
Proof of (i). This is an immediate implication of the statement proven above.
Proof of (ii). Consider two societies A and B with H (A) < H (B ) for which the
generalized condition C1 holds.
Society A is such that H (A) = 0.4 and all its individuals die in their ﬁrst year of
life, which implies that LE (A) = 1 and LGEa ˆ (A) = aˆ − 1. Therefore, society A is
ˆ −1
a 0.6
such that ED0ˆ a (A) = a ˆ and ED1ˆ a (A) = 1 − a ˆ for all a ˆ. Society B is such
ˆ≥a
that H (B ) = 0.5 and all its individuals die at the maximal age a∗ , which implies that
LE (B ) = a∗ + 1 and LGEa
ˆ (B ) = 0. Therefore, society B is such that ED0ˆ
a (B ) = 0
and ED1ˆ ˆ ∈ { 2 , . . . , a∗ } .
a (B ) = 0.5 for all a
By the statement we have proven above, we have EDθa ˆ (B ) for all
ˆ (A) > EDθ a
ˆ ∈ {2, . . . , a∗ } because
θ ∈ [0, 1] and all a
ED0ˆ
a (A) > ED0ˆ ˆ ∈ { 2 , . . . , a∗ }
a (B ) for all a
ˆ −1
as a
ˆ
a ˆ ∈ {2, . . . , a∗ }, and
> 0 for all a
ED1ˆ
a (A) > ED1ˆ ˆ ∈ { 2 , . . . , a∗ }
a (B ) for all a
as a ˆ ∈ { 2 , . . . , a∗ } .
ˆ > 1 for all a
By (i), there remains to show that EDθa ˆ (B ) for all θ ∈ [0, 1] and all
ˆ (A) > EDθ a
ˆ > a∗ . We have shown that EDθa∗ (A) > EDθa∗ (B ) for all θ ∈ [0, 1], which implies
a
by Proposition 3 that P ALEθ (A) < P ALEθ (B ) for all θ ∈ [0, 1]. By Proposition 3
again, P ALEθ (A) < P ALEθ (B ) for all θ ∈ [0, 1] implies that EDθa ˆ (B )
ˆ (A) > EDθ a
ˆ > a∗ , the desired result.
for all θ ∈ [0, 1] and all a
5.5 Proof of Proposition 5
The proof builds on the complete framework presented in Appendix 5.2.
We ﬁrst derive expression (7). As society St is stationary, we have that nt (a) =
nt−a (a) and nt (a + 1) = nt−a (a + 1) for all a ∈ {0, . . . , a∗ − 1}. We can thus
33
successively write
ˆ −1
a
nt (a) − nt (a + 1) nt (a)
ˆ (St ) =
LGEa a − (a + 1)) ∗
(ˆ ∗ ,
a=0
nt (a) nt
ˆ −1
a
nt−a (a) − nt−a (a + 1)
= a − (a + 1)) ∗
(ˆ ∗ Vt (a).
a=0
nt−a (a)
a− 1 t
As society St is stationary, Lemma 1 applies and we have Vt (a) = Πk =0 (1 − µk ) (Eq.
(9)). This result follows from our deﬁnition of the age-speciﬁc mortality rate, where
nt−a (a)−nt−a (a+1)
µt
a = nt−a (a) .
We now prove that GDθa ˆ (St ). As society St is stationary, Lemma
ˆ (St ) = EDθ a
1 applies and Nt = nt LEt (Eq. (10)). Substituting this expression for Nt into the
ˆ proves our result, provided Y Lt = nt LGEa
deﬁnition of GDθa ˆ , which remains to
a ( a)
be shown. As society St is stationary, Lemma 1 applies and we have nt− nt =
a− 1
k=0 (1− µt
k ) (Eq. (9)). Substituting this expression for nt−a (a) into the deﬁnition
ˆ −2
a
of Y Lt , with Y Lt = a=0 nt−a (a) ∗ µt a − (a + 1)), gives:
a ∗ (ˆ
ˆ −2
a a− 1
Y L t = nt a − (a + 1)) ∗ µt
(ˆ a∗ (1 − µt
k ),
a=0 k=0
ˆ (see Eq. (7) and recall that a
which shows that Y Lt = nt LGEa ˆ − (a + 1) = 0 when
ˆ − 1), the desired result.
a=a
6 ˆ and GDθa
Appendix: EDθa ˆ under a transitory shock:
An illustration
ˆ and GDθ a
We illustrate this diﬀerence between EDθa ˆ in their reaction to a transitory
mortality shock with the help of a simple example. Consider a population with a
ﬁxed natality nt (0) = 2 for all periods t. At each period, all alive individuals are
non-poor, implying that Ht = 0. For all t < 0, we assume a constant mortality vector
µt = µ∗ = (0, 1, 1, 1), so that each individual lives exactly two periods. Let us assume
ˆ = 4, so that an individual dies prematurely if she dies before her fourth period of
a
life. Before period t = 0, the population pyramid is stationary, and the two indices
are equal to 1/2 because there is no poor and individuals live for two periods instead
of four. Consider now a permanent shock starting from period 0 onwards, such
that half of the newborns die after their ﬁrst period of life: µ0 = (1/2, 1, 1, 1). The
population pyramid returns to its stationary state in period 1, after a (mechanical)
transition in period 0. This example is illustrated in Figure 11.
Consider ﬁrst GDθa ˆ . In period 0, the actual population pyramid is not stationary
because of the mortality shock. The premature death of one newborn leads to the
loss of three years of life. Also, two one-year old individuals die in period 0, each
losing two years of life. There are thus 7 years of life prematurely lost in period 0,
and GDθa ˆ takes value 7/11. In period 1, the population pyramid is stationary, and
GDθaˆ is equal to 5/8 from then on.
We now turn to EDθa ˆ . Even if the actual population pyramid is not stationary in
ˆ is immediately equal to 5/8 since it records premature mortality as if
period 0, EDθa
34
/ {0, 1, 2}
t∈ t=0 t=1
7
GD = 11
5
Number GD = 4
Number Number GD = 8
8 ∗ ∗ ∗
indiv. indiv. indiv.
a
ˆ a
ˆ a
ˆ
2 2 2
∗
1 1 1
GD
Age Age Age
1 2 3 4 1 2 3 4 1 2 3 4
Number 4 Number
ED = 5
indiv. 8 indiv. ED = 8
a
ˆ a
ˆ
2 2
LGEa
ˆ
1 1
ED LE
Age Age
1 2 3 4 1 2 3 4
Figure 11: Response of GDθa ˆ to a permanent mortality shock in t = 0.
ˆ and EDθ a
The years prematurely lost are shaded.
ˆ focuses
the population pyramid had already reached its new stationary level. EDθa
on the newborn and the one-year old who die prematurely, ignoring that there are
two one-year old dying in the actual population pyramid in period 0 (which is a
legacy of the past).
7 Appendix: Characterization of the EDθa
ˆ index
We ﬁrst introduce the set-up provided by Baland et al. (2021), which we will use to
ˆ.
charcterize EDθa
Each individual i is associated to a birth year bi ∈ Z. In period t, each individual
i with bi ≤ t is characterized by a bundle xi = (ai , si ), where ai = t − bi is the age
that individual i would have in period t given her birth year bi , and si is a categorical
variable capturing individual status in period t, which can be either alive and non-
poor (N P ), alive and poor (AP ) or dead (D), i.e. si ∈ S = {N P, AP, D}. In the
following, we often refer to individuals whose status is AP as “poor”. We consider
here that births occur at the beginning while deaths occur at the end of a period.
As a result, an individual whose status in period t is D died before period t.27
An individual “dies prematurely” if she dies before reaching the minimal lifespan
ˆ ∈ N. Formally, period t is “prematurely lost” by any individual i with si = D and
a
ˆ. A distribution x = (x1 , . . . , xn(x) ) speciﬁes the age and the status in period
ai < a
t of all n(x) individuals. Excluding trivial distributions for which no individual is
alive or prematurely dead, the set of distributions in period t is given by:
X = {x ∈ ∪n∈N (Z × S )n | there is i for whom either si = D or si = D and a
ˆ > t − bi }.
Baland et al. (2021) show that the most natural consistent index to rank distri-
ˆ ). Let d(x) denote the number
butions in X is the inherited deprivation index (IDθa
27 All newborns have age 0 during period t and some among these newborns may die at the end
of period t. This implies that bi = t ⇒ si = D .
35
of prematurely dead individuals in distribution x, which is the number of individuals
i for whom si = D and a ˆ > t − bi , p(x) the number of individuals who are poor and
f (x) the number of alive and non-poor individuals. The IDθa ˆ index is deﬁned as:
d(x) p(x)
ˆ (x) =
IDθa +θ , (13)
f (x) + p(x) + d(x) f (x) + p(x) + d(x)
quantity deprivation quality deprivation
where θ ∈ [0, 1] is a parameter weighing the relative importance of alive deprivation
and lifespan deprivation. An individual losing prematurely period t matters 1/θ
times as much as an individual spending period t in alive deprivation.
We introduce additional notation for the mortality taking place in period t. Con-
sider the population pyramid in period t, and let na (x) be the number of alive indi-
viduals of age a in distribution x, i.e. the number of individuals i for whom ai = a
and si = D. (The deﬁnition of na (x) corresponds to nt−a (a) in the notation used in
the main text of the paper. In this section, we adopt the notation of Baland et al.
(2021), which does not require to mention period t.) The age-speciﬁc mortality rate
µa ∈ [0, 1] denotes the fraction of alive individuals of age a dying at the end of period
t: the number of a-year-old individuals dying at the end of period t is na (x) ∗ µa .
Letting a∗ ∈ N stand for the maximal lifespan (which implies µa∗ −1 = 1), the vector
of age-speciﬁc mortality rates in period t is given by µ = (µ0 , . . . , µa∗ −1 ). Vector
µ summarizes mortality in period t, while distribution x summarizes alive depriva-
tion in period t as well as mortality before period t. The set of mortality vectors is
deﬁned as:
∗
M = µ ∈ [0, 1]a µa∗ −1 = 1 .
We consider pairs (x, µ) for which the distribution x is a priori unrelated to
vector µ. We assume that the age-speciﬁc mortality rates µa must be feasible given
the number of alive individuals na (x). Given that distributions have ﬁnite numbers
of individuals, mortality rates cannot take irrational values, i.e. µa ∈ [0, 1] ∩ Q, where
Q is the set of rational numbers. The set of pairs considered is given by:
ca
O= (x, µ) ∈ X × M for all a ∈ {0, . . . , a∗ } we have µa = for some ca ∈ N .
na (x)
Letting da (x) be the number of dead individuals born a years before t in dis-
tribution x, the total number of individuals born a years before t is then equal to
na (x) + da (x). Formally, the pair (x, µ) is stationary if, for some n∗ ∈ N and all
a ∈ {0, . . . , a∗ }, we have:
• na (x) + da (x) = n∗ ∈ N (constant natality),
• na+1 (x) = na (x) ∗ (1 − µa ) (identical population pyramid in t + 1).
In a stationary pair, the population pyramid is such that the size of each cohort can
be obtained by applying to the preceding cohort the current mortality rate. The
pair associated to a stationary society (as deﬁned in the main text) is stationary. An
ˆ) to P (x, µ)
index is a function P : O × N → R+ . We simplify the notation P (x, µ, a
ˆ is assumed.
as a ﬁxed value for a
36
ˆ . IDθ a
We now introduce the properties characterizing EDθa ˆ Equivalence requires
that, as the current mortality (in period t) is the same as the mortality prevailing in
the previous periods in stationary societies, any index deﬁned on current mortality
28
ˆ in the case of a stationary pair:
rates is equivalent to IDθa
ˆ Equivalence).
Deprivation axiom 1 (IDθa There exists some θ ∈ (0, 1] and a ˆ
ˆ≥a
ˆ (x).
such that for all (x, µ) ∈ O that are stationary we have P (x, µ) = IDθa
Independence of Dead requires that past mortality does not aﬀect the index.
More precisely, the presence of an additional dead individual in distribution x does
not aﬀect the index:
Deprivation axiom 2 (Independence of Dead). For all (x, µ) ∈ O and i ≤ n(x),
if si = D, then P ((xi , x−i ), µ) = P (x−i , µ).
Independence of Birth Year requires that the index does not depend on the birth
year of individuals, i.e. only their status matters. As Independence of Dead requires
to disregard dead individuals, the only relevant information in x is whether an alive
individual is poor or not.
Deprivation axiom 3 (Independence of Birth Year). For all (x, µ) ∈ O and
′ ′
i ≤ n(x), if si = si , then P ((xi , x−i ), µ) = P ((xi , x−i ), µ).
Replication Invariance requires that, if a distribution is obtained by replicating
another distribution several times, they both have the same deprivation when asso-
ciated to the same mortality vector. By deﬁnition, a k -replication of distribution x
is a distribution xk = (x, . . . , x) for which x is repeated k times.
Deprivation axiom 4 (Replication Invariance). For all (x, µ) ∈ O and k ∈ N,
k
P (x , µ) = P (x, µ).
Proposition 7 shows that these properties jointly characterize the EDθa
ˆ index.
ˆ ).
Proposition 7 (Characterization of EDθa
P = EDθa ˆ Equivalence,
ˆ if and only if P satisﬁes Independence of Dead, IDθ a
Replication Invariance and Independence of Birth Year.
ˆ index satisﬁes Independence
Proof. We ﬁrst prove suﬃciency. Proving that the EDθa
of Dead, Replication Invariance and Independence of Birth Year is straightforward
ˆ index satisﬁes IDθ a
and left to the reader. Finally, EDθa ˆ Equivalence because EDθ a
ˆ
ˆ in stationary populations (Proposition 5) and GDθ a
is equal to GDθa ˆ satisﬁes IDθ a
ˆ
Equivalence (Proposition 2 in Baland et al. (2021)). (The pairs associated to sta-
tionary societies are stationary).
We now prove necessity. Take any pair (x, µ) ∈ O. We construct another pair
(x′′′ , µ) that is stationary and such that P (x′′′ , µ) = P (x, µ) and EDθa ′′′
ˆ (x , µ) =
′′′
EDθa ˆ (x, µ). Given that (x , µ) is stationary, we have by IDθ a ˆ Equivalence that
P (x′′′ , µ) = IDθa ′′′
ˆ (x , µ) for some θ ∈ (0, 1]. As IDθ a
ˆ = GDθ a ˆ = EDθ aˆ for stationary
′′′ ′′′
pairs, we have P (x , µ) = EDθa ˆ (x , µ) for some θ ∈ (0, 1]. If we can construct such
pair (x′′′ , µ), then P (x, µ) = EDθa
ˆ (x, µ) for some θ ∈ (0, 1], the desired result.
28 Recall that past mortality is recorded in distribution x while current mortality is recorded in
vector µ. As vector µ is redundant in stationary pairs, in the sense that µ can be inferred from the
population pyramid, the index can be computed on distribution x only. See Baland et al. (2021)
for a complete motivation for this axiom.
37
We turn to the construction of the stationary pair (x′′′ , µ), using two intermediary
pairs (x′ , µ) and (x′′ , µ). One diﬃculty is to ensure that the mortality rates µa can be
achieved in the stationary population given the number of alive individuals na (x′′′ ),
that is µa = na (c
x′′′ ) for some c ∈ N.
We ﬁrst construct a n′ −replication of x that has suﬃciently many alive individuals
to meet this constraint. For any a ∈ {0, . . . , a∗ − 1}, take any naturals ca and ea
ca a∗ − 1 ′ a− 1 cj ′ a∗ −1 ′ 29
such that µa = e a
. Let e = j =0 e j , n a = e j =0 (1 − ej
) and n = j =0 nj .
a∗ − 1
Let x′ be a n′ −replication of x. Letting nx = j =0 nj (x) be the number of alive
individuals in distribution x, we have that x′ has n′ ∗ nx alive individuals. We have
P (x′ , µ) = P (x, µ) by Replication Invariance.
We deﬁne x′′ from x′ by changing the birth years of alive individuals in such a
way that (x′′ , µ) has a population pyramid that is stationary. Formally, we construct
x′′ with n(x′′ ) = n(x′ ) such that
• dead individuals in x′ are also dead in x′′ ,
• alive individuals in x′ are also alive in x′′ and have the same status,
• the birth year of alive individuals are changed such that, for each a ∈ {0, . . . , a∗ −
a−1 cj
j =0 (1− ej )
1}, the number of a-years old individuals is n′ ∗ nx ∗ ∗
a −1 k −1 cj .30
k=0 j =0 (1− ej )
One can check that (x′′ , µ) has a population pyramid corresponding to a station-
ary population and that each age group has a number of alive individuals in N. We
have P (x′′ , µ) = P (x′ , µ) by Independence of Birth Year.
Deﬁne x′′′ from x′′ by changing the number and birth years of dead individuals in
such a way that (x′′′ , µ) is stationary. To do so, place exactly n0 (x′′ ) − na (x′′ ) dead
individuals in each age group a. We have P (x′′′ , µ) = P (x′′ , µ) by Independence of
Dead.
Together, we have that P (x′′′ , µ) = P (x, µ). Finally, by construction we have
H (x′′′ ) = H (x), which implies that EDθa ′′′
ˆ (x, µ).
ˆ (x , µ) = EDθ a
29 These numbers imply that a constant natality of e newborns leads to a stationary population
of n′ alive individuals.
30 Observe that a ∗ −1 k −1 cj n′ ∗nx
k=0 j =0 (1 − e ) = LE , implying that e = a∗ −1 k −1 cj .
j =0 (1− ej )
j
k=0
38