metadata: authors: 0: author: Baland,Jean-Marie
1: author: Cassan,Guilhem
2: author: Decerf,Benoit Marie A
count: World
docna: 0: docna: Integrating Mortality into Poverty
            Measurement through the Poverty Adjusted Life Expectancy Index
docty: Policy Research Working Paper
abstracts: cdata!: Poverty measures typically do not
            account for mortality, resulting in counter-intuitive
            evaluations. The reason is that they (i) suffer from a
            mortality paradox and (ii) do not attribute intrinsic value
            to the lifespan. The paper proposes the first poverty index
            that always attributes a positive value to lifespan and does
            not suffer from the mortality paradox. This index, called
            the poverty-adjusted life expectancy, follows an expected
            lifecycle utility approach a la Harsanyi and is based on a
            single normative parameter that transparently captures the
            trade-o&#xfb00; between poverty and mortality. This
            indicator can be straightforwardly generalized to account
            for unequal lifespans. Empirically, we show that accounting
            for mortality substantially changes cross-country
            comparisons and trends. The paper also quantifies the
            fraction of these comparisons that are robust to the choice
            of the normative parameter.
display_title: Integrating Mortality into Poverty
            Measurement through the Poverty Adjusted Life Expectancy Index
listing_relative_url: /research/2022/07/33875122/integrating-mortality-poverty-measurement-through-poverty-adjusted-life-expectancy-index
topicv3: Multi-dimensional Poverty m1326485 1151,Poverty Measurement and Analysis m1326941 707,Mortality m1504365 3661
publishtoextweb_dt: 2022-07-29T00:00:00Z
dois: 10.1596/1813-9450-10133
origu: Off of Sr VP Dev Econ/Chief Econ (DECVP)
no_of_pages: 43
author_keys: 0: author_key: 1018161
1: author_key: NA
2: author_key: 1947055
model_extractions: Policy Research Working Paper 10133 Integrating Mortality into Poverty Measurement through the Poverty Adjusted Life Expectancy Index _Jean-Marie Baland Guilhem Cassan Benoit Decerf_ Development Economics Development Research Group July 2022
Policy Research Working Paper 10133

# **Abstract**

Poverty measures typically do not account for mortality, resulting in counter-intuitive evaluations. The reason is that they (i) suffer from a mortality paradox and (ii) do not attribute intrinsic value to the lifespan. The paper proposes the first poverty index that always attributes a positive value to lifespan and does not suffer from the mortality paradox. This index, called the poverty-adjusted life expectancy, follows an expected lifecycle utility approach a la Harsanyi and is based on a single normative parameter that transparently captures the trade-off between poverty and mortality. This indicator can be straightforwardly generalized to account for unequal lifespans. Empirically, we show that accounting for mortality substantially changes cross-country comparisons and trends. The paper also quantifies the fraction of these comparisons that are robust to the choice of the normative parameter.

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 issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
Integrating mortality into poverty measurement through the Poverty Adjusted Life Expectancy index.<sup>∗</sup> Jean-Marie Baland,<sup>†</sup> Guilhem Cassan,<sup>‡</sup> Benoit Decerf<sup>§</sup> Originally published in the Policy Research Working Paper Series on _July 2022._ This version is updated on _December 2023._ To obtain the originally published version, please email <u>prwp@worldbank.org.</u>

# **JEL** : D63, I32, O15.

**Keyworks** : Multidimensional poverty, Poverty, Mortality, Mortality paradox.

> ∗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<sup>_◦_</sup> 33665820 and Excellence of Science (EOS) Research project of FNRS n<sup>_◦_</sup> 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**

Poverty measures are widely used for monitoring progress and guiding policies. However, most poverty measures do not account for the impact that mortality has on longevity.<sup>1</sup> The orders of magnitudes involved are staggering. As illustrated in Figure 1, in 2019, a new born expects to lose 7 years of life due to premature death<sup>2</sup> and to spend 7 years of life in poverty according to our Expected Deprivation index (which we define below). This represents, overall, 1.5 billion years of life either spent in extreme poverty or lost to premature death in that year.

**Figure 1:** Expected number of years spent in extreme poverty and prematurely lost for a newborn worldwide according to _ED_ 1 _,_ 70, 1990-2019.

<!-- Start of picture text --> 30<br>25<br>20<br>15<br>10<br>5<br>Year<br>Years spent in Extreme Poverty Years prematurely lost<br>Expected Number of Years<br>1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020<br><!-- End of picture text --> Reading: in 1990, a newborn expected to spend 28 years in poverty and to lose 13 years due to premature death.

Mortality should be integrated into poverty measurement for several reasons. First, mortality reduces the lifespan of the deceased. As lifespan is a key resource, it should be attributed a positive _intrinsic_ value. Second, mortality has a perverse _instrumental_ impact on poverty measures. As observed by Kanbur and Mukherjee (2007), poverty measures face a “mortality paradox”, since the death of poor individuals is measured as an improvement. Finally, an integrated indicator may be useful to guide policy decisions that require trading-off poverty and mortality. How to allocate a fixed budget between poverty alleviation and premature mortality reduction? How much should be spent on AIDS prevention programs? An integrated indicator that meaningfully reflects the relative impacts that poverty and mortality have on

> 1This remark also applies to measures of multidimensional poverty, which always ignore the impact of mortality on the deceased. While we refer to income poverty throughout the paper, our argument can be applied to these measures as well.

> 2We define a death as premature if it occured before 70 years old, the average life expectancy in 2019 in our data.

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# well-being may prove useful in guiding such choices.

However, integrating mortality in a meaningful way is not straightforward. Indeed, mortality reduces the _quantity_ of life while all other forms of deprivation reduce the _quality_ of life. The difficulty is that poverty – which reduces the quality of life – is typically measured _in a given year_ while properly accounting for mortality – which reduces the quantity of life – requires taking a _life-cycle perspective_ . As a result, an indicator that aggregates the poverty and the mortality outcomes for a given year should also reflect a life-cycle perspective. An added difficulty comes from the fact that mortality shocks generate long-lasting dynamic mechanical adjustments to population pyramids, which may blur normative comparisons. A related reason why integrating mortality into poverty measures requires a specific aggregation is that mortality necessarily excludes other forms of deprivation: individuals, once dead, cannot suffer from other forms of deprivation.

There are two views on the intrinsic value that poverty measures should attribute to lifespan and thus to mortality. The “minimalist” view holds that individuals whose death is too premature should be considered lifespan deprived. Hence, mortality matters in so far as it occurred below a given age threshold, which defines a minimally acceptable lifespan. The “maximalist” view holds that being alive is the most fundamental component of well-being. Therefore, death, no matter at which age, should always have a negative impact.<sup>3</sup> In this paper, we propose a new index, the poverty-adjusted life expectancy ( _PALEθ_ ), that meaningfully integrates the poverty and mortality observed in a given year under a “maximalist” view. We derive the conditions under which this index does not suffer from the mortality paradox. We then show that this index can be generalized to define a new poverty index consistent with the “minimalist” view, thereby encompassing the two different views on the integration of mortality into poverty measurement. We also study the conditions under which comparisons based on our indices are robust to all plausible values of their parameters. Our empirical application shows that our indicators substantially change poverty comparisons and quantifies the cases for which reversed comparisons are robust.

Our main indicator, _PALEθ_ , is normatively grounded on the expected lifecycle utility, the measure of social welfare proposed by Harsanyi (1953).<sup>4</sup> _PALEθ_ normalizes the expected lifecycle utility of a newborn who _assumes_ she will be confronted throughout her lifetime to the poverty and mortality prevailing in the current period.<sup>5</sup> This index simply counts the number of years that such newborn expects to live but weighs down the periods that she expects to live in poverty. Mathematically, our index is obtained by multiplying life expectancy at birth by a factor one minus the fraction of poor, with a lower weight being given to the latter. This (normative)

> 3Note how these two views, while conceptually different, may in practice differ only parametrically: a “minimalist” approach using a very large age threshold is in practice “maximalist”.

> 4Following Harsanyi, social welfare in a given period can be understood as the lifecycle utility expected by a newborn when drawing at random a life that reflects the outcomes observed in that particular period.

> 5As we make clear later, our index is closely related to the concept of life expectancy, and its interpretation is based on similar assumptions. In particular, our index is not a forecast or a record of the actual average lifecycle utility of the cohort born in a particular period.

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weight _θ >_ 0, which captures the trade-off between poverty and mortality, corresponds to the fraction of the period utility lost when poor. When being poor has no utility cost, _θ_ takes the value zero and _PALE_ 0 corresponds to life expectancy at birth. When being poor (for one year) is as bad as losing one year of life, _θ_ = 1 and our index _PALE_ 1 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. _PALEθ_ does not suffer from the mortality paradox as long as _θ ≤_ 1. Besides its theoretical properties, _PALEθ_ enjoys two practical advantages. First, its data-requirement are minimal as only the life-expectancy at birth and the poverty head-count ratio are necessary. Second, _PALEθ_ has a simple interpretation, as it measures the equivalent number of years of life spent out of poverty.

In a second step, we show how to generalize _PALEθ_ to account for the distribution of lifespans. More precisely, _PALEθ_ lends itself to the definition of a new indicator, which is consistent with the minimalist view. This requires the introduction of a normative age threshold ˆ _a_ , which corresponds to our definition of premature mortality. This new index, which we call the _expected_ deprivation index ( _EDθa_ ˆ), is the weighted sum of the number of years that a newborn expects to lose prematurely or to spend in poverty, using the same weight as in _PALEθ_ .<sup>6</sup> We call the index _expected_ deprivation, given its proximity to the concept of life expectancy in its construction, interpretation and assumptions.

To measure the real world relevance of our indexes, we combine data sets provided by the World Bank data on income poverty (Poverty and Inequality Platform (World Bank, 2023)) and an internationally comparable data set on mortality data (Global Burden of Disease Collaborative Network, 2020) from 1990 to 2019. We show that mortality is growing in relative importance and substantially affects global poverty comparisons: during the 2005-2019 period, at least 34% of PALE’s growth was due to the growth of life expectancy, as opposed to 17% from 1991 to 2004. For all possible values of _θ_ , _PALEθ_ is able to solve in 2019 about half of the between country and 40% of the within country comparisons when focusing on these comparisons for which life expectancy and headcount are conflicting.

**Literature review** Baland et al. (2021) proposed a way of integrating mortality into poverty measurement that is consistent with the minimalist view. They observe that an integrated indicator should at least meaningfully compare _stationary_ societies, for which natality, mortality and poverty are constant over time. (In stationary societies, the outcomes observed in a given year completely reflect the life-cycle effects of their mortality.) They show that such indicators satisfy a set of basic axioms when based on a weighted sum of a number of years of life prematurely lost and a number of years of life spent in poverty.

We improve on _GDθa_ ˆ, the main indicator proposed by Baland et al. (2021), along the following dimensions. First, given an age threshold ˆ _a_ , the death of a poor individˆ ual above the age of _a_ is considered by _GDθa_ ˆ as an improvement. The “minimalist” view taken by this measure therefore implies a form a mortality paradox. Second, _GDθa_ ˆ is not straightforward to interpret, preventing its widespread diffusion in pub-

> 6Again, this implies that a newborn is expected to be exposed throughout her lifespan to the poverty and mortality observed in the current period.

4
lic debates. Third, the indicators they consider respond to mortality shocks with considerable inertia, reflecting long run adjustments in the population pyramid.<sup>7</sup> Inertia is not, in general, a desirable feature for poverty measures. Analyzing the relations between _EDθa_ ˆ, _PALEθ_ and _GDθa_ ˆ, we show that _EDθa_ ˆ is more reactive to shocks than _GDθa_ ˆ and lends itself to straightforward intepretations.

Our indicators differ from those proposed in the mortality paradox literature (Kanbur and Mukherjee (2007); Lefebvre et al. (2013)), which aimed at neutralizing the instrumental impact mortality has on poverty measurement (Decerf, 2023). By doing so, however, they do not attribute an intrinsic value to lifespan. As a result, an general increase in the life-cycle utility of a population is not necessarily considered as an improvement.

The poverty-adjusted life expectancy is reminiscent of several indicators proposed in health economics, like the quality-adjusted life expectancy (QALE) or the qualityadjusted life year (QALY).<sup>8</sup> Following Sullivan (1971), these two indicators account for the quality and quantity of life, by weighting down the quantity of life for periods with low quality. We show that they directly follow from the expected life-cycle utility approach in stationary societies, and correspond to well defined properties. Our index, however, accounts for a major dimension of well-being other than health, which is poverty.

The remainder of the paper is organized as follows. In Section 2, we shortly present a way to integrate poverty and mortality. In Section 3, we present the theory supporting our indicators. In Section 4, we present our global empirical application. Section 5 concludes by discussing a key limitation of our indicators, namely that neither _PALEθ_ nor _EDθa_ ˆ account for the unequal distribution of lifecycle utilities when the same individuals _cumulate_ poverty and premature mortality.

# **2 Aggregating poverty and mortality as time units: a minimalist approach**

Integrating mortality into poverty measurement is challenging. One reason is that poverty is measured in a given year while accounting for the direct impact of mortality requires a lifecycle perspective. Another difficulty is that, to be policy relevant, the combined index should account for the mortality that takes place _in the same year_ as the year in which poverty is measured. In particular, the evaluation of current policies should not be affected by past mortality shocks. A solution, pioneered by Baland et al. (2021) is to express poverty and mortality outcomes in terms of years of human life, using time units to account for the life-cycle effects of mortality and poverty.<sup>9</sup>

> 7In particular, following a permanent mortality shock, they show that _GDθa_ ˆ may follow a nonmonotonic trend.

> 8See for instance Whitehead and Ali (2010) for an economic interpretation of QALYs, or Heijink et al. (2011); Jia et al. (2011) for applications of the QALE index to comparisons of health outcomes across populations.

> 9Under the minimalist view, an alternative solution could be to weight the fraction of individuals who are poor in year _t_ with the fraction of individuals who will die prematurely given the mortality observed in year _t_ . For instance, the Human Poverty Index is defined in this way (Watkins, 2006). Unfortunately, this seemingly natural solution suffers from the mortality paradox and yields counter-

5
Consider two stationary societies A and B, in which two individuals are born every year, one in the poor dynasty and one in the rich dynasty. Individuals born in the poor (resp. rich) dynasty remain poor (resp. non-poor) throughout their lives. The lifespan of a rich individual is four years. The only difference between societies A and B is the lifespan of a poor individual, which is one year in society A and three years in society B. These societies are stationary in the sense that natality is constant and the lifecycle outcomes of two individuals born in the same dynasty are the same.<sup>10</sup> The relevant outcomes in society A and B are summarized in Table 1. Consider society A. In any arbitrary year _t_ , one individual is poor (P) and four individuals are non-poor (NP). The head-count ratio, which we denote by _H_ , is thus 1 _/_ 5. The minimalist view defines premature mortality using an age threshold ( _a_ ˆ), which we assume is three years. A dead individual is considered prematurely dead (PD) if she is born less than 3 years before _t_ (and considered dead (D) if born at least 3 years before _t_ ).

**Table 1:** Comparison of stationary societies A and B under a lifecycle perspective.

|Age in year _t_|0|1|2|3|
|---|---|---|---|---|
|Birth year|_t_|_t −_1|_t −_2|_t −_3|
|Poor dynasty A|_P_|_PD_|_PD_|_D_|
|Non-poor dynasty A|_NP_|_NP_|_NP_|_NP_|
|Poor dynasty B|_P_|_P_|_P_|_D_|
|Non-poor dynasty B|_NP_|_NP_|_NP_|_NP_|

When mortality is ignored, the poverty comparison of societies A and B reveals a mortality paradox, since the head count ratio is larger in society B than in society A: _H_ ( _A_ ) = 1 _/_ 5 and _H_ ( _B_ ) = 3 _/_ 7. The longer lifespan of the poor dynasty in society B is thus recorded by _H_ as a worsening. The problem is that, in society A, _H_ does not take into account the two poor individuals born in _t −_ 1 and _t −_ 2 who died prematurely and miss year _t_ .

Properly accounting for missing (dead) individuals allows for a more sensible comparison. Following the minimalist view, Baland et al. (2021) propose the inherited deprivation index ( _IDθa_ ˆ): where # _P_ , # _NP_ and # _PD_ respectively denote the number of poor, non-poor and prematurely dead individuals in year _t_ and parameter _θ >_ 0 captures the normative trade-off between one poor individual and one prematurely dead individual. For _θ_ = 1, one poor individual contributes the same to _IDθa_ ˆ as one prematurely dead individual. Note that the reference population in the denominator accounts for the prematurely dead individuals, which prevents from making inconsistent trade-offs between a poor and a prematurely dead individual.

However, _IDθa_ ˆ is not policy relevant because this index depends on _past_ mortality: it indeed measures the extent of deprivation _inherited_ from the past. In our example intuitive comparisons (e.g., concluding that society A is better off than society C, see below). 10Stationary societies are more formally defined in Appendix A.

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above, the two prematurely dead individuals respectively died in years _t−_ 1 and _t−_ 2. One may prefer, given the need of policy relevance, indicators that only depend on current mortality – mortality in year _t_ . One example is the generated deprivation index ( _GDθa_ ˆ) (Baland et al., 2021), which collects all the future years of life lost to premature deaths in year _t_ and attributes them to year _t_ : where _Y LL_ denotes the total number of years of life prematurely lost due to mortality in year _t_ where _Na_ is the number of alive individuals who have age _a_ and _μa_ is the mortality rate observed for individuals who have age _a_ . For example, in society A, the only premature death that takes place in year _t_ is that of the newborn in the poor dynasty. Her premature death implies that this newborn will prematurely lose two years of life, respectively in _t_ + 1 and _t_ + 2, and thus _Y LL_ = 2. Note that in society A, we have _Y LL_ = # _PD_ . That is, _GDθa_ ˆ = _IDθa_ ˆ in stationary societies. This equality reflects the fact that each cell in Table 1 represents an individual as well as a unit of time.<sup>11</sup> The key difference is that # _PD_ captures mortality before _t_ while _Y LL_ captures mortality in _t_ . By relying on units of time ( _Y LL_ ), _GDθa_ ˆ only depends on current mortality and compares the stationary societies A and B in a meaningful way.

As it is based on the minimalist view, _GDθa_ ˆ is still affected by the mortality paradox: any death of a poor occurring above the age threshold _a_ ˆ is recorded as an improvement. To see this, consider the stationary society C whose only difference with society B is that the individuals born in the poor dynasty live for four periods. Hence, poor individuals live one year longer in C than in B. We have _GDθ_ 3( _C_ ) = 4 _θ/_ 8 and _GDθ_ 3( _B_ ) = 3 _θ/_ 7, which shows that _GDθa_ ˆ can improve with the death of a poor individual. In comparison, our approach also builds on an aggregation based on time units, considers only mortality in year _t_ but takes a “maximalist” view.<sup>12</sup>

# **3 Theory**

In this section, we first define the poverty-adjusted life expectancy ( _PALEθ_ ) index and relate it to the social welfare approach proposed by Harsanyi. We then introduce the expected deprivation ( _EDθa_ ˆ) index and characterize the conditions under which _EDθa_ ˆ avoids the mortality paradox. We then explore the connections between _EDθa_ ˆ, _PALEθ_ , _IDθa_ ˆ and _GDθa_ ˆ (Baland et al., 2021). Finally, we study the conditions

> 11Summing _Y LL_ with # _P_ and # _NP_ in the denominator may seem strange until one realizes that all three terms capture years of human life, respectively prematurely lost, spent in poverty and spent out of poverty. Indeed, # _P_ captures the number of individuals who spent one year – year _t_ – in poverty and thus # _P_ is a number of poverty years. Each unit of time can be categorized as non-poor, poor, prematurely dead or dead, allowing for a proper account of the life-cycle effects of mortality. Under this approach, it is natural to compute the share of units of time spent in deprivation among the total amount of units one ought to live out of deprivation.

> 12Our measure will be compared to _IDθa_ ˆ and _GDθa_ ˆ in Subsection 3.2.

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under which comparisons by _PALEθ_ and _EDθa_ ˆ are robust to all admissible values for their parameters.

# **3.1 The Poverty-Adjusted Life Expectancy index**

## **Definition of** _PALEθ_

The poverty-adjusted life expectancy index is defined as where _θ >_ 0 captures the normative trade-off between one year spent in poverty and one year of life lost, _H_ denotes the poverty head-count ratio and _LE_ denotes life _a−_ 1 expectancy at birth, i.e., _LE_ =<sup>�</sup><sup>_a_</sup> _a_<sup>_∗_</sup> =0<sup>_−_1</sup> � _k_ =0<sup>(1</sup><sup>_−μ_</sup> _k_<sup>_t_)where</sup><sup>_a∗_denotesthemaximal</sup> lifespan than can be reached and thus _μa∗−_ 1 = 1.

_PALEθ_ is the weighted sum of a number of years spent in poverty and a number of years spent out of poverty. Indeed, its mathematical expression can be written as _LE_ (1 _− H_ ) + (1 _− θ_ ) _LE H_ . For a newborn who expects to face throughout her life the poverty and mortality observed in year _t_ , the term _LE_ (1 _− H_ ) captures the number of years she expects to live out of poverty and the term _LE H_ captures the number of years she expects to live in poverty.<sup>13</sup> Years out of poverty receive weight 1 and years in poverty receive weight (1 _− θ_ ).

_PALEθ_ encapsulates the maximalist view according to which all death matters. Indeed, _PALEθ_ is based on life expectancy at birth, which depends on mortality rates at all ages. Its data requirements are therefore limited, as it is based simply on life expectancy at birth and the poverty head-count ratio. Two special cases are worth noting: _PALE_ 0 corresponds to life expectancy at birth and _PALE_ 1 corresponds to the Poverty Free Life Expectancy (PFLE), an indicator proposed by Riumallo-Herl et al. (2018).

## **Relationship with social welfare a la Harsanyi**

We show that, under two assumptions, _PALEθ_ corresponds to social welfare a la Harsanyi. According to Harsanyi (1953), social welfare in a given year _t_ corresponds to the lifecycle utility expected by a newborn given the outcomes observed in year _t_ . Behind the veil of ignorance, the newborn faces a lottery whereby she ignores whether and when she will be poor and for how long she will live. When evaluating her lifecycle utility,<sup>14</sup> 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

> 13As we explain below, _PALEθ_ is not a forecast on the life of a newborn. Rather, its purpose is to jointly assess the mortality and poverty taking place in a given year.

> 14The rationality requirements of decision theory provide a structure on admissible 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 _U_ =<sup>∑</sup><sup>_d_</sup> _a_ =0<sup>_βau_(</sup><sup>_ca_)where</sup><sup>_d∈_Nistheageatdeath,</sup><sup>_β∈_[0</sup><sup>_,_1]isthediscountfactor,</sup><sup>_ca_is</sup> consumption at age _a_ and _u_ is the period utility function.

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where _β ∈_ [0 _,_ 1] is the discount factor, _ca ≥_ 0 is consumption at age _a_ , _u_ is the period utility function, _V_ ( _a_ ) is the (unconditional) probability that the newborn survives to age _a_ , _a_<sup>_∗_</sup> is the maximal lifespan one can reach and the expectation operator E applies to the uncertainty with respect to _ca_ . The period utility when being dead is normalized to zero, i.e., _u_ ( _D_ ) = 0. As a result, mortality is valued through its opportunity cost: death reduces the number of periods during which a newborn can enjoy consumption.

Under two assumptions, Eq. (4) simplifies into _PALEθ_ . Assumption A1 is to ignore discounting, i.e. _β_ = 1. Such assumption is necessary in order to assign equal weights to all individuals, regardless of their age.<sup>15</sup> Assumption A2 is to transform consumption into a binary variable, i.e., _ca_ can be either being non-poor ( _NP_ ) or being poor ( _P_ ). This strong assumption implies that the impact on period utility of consumption differences within these two categories is ignored.<sup>16</sup> We denote the period utilities associated to being poor and being non-poor respectively by _uP_ = _u_ ( _P_ ) and _uNP_ = _u_ ( _NP_ ).

Proposition 1 shows that, under A1 and A2, in any stationary society, _PALEθ_ corresponds to expected life-cycle utility as expressed in Eq. (4).

**Proposition 1** (Correspondence between Harsanyi and _PALEθ_ ) **.**

_For any stationary society, assumptions A1 and A2 imply that_ _and thus PALEθ is ordinally equivalent to EU ._ _Proof._ The proof is provided in Appendix B.

Proposition 1 calls for several remarks. First, this result provides a mathematical expression for parameter _θ_ . Parameter _θ_ captures the fraction of her period utility that a non-poor individual looses when she becomes poor. This mathematical expression allows calibrating a value for parameter _θ_ when selecting a period utility function and computing its value for the typical consumption of the poor and the non-poor, as we show in Appendix C.

Second, Proposition 1 holds even when mortality is selective, that is when mortality rates affect differently poor and non-poor individuals, as in the case of society A and B. The reason why _PALEθ_ is a simple normalization of _EU_ even when mortality is selective is that _EU_ is a _risk-neutral_ social welfare function. Being risk-neutral, _EU_ is unaffected by the distribution across individuals of periods spent in poverty or lost to mortality. A social planner who cares for unequal lifespans will not evaluate welfare on the basis of Eq. (4), and may prefer _EDθa_ ˆ, which we define below and accounts for unequal lifespans. In the conclusion, we discuss the more general case of a social planner who cares for unequal lifecycle _utilities_ , when some individuals

> 15Indeed, Eq. (4) 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.

> 16Assumption A2 allows us to use the head-count ratio, the simplicity of which largely explains its popularity. The headcount ratio remains however a crude indicator of poverty with well-known limitations (Sen, 1976).

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combine poverty and premature mortality. Information on such individuals is often not available. As a result, a social planner who cares for unequal utilities may not do better than integrating mortality into poverty measurement through indicators like _PALEθ_ or _EDθa_ ˆ.

Third, populations are in practice not stationary and we cannot in general interpret _PALEθ_ as the expected life-cycle utility of a newborn. Indeed, the poverty and mortality observed at birth are not necessarily good predictors of the future. Therefore, _PALEθ_ should _not_ in general be interpreted as a projection or a forecast for _EU_ . However, the validity of _PALEθ_ to evaluate a society in period _t_ does _not_ rely on its capacity to correctly forecast the future. Indeed, our objective was to aggregate the mortality and poverty observed in period _t_ in a consistent manner, using a lifecycle perspective. This aggregation should not depend on the future evolutions of poverty and mortality.<sup>17</sup> Rather, one way to do so is to take the perspective of a newborn who _assumes_ that she is born in a stationary society, i.e. that the poverty and mortality observed at the time of her birth remain unchanged during her whole life. 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 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 is widely accepted as a meaningful measure of period mortality.

# **3.2 The Expected Deprivation index**

## **Definition of** _EDθa_ ˆ

We define a new indicator – the expected deprivation index ( _EDθa_ ˆ) – which generalizes _PALEθ_ under a minimalist view. Under this view, one should only give a (negative) intrinsic value to the years of life lost _before_ reaching a minimal age threshˆ old _a_ . _EDθa_ ˆ accounts for mortality through the lifespan gap expectancy ( _LGEa_ ˆ), which measures the number of years that a newborn expects to lose prematurely.<sup>18</sup> We illustrate in Figure 2 the close connection between _LGEa_ ˆ and _LE_ . The figure depicts for each age the fraction of newborns that are expected to still be alive at age _a_ , assuming again that age-specific mortality rates are fixed. These fractions define a normalized counterfactual population pyramid. Indeed, the population pyramid of a stationary society confronted to these fixed mortality rates is obtained by multiplying these fractions by the fixed number of newborns.<sup>19</sup> In the left panel of Figure 2, _LE_ is proportional to the area below the normalized population pyramid. By contrast,

> 17For 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 realized life-cycle utility of newborns can therefore be negligible, or nil if the shock did not affect the mortality rates of the newborns.

> 18 _LGEa_ ˆ is a particular version of the Years of Potential Life Lost, an indicator used in medical research in order to quantify and compare the burden on society due to different causes of death (Gardner and Sanborn, 1990).

> 19In a stationary society, the current population pyramid can be obtained by successively applying the current age-specific mortality rates to each age group.

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_LGEa_ ˆ is equal to the area between this normalized population pyramid and the age threshold. The right panel illustrates the property that, for large enough age ˆ thresholds, _LGEa_ ˆ is the complement of _LE_ . Formally, when _a ≥ a_<sup>_∗_</sup> , where _a_<sup>_∗_</sup> is the maximal lifespan, we have _LGEa_ ˆ = ˆ _a − LE_ .

**Figure 2:** Life Expectancy and Lifespan Gap Expectancy

<!-- Start of picture text --> Fraction ˆ Fraction<br>newborns a newborns<br>alive alive aˆ<br>1 1<br>LGEaˆ<br>LGEaˆ<br>LE LE<br>Age Age<br>0 1 2 3 4 0 1 2 3 4 5<br><!-- End of picture text --> Note: In the Left panel, the light green area below the normalized “stationary” population pyramid is equal to _LE_ and the dark pink area is equal to _LGEa_ ˆ.

The expected deprivation index ( _EDθa_ ˆ) aggregates the poverty and mortality observed in year _t_ by taking the perspective of a newborn who expects to be confronted, throughout her life-cycle, to the poverty and mortality prevailing at the time of her birth.

with the same parameters _θ >_ 0 and _a_ ˆ _≥_ 2. The two normative parameters _θ_ and _a_ ˆ jointly define the respective importance attributed to poverty and mortality. Parameter _θ_ determines the relative weights of being dead or being poor _for one period_ . In contrast, parameter _a_ ˆ determines the number of periods for which “being prematurely dead” is accounted for. Hence, _a_ ˆ affects the relative size of the deprivation coming from mortality versus the deprivation coming from poverty.

Both terms have the same denominator, which measures a normative lifespan 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 1. The numerator of each term measures the expected number of years characterized by one of the two dimensions of deprivation, again assuming that the society is stationary. The numerator of the mortality term measures the number of years that a newborn expects to lose ˆ prematurely (when observing mortality in the period) given the age threshold, _a_ . The numerator of the poverty term measures the number of years that a newborn expects to spend in poverty.

# **Relationship between** _EDθa_ ˆ _IDθa_ ˆ **and** _GDθa_ ˆ

Like _GDθa_ ˆ, _EDθa_ ˆ only depends on current mortality and is thus policy relevant. Proposition 2 shows that _EDθa_ ˆ also compares stationary societies in the same way 11
# as _IDθa_ ˆ and _GDθa_ ˆ.

**Proposition 2** ( _EDθa_ ˆ, _GDθa_ ˆ and _IDθa_ ˆ are identical in stationary societies) **.** _For any stationary society, we have EDθa_ ˆ = _GDθa_ ˆ = _IDθa_ ˆ _._

# _Proof._ See Appendix D.

We now discuss more systematically the differences between _EDθa_ ˆ and _GDθa_ ˆ. _EDθa_ ˆ and _GDθa_ ˆ rank non-stationary societies differently. The main difference between _EDθa_ ˆ and _GDθa_ ˆ comes from the way the two indices account for the number of years prematurely lost. _GDθa_ ˆ records the number of years prematurely lost over all premature deaths actually taking place in year _t_ . _EDθa_ ˆ also counts the number of years prematurely lost but, instead of being computed on the actual population pyramid, _EDθa_ ˆ 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 _GDθa_ ˆ. Consider a permanent mortality shock. The population dynamics 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 stationary population pyramid is reached, typically after _a_<sup>_∗_</sup> periods. _GDθa_ ˆ records each step of this transition and therefore exhibits inertia in its response to a permanent mortality shock.<sup>20</sup> By contrast, _EDθa_ ˆ immediately refers to the new stationary population pyramid and disregards the inertia caused by these transitory demographic adjustments. We provide an illustration of this property in Appendix E.

Finally, Baland et al. (2021) show that _GDθa_ ˆ is essentially the only index decomposable into subgroups to compare stationary societies in a way that satisfies some basic properties. As a result, _EDθa_ ˆ cannot be decomposable into subgroups.<sup>21</sup> This is no surprise given that _EDθa_ ˆ is based on life expectancy, which cannot be decomposed into subgroups. In Appendix F, we also show that _EDθa_ ˆ is the only index that is independent on the actual population pyramid (no inertia) and compares stationary populations in a way that respects basic properties.

# **Relationship between** _EDθa_ ˆ **and** _PALEθ_

We show that _PALEθ_ is a version of _EDθa_ ˆ that encapsulates the maximalist view. Indeed, as stated in Proposition 3, _EDθa_ ˆ ranks societies exactly in the same way as _PALEθ_ as long as its age threshold _a_ ˆ is at least as large as the maximal lifespan _a_<sup>_∗_</sup> . For such values, the age threshold is not binding, and all deaths become relevant in terms of deprivation.

> 20For instance, assume that society A (see Table 1) undergoes a permanent mortality shock such that society A is subjected to her mortality vector in all years before _t_ , namely ( _μ_<sup>_A_</sup> 0<sup>_, μA_</sup> 1<sup>_, μA_</sup> 2<sup>_, μA_</sup> 3<sup>) =</sup> (1 _/_ 2 _,_ 0 _,_ 0 _,_ 1), but from year _t_ onwards society A is subjected to the mortality vector of society B, namely ( _μ_<sup>_B_</sup> 0<sup>_, μB_</sup> 1<sup>_, μB_</sup> 2<sup>_, μB_</sup> 3<sup>)=(0</sup><sup>_,_0</sup><sup>_,_1</sup><sup>_/_2</sup><sup>_,_1).Thereisamechanicaladjustmenttothepopulation</sup> pyramid, such that only two poor individuals live in year _t_ + 1. Only in year _t_ + 2 does the population pyramid reach the new equilibrium, with three poor individuals. The inertia of _GDθa_ ˆ may be deemed undesirable because it may complicate the analysis. Baland et al. show that the mechanical adjustments following a permanent mortality shock may lead to a non-monotonic trend in _GDθa_ ˆ.

> 21In other words, if decomposability into subgroups is seen as a key property, one should use _GDθa_ ˆ. Indeed, this index yields the same ranking as _EDθa_ ˆ in stationary populations. In those populations, _GDθa_ ˆ thus yields the same ranking as _PALEθ_ when all deaths are normatively relevant ( _a_ ˆ _≥ a_<sup>_∗_</sup> ).

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# **Proposition 3** ( _EDθa_ ˆ generalizes _PALEθ_ ) **.**

ˆ ˆ _For all a ≥ a_<sup>_∗_</sup> _we have PALEθ_ = _a_ (1 _− EDθa_ ˆ) _, which implies that, for any two societies A and B,_

# _PALEθ_ ( _A_ ) _≥ PALEθ_ ( _B_ ) _⇔ EDθa_ ˆ( _A_ ) _≤ EDθa_ ˆ( _B_ ) _._

_Proof._ See Appendix G.

When the age threshold is binding (smaller than the maximal age _a_<sup>_∗_</sup> ), the rankings obtained under _EDθa_ ˆ may not correspond to the rankings obtained under _PALEθ_ . In Appendix H, we contrast the impact of mortality shocks on _PALEθ_ and _EDθa_ ˆ.

# **Escaping the mortality paradox with** _PALEθ_

As observed in Section 2, _GDθa_ ˆ is not immune to the mortality paradox. A paradoxfree index should record an improvement when a stationary society is obtained from another stationary society by _an increment to the lifespan of a poor person_ .<sup>22</sup> For instance, in Table 1, stationary society B is obtained from stationary society A by two successive increments to the lifespan of individuals in the poor dynasty.

**Definition 1** (Paradox-free) **.**

_Mθa_ ˆ _∈{EDθa_ ˆ _, GDθa_ ˆ _, IDθa_ ˆ _} is paradox-free if for any two stationary societies A and B such that B is obtained from A by an increment to the lifespan of a poor person we have Mθa_ ˆ( _A_ ) _≥ Mθa_ ˆ( _B_ ) _._ Proposition 4 identifies the values for the two parameters _θ_ and _a_ ˆ under which _EDθa_ ˆ is Paradox-free. First, all deaths should matter, which implies that the age ˆ threshold _a_ should be at least as large as the maximal lifespan _a_<sup>_∗_</sup> . Second, one year of life prematurely lost should be at least as bad as one year of life spent in poverty, which implies that _θ ≤_ 1 and thus _uNP − uP ≤ uNP − uD_ .

**Proposition 4** ( _EDθa_ ˆ and the mortality paradox) **.**

ˆ _Mθa_ ˆ _∈{EDθa_ ˆ _, GDθa_ ˆ _, IDθa_ ˆ _} is Paradox-free if and only if θ ≤_ 1 _and a ≥ a_<sup>_∗_</sup> _._ _Proof._ See Appendix I.

An immediate corollary for Propositions 3 and 4 follows: the only way for _EDθa_ ˆ to be Paradox-free is to be ordinally equivalent to _PALEθ_ with _θ ≤_ 1.

**Corollary 1** ( _PALEθ_ and the mortality paradox) **.**

_EDθa_ ˆ _is Paradox-free if and only if EDθa_ ˆ _is ordinally equivalent to PALEθ with θ ≤_ 1 _._ These results show that indicators that embody the minimalist view cannot avoid the mortality paradox. The mortality paradox can only be avoided when the deaths taking place at older age are also attributed negative intrinsic value. Corollary 1 thus shows that the mortality paradox provides a justification for _PALEθ_ .

> 22We define more formally the notion of an increment to the lifespan of a poor person in this footnote. Following our formal framework presented in Appendix A, the life of an individual _i_ is a list of poverty statuses _li_ = ( _li_ 0 _, . . . , lidi_ ) that she experiences between age 0 and the age at which she dies _di ∈{_ 0 _, . . . , a_<sup>_∗_</sup> _−_ 1 _}_ , where _lia ∈{NP, P }_ . We say that stationary society B is obtained from stationary society A by an increment to the lifespan of a poor person when both societies have the same natality, _li_<sup>_A_=</sup><sup>_l_</sup> _i_<sup>_B_</sup> for all individuals _i_ except for some individual _j_ such that _d_<sup>_B_</sup> _j_<sup>=</sup><sup>_dA_</sup> _j_<sup>+ 1,</sup> _lja_<sup>_A_=</sup><sup>_P_forall</sup><sup>_a ≤dA_</sup> _j_<sup>and</sup><sup>_l_</sup> _ja_<sup>_B_=</sup><sup>_P_forall</sup><sup>_a ≤dB_</sup> _j_<sup>.</sup>

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# **3.3 Robust comparisons**

We study the conditions under which comparisons by _PALEθ_ are robust to the plausible values for parameter _θ_ .

By Corollary 1, _PALEθ_ is Paradox-free when _θ ∈_ (0 _,_ 1]. Yet, the comparison of two societies with _PALEθ_ may depend on the particular value assigned to _θ ∈_ (0 _,_ 1]. We show that a nontrivial part of these comparisons does not depend on the value for _θ 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 higher mortality that are _robustly_ ranked by _PALEθ_ , that is, in the same way for all values of _θ ∈_ (0 _,_ 1].

We illustrate this property in Figure 3. 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-PALE _θ_ 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. However, the iso-PALE1 curve (associated to _θ_ = 1) is _not_ a horizontal line. This defines two additional areas for which welfare can be robustly compared with that of society A. The iso-PALE _θ_ 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 robustly higher social 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 robustly lower social 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 _PALE_ 1 at A. For society A and _PALE_ 1, this marginal rate of substitution is given by<sup>_LE_</sup><sup><u>(</u></sup> (<sup>_A_</sup> _LE_<sup><u>)(1</u></sup> ( _A_<sup>_−_</sup> ))<sup>_H_2(</sup><sup>_A_</sup><sup><u>))</u></sup> . 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.These additional robust comparisons follow from (i) the fact that expected life-cycle utility sums period utilities and (ii) the assumption that a year of life spent in poverty is considered not worse than a year of life lost (i.e., 1 _≥ θ_ , which is _uP ≥ uD_ ).

As an illustration, Table 2 below reports the situation of Pakistan and Bangladesh in 2019. Note that Life Expectancy can trivially be decomposed into Poverty Expectancy (LE*H) and Poverty Fee Life Expectancy (LE*(1-H)). Pakistan has a lower headcount ratio than Bangladesh, but life expectancy is also lower in Pakistan. Therefore, it is a priori difficult to rank those two societies. Assuming that poverty and mortality remain unchanged, an individual 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, _PALEθ_ ranks Bangladesh above Pakistan for all _θ ∈_ (0 _,_ 1].

In the absence of domination (NE and SW quadrants in Figure 3), ignoring mortality, i.e., comparing two societies based on _H_ , may lead to robustly erroneous comparisons. This happens when the ranking provided by _PALEθ_ is robust but 14
<!-- Start of picture text --> H<br>θ = 0<br>1<br>0 < θ < 1<br>θ = 1<br>Dominated by A<br>Larger EU<br>B than A<br>A<br>H(A)<br>Smaller EU<br>Dominates A<br>than A<br>LE<br>0<br>LE(A)<br><!-- End of picture text -->

**Figure 3:** A and B are robustly ranked even though _H_ ( _A_ ) _< H_ ( _B_ ) and _LE_ ( _A_ ) _< LE_ ( _B_ ).

**Table 2:** An example of robust comparison: Pakistan and Bangladesh in 2019.

||Headcount|Life|Poverty|Poverty Free|
|---|---|---|---|---|
||ratio|Expectancy|Expectancy|Life Expectancy|
||||(_LE ∗H_)|_LE ∗_(1_−H_) =_PALE_1|
|Pakistan|4.3%|64.8|2.8|62.1|
|Bangladesh|6.7%|73.6|4.9|68.8|

differs from the ranking provided by _H_ . Proposition 5 describes the conditions under which _PALEθ_ comparisons are robust.

**Proposition 5.** _(Robust comparisons with PALEθ)_

_(i) For any two societies A and B, PALEθ_ ( _A_ ) _≤ PALEθ_ ( _B_ ) _for all θ ≤_ 1 _if and only if_ _(ii) There exist societies A and B for which PALEθ_ ( _A_ ) _≤ PALEθ_ ( _B_ ) _for all θ ≤_ 1 _even though H_ ( _A_ ) _< H_ ( _B_ ) _. These societies are such that H_ ( _A_ ) _< H_ ( _B_ ) _and LE_ ( _A_ ) _< LE_ ( _B_ ) _._ _Proof._ See Appendix J.

In Appendix K, we study the conditions under which comparisons by _EDθa_ ˆ are robust to the plausible values for its parameters.

# **4 Real world implications**

We now turn to data on poverty and life expectancy spanning the period 1990-2019. The data come respectively from the World Bank’s Poverty and Inequality Platform (World Bank, 2023) and the Global Burden of Disease Project (Global Burden of 15
Disease Collaborative Network, 2020).<sup>23</sup> Appendix N presents a practictioner guide to the construction of our index.

# **4.1 A case study of South Africa**

We first illustrate the relevance of our indices with the case of South Africa. Figure 4 reports the evolution of life expectancy, poverty rate and _PALE_ 1 for South Africa from 1990 to 2019. From the perspective of poverty, the progress of South Africa is impressive, with poverty rates decreasing from 31% to 21% over the period. However, life expectancy shows a different pattern. Following the AIDS epidemic, life expectancy decreased from the mid 90s onwards, to revert back to the pre-AIDS levels after 2013. Thus, in 2007, poverty rates are low, at 19%, but life expectancy is also low, at only 53 years. How then do we compare South Africa in 2007 to South Africa in 1990 ? _PALE_ 1 indicates that deprivation is higher in 2007 than in was in 1990. Indeed, _PALE_ 1 is equal to 44 years in 1990 as opposed to 42 years in 2007. We discuss in Section 4.3 the sensitivity of the comparisons made under _PALEθ_ to the choice of _θ_ .

**Figure 4:** South Africa Evolution of _PALE_ 1 and Life Expectancy, 1990-2019

<!-- Start of picture text --> 80 70<br>70 60<br>60<br>50<br>50<br>40<br>40 %<br>30<br>30<br>20<br>20<br>10 10<br>0 0<br>Year<br>Headcount ratio (right axis)<br>PALE Life Expectancy<br>Years<br>1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020<br><!-- End of picture text --> Reading: in 1990, life expectancy was 63 years, 31% of the population was living below the poverty line and poverty adjusted life expectancy was 44 years.

# **4.2 Life expectancy and poverty in the World, 1990-2019**

At the world level, Figure 5 presents the evolution of life expectancy, the headcount ratio and _PALEθ_ between 1990 and 2019. Throughout this period, life expectancy

> 23See Appendix M for the list of countries in the database as well as their descriptive statistics for the year 2019.

16
**Figure 5:** Evolution of _PALE_ and Life Expectancy, 1990-2019

<!-- Start of picture text --> 70 60 70 60<br>60 50 60 50<br>50 50<br>40 40<br>40 40<br>30 % 30 %<br>30 30<br>20 20<br>20 20<br>10 10 10 10<br>0 0 0 0<br>Year Year<br>Life Expectancy PALE Life Expectancy PALE<br>Headcount ratio (right axis) Headcount ratio (right axis)<br>(a) PALE 1 (b) PALE 0 . 5<br>Years Years<br>1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020<br><!-- End of picture text --> Reading: in 1990, Poverty-Adjusted Life Expectancy was about 30 years according to _PALE_ 1 and 48 years according to _PALE_ 0 _._ 5.

increased from 62 to 71 but the decrease in poverty expectancy is even more spectacular, from 30 years in 1990 to only 7 years in 2019. This decrease in poverty combined with an increase in life expectancy resulted in a large increase in _PALE_ 1, from 32 in 1990 to 64 years in 2019. For _θ <_ 1, the corresponding _PALEθ_ curves all lie between life expectancy and the _PALE_ 1 curve. We show _PALE_ 0 _._ 5 in the right hand panel of Figure 5. By construction, _PALE_ 0 _._ 5 is higher in absolute value (47 years in 1990). However, its evolution is much slower than that of _PALE_ 1: from 1990 to 2019, _PALE_ 0 _._ 5 increased by 44% as opposed to 101% for _PALE_ 1. Indeed _θ_ = 1 implies that one year spent in poverty is equivalent to one year spent dead. When instead one assumes that a year spent in poverty is equivalent to half a year lost to death, life expectancy has more weight in PALE. As the progress on life expectancy have been much slower than those against poverty, _PALE_ 0 _._ 5 growth is slower.

Note that PALE can not be directly decomposed into each of its components. However, it is possible to decompose its growth into the contribution of that of each of its component. Indeed, the growth rate of _PALEθ_ can be decomposed as follows: where _εLE_ = 1 and _εH_ = (1 _−−θHθH_ )<sup>representtheelasticitiesof</sup><sup>_PALEθ_tolifeex-</sup> pectancy and poverty, respectively. Figure 6 shows the share of the growth of _PALEθ_ explained by changes in life expectancy, from 1991 to 2019. First, note that the choice of _θ_ = 1 is conservative: the contribution of life expectancy to PALE is on average 16 percentage point smaller when _θ_ = 1 than when _θ_ = 0 _._ 5. Second, irrespective of the precise value given to _θ_ , the contribution of life expectancy is growing over time. Life expectancy contributes on average to 17 (resp. 32) percent of PALE’s growth from 1991 to 2004, as opposed to 34% (resp. 51%) from 2005 onwards. Even though mortality has not decreased as much as poverty, the changes in mortality still play a substantial role in the trend of _PALEθ_ .

17
**Figure 6:** Share of the growth of LE in the growth of _PALE_ 1 and _PALE_ 0 _._ 5, 19902019

<!-- Start of picture text --> Year<br>LE growth contribution, θ = 1 LE growth contribution, θ = 0.5<br>80<br>70<br>60<br>50<br>%<br>40<br>30<br>20<br>10<br>0<br>1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020<br><!-- End of picture text --> Reading: in 1991, the growth of life expectancy contributed to 10% of the growth of _PALE_ 1 and to 25% to that of _PALE_ 0 _._ 5.

Turning now to deprivation as captured by the _EDθa_ ˆ index, Figure 7 reports the evolution of _ED_ 1 _,_ 70 in the world during the same period.<sup>24</sup> The evolution of _ED_ 1 _,_ 70 illustrates the vast progress made against deprivation at the world level. While a newborn expected to have 58% of its normative lifespan either lost to mortality or spent in poverty in 1990, this expected loss falls to 18% in 2019. Unlike PALE, ED can be directly decomposed into each of its components, as indicated in the graph. In particular, the share of the lifespan deprivation component increased over time: from 30% in 1990 to 51% in 2019. As a result, the neglect of mortality in most poverty measures amounts to missing an increasing and significant share of total deprivation.

# **4.3 Improving on comparisons based on the headcount only**

Our indices also allow for comparing countries in which life expectancy and poverty evolve in opposite direction, as in the Pakistan-Bangladesh comparison presented in Table 2 or in the comparison between the years 1990 and 2007 in South Africa (Figure 4). However, the choice of _θ_ may not be innocuous in these comparisons. We now focus on these cases. For these cases, we discuss the extent to which _PALEθ_ offers comparisons that are robust, that is, for which the ranking is not affected by the choice of _θ_ . Note that if the ranking proposed by _PALEθ_ is robust to the choice of _θ_ , this implies that _PALE_ 0 and _PALE_ 1 yield the same ranking. Since _PALE_ 0 corresponds to life expectancy, a measure solely based on the headcount provides a

> 24We take 70 as the lifespan deprivation threshold since world life expectancy is 71 years in 2019. This threshold is therefore a reasonable choice given our maximalist perspective: dying before this age means dying below the average expected age of death at the world level.

18
ˆ **Figure 7:** Evolution of _EDθa_ ˆ and H, 1990-2019 (where _θ_ = 1 and _a_ = 70).

<!-- Start of picture text --> 60 50<br>50<br>40<br>40<br>30<br>% 30 %<br>20<br>20<br>10<br>10<br>0 0<br>Year<br>Income Deprivation Lifespan Deprivation (70)<br>Headcount ratio ED (70)<br>Lifespan Deprivation share in ED (right axis)<br>1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020<br><!-- End of picture text --> Reading: in 1990, a newborn expected to be deprived of 58% of its life: 17 percentage points lost because of lifespan deprivation and 41 because of income deprivation. Lifespan deprivation accounted for 30% of expected deprivation in 1990.

wrong ranking whenever life expectancy and headcount diverge (which is the case we focus on). The main interest of robust comparisons is to measure the extent to which PALE allows to improve on a ranking based on the headcount only: in all these situations, irrespective of the value given to _θ_ , the ranking under PALE contradicts the poverty ranking.

# **Inter country comparisons**

To what extent does _PALEθ_ help in robustly ranking countries, as compared to a simple headcount? Figure 8 reports the proportion of all country-pairs comparisons whose ranking based on life expectancy and headcount ratio differs. There are 23% of them.<sup>25</sup> The share of these ambiguous cases for which _PALEθ_ provides a robust answer is equal to 37 percent, independently of the value given to _θ_ . In other words, 37% of these “ambiguous” cases are wrongly classified by the headcount ratio. Note also that the share of ambiguous comparisons that our index unambiguously solves strongly increases over time, owing to the falling incidence of absolute poverty in many countries.<sup>26</sup>

> 25These are the only situations in which PALE can offer a different ranking than the headcount. 26The 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.

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**Figure 8:** Evolution of the resolution of ambiguous inter-country comparisons, 19902019

<!-- Start of picture text --> 50<br>40<br>%<br>30<br>20<br>1990 2000 2010 2020<br>Year<br>Share of ambiguous comparisons<br>Minimal share of ambiguous comparisons solved with PALE<br><!-- End of picture text --> Reading: in 1990, countries had on average 23% of ambiguous comparisons, out of which at least 26% were solved by the use of PALE.

## **Countries’ trajectories**

We now turn to individual trajectories of all countries, such as the South African case discussed earlier. For each country in our data, we computed the growth rate of H and LE over each 5-years period. Figure 9 presents the evolution of the share of ambiguous intra-country comparisons as well as the share that is robustly resolved by _PALEθ_ . Over the period, the share of ambiguous trajectories oscillates between 20 and 40% of all cases. The share of these cases that _PALEθ_ ranks unambiguously varies between between 20 and 40% for the period 1995-2005 up to 40 to 60% in the 2005-2015 period. As above, _PALEθ_ corrects an increasing share of the rankings proposed by the headcount. In Appendix O, we present each country’s evolution for the period 1990-2019 and its resolution in a graphical format reminiscent of the theoretical Figure 3.

# **5 Concluding remarks**

An important limitation of the two indices proposed in this paper, _PALEθ_ and _EDθa_ ˆ, is that they account for the distribution of outcomes “dimension-by-dimension”. More precisely, they account for the distribution of quality of life and for the distribution of quantity of life, but not for the distribution of life-cycle utilities. 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 implications for the distribution of life-cycle utilities. When poor individuals die early, they cumulate low achievements 20
**Figure 9:** Evolution of the resolution of ambiguous countries’ trajectories, 1990-2019

<!-- Start of picture text --> 80<br>60<br>% 40<br>20<br>0<br>1995 2000 2005 2010 2015 2020<br>Year<br>Share of ambiguous comparisons<br>Minimal share of ambiguous comparisons solved with PALE<br><!-- End of picture text --> Reading: in the 1995, 35% of countries’ trajectories was ambiguous. Among these, 29% can be assessed with PALE.

in the two dimensions and the difference between their life-cycle utility and that of non-poor individuals increases.

Accounting for the distribution of lifecycle utilities requires data that are typically not available. The necessary data include not only information on the correlation between poverty and premature mortality, but also information on mobility in and out of poverty. When such data is not available, _PALEθ_ and _EDθa_ ˆ can be used as a second-best solution, as they improve over the widespread practice of entirely ignoring the impact of mortality on longevity. This is particularly relevant for societies in which premature mortality is highly selective, affecting disproportionately poorer individuals. In particular, the premature mortality term of _EDθa_ ˆ essentially captures these negative outcomes.

If one cares about the distribution of life-cycle utilities and the necessary data is available, our indicators would need to be adjusted. Let us define as “life-cycle poor” the 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 no period of poverty. One index combining mortality and poverty that would account for the distribution of life-cycle utilities is the expected fraction of newborns who will be “life-cycle poor”, again assuming constant poverty and mortality.<sup>27</sup> Our paper calls for future research on the value that the normative parameter _θ_ should take. Its mathematical expression based on social welfare a la Harsanyi allows calibrating its value, as we show in Appendix C. However, the calibrated

> 27Note that the indicators proposed in the literature on the mortality paradox are typically not appropriate to capture the distribution of life-cycle utilities, as they do not attribute an intrinsic value to the quantity of life. The may therefore miss improvements when the lifecycle utility of all individuals increase, for instance if the lifespan of all individuals is multiplied by a common factor.

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