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, is based on a single normative parameter that transparently resulting in counter-intuitive evaluations. The reason is that captures the trade-off between poverty and mortality. This they (i) suffer from a mortality paradox and (ii) do not indicator can be straightforwardly generalized to account attribute intrinsic value to the lifespan. The paper proposes for unequal lifespans. Empirically, we show that accounting the first poverty index that always attributes a positive value for mortality substantially changes cross-country compar- to lifespan and does not suffer from the mortality paradox. isons and trends. The paper also quantifies the fraction of This index, called the poverty-adjusted life expectancy, fol- these comparisons that are robust to the choice of the nor- lows an expected lifecycle utility approach a la Harsanyi and mative 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.∗ † ‡ Jean-Marie Baland, Guilhem Cassan, Benoit Decerf§ 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 prwp@worldbank.org. 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◦ 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 Poverty measures are widely used for monitoring progress and guiding policies. How- ever, most poverty measures do not account for the impact that mortality has on longevity.1 The orders of magnitudes involved are staggering. As illustrated in Fig- ure 1, in 2019, a new born expects to lose 7 years of life due to premature death2 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 ED1,70 , 1990-2019. 30 Expected Number of Years 25 20 15 10 5 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 Year 2020 Years spent in Extreme Poverty Years prematurely lost 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 individ- uals 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 allo- cate 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 1 This 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. 2 We define a death as premature if it occured before 70 years old, the average life expectancy in 2019 in our data. 2 well-being may prove useful in guiding such choices. However, integrating mortality in a meaningful way is not straightforward. In- deed, 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 re- sult, 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 mat- ters 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 funda- mental component of well-being. Therefore, death, no matter at which age, should always have a negative impact.3 In this paper, we propose a new index, the poverty-adjusted life expectancy (P ALEθ ), 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, P ALEθ , is normatively grounded on the expected lifecycle utility, the measure of social welfare proposed by Harsanyi (1953).4 P ALEθ normal- izes 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 pe- riod.5 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) 3 Note how these two views, while conceptually different, may in practice differ only parametri- cally: a “minimalist” approach using a very large age threshold is in practice “maximalist”. 4 Following 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. 5 As 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. 3 weight θ > 0, which captures the trade-off between poverty and mortality, corre- sponds to the fraction of the period utility lost when poor. When being poor has no utility cost, θ takes the value zero and P ALE0 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 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. P ALEθ does not suffer from the mortality paradox as long as θ ≤ 1. Besides its theoretical properties, P ALEθ 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, P ALEθ 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 P ALEθ to account for the distri- bution of lifespans. More precisely, P ALEθ lends itself to the definition of a new indicator, which is consistent with the minimalist view. This requires the introduc- ˆ, which corresponds to our definition of premature tion of a normative age threshold a ˆ ), is mortality. This new index, which we call the expected deprivation index (EDθa 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 P ALEθ .6 We call the index expected deprivation, given its proximity to the concept of life expectancy in its con- struction, 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, P ALEθ 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 soci- eties, 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 ef- fects 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. ˆ , the main indicator proposed by Baland et al. (2021), along We improve on GDθa ˆ, the death of a poor individ- the following dimensions. First, given an age threshold a ˆ is considered by GDθa ual above the age of a ˆ as an improvement. The “minimalist” view taken by this measure therefore implies a form a mortality paradox. Second, GDθaˆ is not straightforward to interpret, preventing its widespread diffusion in pub- 6 Again, 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.7 Inertia is not, in general, a desirable feature for poverty measures. Analyzing the relations between EDθaˆ , P ALEθ and GDθ a ˆ is more reactive to ˆ , we show that EDθ a 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 quality- adjusted life year (QALY).8 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 ˆ account for the unequal distribution of lifecycle utilities neither P ALEθ nor EDθa 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.9 7 In particular, following a permanent mortality shock, they show that GD ˆ may follow a non- θa monotonic trend. 8 See 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. 9 Under 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.10 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 ˆ), which we minimalist view defines premature mortality using an age threshold (a 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 com- parison. Following the minimalist view, Baland et al. (2021) propose the inherited ˆ ): deprivation index (IDθa #P D #P ˆ = IDθa +θ , (1) #P + # N P + # P D #P + # N P + # P D mortality term poverty term where #P , #N P and #P D 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 ˆ as one prematurely dead θ = 1, one poor individual contributes the same to IDθa 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. ˆ is not policy relevant because this index depends on past mortality: However, IDθa 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). 10 Stationary societies are more formally defined in Appendix A. 6 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 (GDθaˆ ) (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: Y LL #P ˆ = GDθa +θ , (2) #P + #N P + Y LL #P + #N P + Y LL mortality term poverty term where Y LL denotes the total number of years of life prematurely lost due to mortality in year t ˆ −2 a Y LL = a − (a + 1)), Na ∗ µa ∗ (ˆ a=0 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 ˆ = IDθ a have Y LL = #P D. That is, GDθa 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.11 The key difference is that #P D captures mortality before t while Y LL ˆ only depends on captures mortality in t. By relying on units of time (Y LL), GDθa current mortality and compares the stationary societies A and B in a meaningful way. ˆ is still affected by the mortality As it is based on the minimalist view, GDθa ˆ is recorded as an paradox: any death of a poor occurring above the age threshold a 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 GDθ3 (C ) = 4θ/8 ˆ can improve with the death of a poor and GDθ3 (B ) = 3θ/7, which shows that GDθa 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.12 3 Theory In this section, we first define the poverty-adjusted life expectancy (P ALEθ ) index and relate it to the social welfare approach proposed by Harsanyi. We then introduce ˆ ) index and characterize the conditions under which the expected deprivation (EDθa ˆ avoids the mortality paradox. We then explore the connections between EDθ a EDθa ˆ, ˆ and GDθ a P ALEθ , IDθa ˆ (Baland et al., 2021). Finally, we study the conditions 11 Summing Y LL with #P and #N P 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. 12 Our measure will be compared to ID ˆ and GDθ a θa ˆ in Subsection 3.2. 7 ˆ are robust to all admissible values under which comparisons by P ALEθ and EDθa for their parameters. 3.1 The Poverty-Adjusted Life Expectancy index Definition of P ALEθ The poverty-adjusted life expectancy index is defined as P ALEθ = LE (1 − θH ). (3) 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 a− 1 expectancy at birth, i.e., LE = a=0 k=0 (1 − µk ) t where a∗ denotes the maximal lifespan than can be reached and thus µa∗ −1 = 1. P ALEθ 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.13 Years out of poverty receive weight 1 and years in poverty receive weight (1 − θ). P ALEθ encapsulates the maximalist view according to which all death matters. Indeed, P ALEθ 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: P ALE0 corresponds to life expectancy at birth and P ALE1 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, P ALEθ 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 life- cycle utility,14 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( c a ) V ( a ) , (4) a=0 13 As we explain below, P ALE is not a forecast on the life of a newborn. Rather, its purpose is θ to jointly assess the mortality and poverty taking place in a given year. 14 The 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 ∑d utility discounted function, which aggregates these streams as a discounted sum of period utilities a=0 β u(ca ) where d ∈ N is the age at death, β ∈ [0, 1] is the discount factor, ca is U = a consumption at age a and u is the period utility function. 8 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∗ 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 P ALEθ . 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.15 Assumption A2 is to transform consumption into a binary variable, i.e., ca can be either being non-poor (N P ) or being poor (P ). This strong assumption implies that the impact on period utility of consumption differences within these two categories is ignored.16 We denote the period utilities associated to being poor and being non-poor respectively by uP = u(P ) and uN P = u(N P ). Proposition 1 shows that, under A1 and A2, in any stationary society, P ALEθ corresponds to expected life-cycle utility as expressed in Eq. (4). Proposition 1 (Correspondence between Harsanyi and P ALEθ ). For any stationary society, assumptions A1 and A2 imply that EU uN P − uP = LE 1 − H uN P uN P − uD θ and thus P ALEθ 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 ex- pression 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 mortal- ity rates affect differently poor and non-poor individuals, as in the case of society A and B. The reason why P ALEθ 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 15 Indeed, 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. 16 Assumption 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). 9 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 P ALEθ or EDθa ˆ. Third, populations are in practice not stationary and we cannot in general in- terpret P ALEθ 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, P ALEθ should not in general be interpreted as a projection or a forecast for EU . However, the validity of P ALEθ to evaluate a society in period t does not rely on its capacity to correctly forecast the future. Indeed, our objective was to ag- gregate 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.17 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 ˆ ) – which gen- We define a new indicator – the expected deprivation index (EDθa eralizes P ALEθ 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 thresh- ˆ. EDθa old a ˆ ), ˆ accounts for mortality through the lifespan gap expectancy (LGEa which measures the number of years that a newborn expects to lose prematurely.18 ˆ −1 a a−1 ˆ = LGEa a∗ a − (a + 1)) ∗ µt (ˆ (1 − µt k ), a=0 k=0 ˆ and LE . The figure We illustrate in Figure 2 the close connection between LGEa 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.19 In the left panel of Figure 2, LE is proportional to the area below the normalized population pyramid. By contrast, 17 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 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 LGE is a particular version of the Years of Potential Life Lost, an indicator used in medical a ˆ research in order to quantify and compare the burden on society due to different causes of death (Gardner and Sanborn, 1990). 19 In a stationary society, the current population pyramid can be obtained by successively applying the current age-specific mortality rates to each age group. 10 ˆ is equal to the area between this normalized population pyramid and the LGEa age threshold. The right panel illustrates the property that, for large enough age thresholds, LGEa ˆ ≥ a∗ , where a∗ is the ˆ is the complement of LE . Formally, when a maximal lifespan, we have LGEa ˆ =aˆ − LE . Figure 2: Life Expectancy and Lifespan Gap Expectancy Fraction ˆ a Fraction newborns newborns alive alive ˆ a 1 1 LGEa ˆ LGEa ˆ LE LE Age Age 0 1 2 3 4 0 1 2 3 4 5 Note: In the Left panel, the light green area below the normalized “stationary” ˆ. population pyramid is equal to LE and the dark pink area is equal to LGEa ˆ ) aggregates the poverty and mortality ob- The expected deprivation index (EDθa served 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. LGEaˆ LE ∗ H ˆ = EDθa +θ , (5) LE + LGEa ˆ LE + LGEa ˆ mortality term poverty term ˆ ≥ 2. The two normative parameters θ and with the same parameters θ > 0 and a ˆ jointly define the respective importance attributed to poverty and mortality. Pa- a rameter θ determines the relative weights of being dead or being poor for one period. ˆ determines the number of periods for which “being prema- In contrast, parameter a ˆ affects the relative size of the deprivation turely dead” is accounted for. Hence, a coming from mortality versus the deprivation coming from poverty. Both terms have the same denominator, which measures a normative lifespan ˆ . This normative lifespan can be in- corresponding to the sum of LE and LGEa terpreted 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 dimen- sions 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 prematurely (when observing mortality in the period) given the age threshold, aˆ. The numerator of the poverty term measures the number of years that a newborn expects to spend in poverty. ˆ IDθ a Relationship between EDθa ˆ and GDθ a ˆ ˆ , EDθ a Like GDθa ˆ only depends on current mortality and is thus policy relevant. ˆ also compares stationary societies in the same way Proposition 2 shows that EDθa 11 ˆ and GDθ a as IDθa ˆ. ˆ , GDθ a Proposition 2 (EDθa ˆ are identical in stationary societies). ˆ and IDθ a ˆ = GDθ a For any stationary society, we have EDθa ˆ. ˆ = IDθ a Proof. See Appendix D. ˆ and GDθ a We now discuss more systematically the differences between EDθa ˆ. ˆ and GDθ a EDθa ˆ rank non-stationary societies differently. The main difference be- ˆ and GDθ a tween EDθa ˆ comes from the way the two indices account for the number ˆ records the number of years prematurely lost over of years prematurely lost. GDθa ˆ also counts the number all premature deaths actually taking place in year t. EDθa of years prematurely lost but, instead of being computed on the actual population ˆ uses a counterfactual population pyramid, which is the one that pyramid, EDθa would prevail in a stationary society characterized by the age-specific mortality rates observed in the period. ˆ is more reactive to policy A major implication of this difference is that EDθa ˆ . Consider a permanent mortality shock. The population dy- changes than GDθa 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.20 By contrast, EDθa ˆ 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 E. 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 21 ˆ cannot be decomposable into subgroups. some basic properties. As a result, EDθa ˆ is based on life expectancy, which cannot be This is no surprise given that EDθa ˆ is the only in- decomposed into subgroups. In Appendix F, we also show that EDθa dex that is independent on the actual population pyramid (no inertia) and compares stationary populations in a way that respects basic properties. ˆ and P ALEθ Relationship between EDθa ˆ that encapsulates the maximalist view. We show that P ALEθ is a version of EDθa ˆ ranks societies exactly in the same way as Indeed, as stated in Proposition 3, EDθa ˆ is at least as large as the maximal lifespan a∗ . P ALEθ as long as its age threshold a For such values, the age threshold is not binding, and all deaths become relevant in terms of deprivation. 20 For 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 (µA A A A 0 , µ1 , µ2 , µ3 ) = (1/2, 0, 0, 1), but from year t onwards society A is subjected to the mortality vector of society B, namely (µB B B B 0 , µ1 , µ2 , µ3 ) = (0, 0, 1/2, 1). There is a mechanical adjustment to the population 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. 21 In other words, if decomposability into subgroups is seen as a key property, one should use GD . θaˆ 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∗ ). 12 ˆ generalizes P ALEθ ). Proposition 3 (EDθa ˆ ≥ a∗ we have P ALE θ = a For all a ˆ(1 − EDθa ˆ ), which implies that, for any two societies A and B, ˆ (A) ≤ EDθ a P ALE θ (A) ≥ P ALE θ (B ) ⇔ EDθa ˆ (B ). Proof. See Appendix G. When the age threshold is binding (smaller than the maximal age a∗ ), the rank- ˆ may not correspond to the rankings obtained under ings obtained under EDθa P ALEθ . In Appendix H, we contrast the impact of mortality shocks on P ALEθ ˆ. and EDθa Escaping the mortality paradox with P ALE θ ˆ is not immune to the mortality paradox. A paradox- As observed in Section 2, GDθa free 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.22 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). ˆ ∈ {ED θ a Mθ a ˆ , GD θ a ˆ } is paradox-free if for any two stationary societies A and ˆ , ID θ a B such that B is obtained from A by an increment to the lifespan of a poor person ˆ ( A) ≥ M θ a we have Mθa ˆ (B ). ˆ under which Proposition 4 identifies the values for the two parameters θ and a ˆ is Paradox-free. First, all deaths should matter, which implies that the age EDθa ˆ should be at least as large as the maximal lifespan a∗ . Second, one year threshold a 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 uN P − uP ≤ uN P − uD . ˆ and the mortality paradox). Proposition 4 (EDθa ˆ ∈ {ED θ a Mθ a ˆ , GD θ a ˆ ≥ a∗ . ˆ } is Paradox-free if and only if θ ≤ 1 and a ˆ , ID θ a 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 P ALE θ with θ ≤ 1. Corollary 1 (P ALE θ and the mortality paradox). ˆ is Paradox-free if and only if ED θ a EDθa ˆ is ordinally equivalent to P ALE θ 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 P ALE θ . 22 We 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 = (li0 , . . . , lidi ) that she experiences between age 0 and the age at which she dies di ∈ {0, . . . , a∗ − 1}, where lia ∈ {N P, 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 A = lB for all individuals i except for some individual j such that dB = dA + 1, i j j lja = P for all a ≤ dA A j and lja = P for all a ≤ dj . B B 13 3.3 Robust comparisons We study the conditions under which comparisons by P ALE θ are robust to the plausible values for parameter θ. By Corollary 1, P ALEθ is Paradox-free when θ ∈ (0, 1]. Yet, the comparison of two societies with P ALEθ 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 P ALEθ , 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-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. 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-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 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 P ALE1 at A. For LE (A)(1−H (A)) society A and P ALE1 , this marginal rate of substitution is given by (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.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 Ex- pectancy (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, P ALEθ 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 P ALEθ is robust but 14 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 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 ) = P ALE1 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 P ALEθ comparisons are robust. Proposition 5. (Robust comparisons with P ALEθ ) (i) For any two societies A and B, P ALEθ (A) ≤ P ALEθ (B ) for all θ ≤ 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 ≤ 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. ˆ are In Appendix K, we study the conditions under which comparisons by EDθa 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).23 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. Fig- ure 4 reports the evolution of life expectancy, poverty rate and P ALE1 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 ? P ALE1 indicates that deprivation is higher in 2007 than in was in 1990. Indeed, P ALE1 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 P ALEθ to the choice of θ. Figure 4: South Africa Evolution of P ALE1 and Life Expectancy, 1990-2019 80 70 70 60 60 50 50 40 Years 40 % 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 Headcount ratio (right axis) PALE Life Expectancy 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 P ALEθ between 1990 and 2019. Throughout this period, life expectancy 23 See 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 P ALE and Life Expectancy, 1990-2019 70 60 70 60 60 50 60 50 50 50 40 40 40 40 Years Years 30 % 30 % 30 30 20 20 20 20 10 10 10 10 0 0 0 0 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 Year Year Life Expectancy PALE Life Expectancy PALE Headcount ratio (right axis) Headcount ratio (right axis) (a) P ALE1 (b) P ALE0.5 Reading: in 1990, Poverty-Adjusted Life Expectancy was about 30 years according to P ALE1 and 48 years according to P ALE0.5 . increased from 62 to 71 but the decrease in poverty expectancy is even more spec- tacular, 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 P ALE1 , from 32 in 1990 to 64 years in 2019. For θ < 1, the corresponding P ALEθ curves all lie between life expectancy and the P ALE1 curve. We show P ALE0.5 in the right hand panel of Figure 5. By construction, P ALE0.5 is higher in absolute value (47 years in 1990). However, its evolution is much slower than that of P ALE1 : from 1990 to 2019, P ALE0.5 increased by 44% as opposed to 101% for P ALE1 . 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, P ALE0.5 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 P ALEθ can be decomposed as follows: ∂P ALE ∂LE ∂H = ϵLE ∗ + ϵH ∗ ∂t ∂t ∂t −θH where ϵLE = 1 and ϵH = (1−θH ) represent the elasticities of P ALEθ to life ex- pectancy and poverty, respectively. Figure 6 shows the share of the growth of P ALEθ 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 P ALEθ . 17 Figure 6: Share of the growth of LE in the growth of P ALE1 and P ALE0.5 , 1990- 2019 80 70 60 40 50 % 30 20 10 0 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 Year LE growth contribution, θ = 1 LE growth contribution, θ = 0.5 Reading: in 1991, the growth of life expectancy contributed to 10% of the growth of P ALE1 and to 25% to that of P ALE0.5 . ˆ index, Figure 7 reports the Turning now to deprivation as captured by the EDθa evolution of ED1,70 in the world during the same period.24 The evolution of ED1,70 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 P ALEθ 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 P ALEθ is robust to the choice of θ, this implies that P ALE0 and P ALE1 yield the same ranking. Since P ALE0 corresponds to life expectancy, a measure solely based on the headcount provides a 24 We 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 and H, 1990-2019 (where θ = 1 and a Figure 7: Evolution of EDθa ˆ = 70). 60 50 50 40 40 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 Income Deprivation Lifespan Deprivation (70) Headcount ratio ED (70) Lifespan Deprivation share in ED (right axis) 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 P ALEθ 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.25 The share of these ambiguous cases for which P ALEθ 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.26 25 These are the only situations in which PALE can offer a different ranking than the headcount. 26 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. 19 Figure 8: 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. 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 P ALEθ . Over the period, the share of ambiguous trajectories oscillates between 20 and 40% of all cases. The share of these cases that P ALEθ 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, P ALEθ 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 ˆ, An important limitation of the two indices proposed in this paper, P ALEθ and EDθa 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 80 60 % 40 20 0 1995 2000 2005 2010 2015 2020 Year Share of ambiguous comparisons Minimal share of ambiguous comparisons solved with PALE 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 ˆ can be used out of poverty. When such data is not available, P ALEθ and EDθa 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 so- cieties in which premature mortality is highly selective, affecting disproportionately ˆ essentially poorer individuals. In particular, the premature mortality term of EDθa 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.27 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 27 Note 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. 21 values are highly sensitive to the parametric values selected for the period utility function. Survey-based estimates for θ may provide a firmer base for narrowing the plausible range of values for this central parameter. References Baland, J.-M., Cassan, G., and Decerf, B. (2021). “too young to die”: Depriva- tion measures combining poverty and premature mortality. American Economic Journal: Applied Economics, 13(4):226–257. 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Population health metrics, 9(1):1–11. Jia, H., Zack, M. M., and Thompson, W. W. (2011). State quality-adjusted life expectancy for us adults from 1993 to 2008. Quality of Life Research, 20(6):853– 863. Jones, C. I. and Klenow, P. J. (2016). Beyond GDP? Welfare across countries and time. American Economic Review, 106(9):2426–2457. Kanbur, R. and Mukherjee, D. (2007). Premature mortality and poverty measure- ment. Bulletin of economic research, (September):339–359. Koopmans, T. C. (1960). Stationary ordinal utility and impatience. Econometrica: Journal of the Econometric Society, pages 287–309. Lefebvre, M., Pestieau, P., and Ponthiere, G. (2013). Measuring poverty without the Mortality Paradox. Social Choice and Welfare, 40(1):285–316. Riumallo-Herl, C., Canning, D., and Salomon, J. A. (2018). Measuring health and economic wellbeing in the sustainable development goals era: development of a poverty-free life expectancy metric and estimates for 90 countries. The Lancet Global Health, 6(8):e843–e858. 22 Sen, A. (1976). Poverty: an ordinal approach to measurement. Econometrica: Jour- nal of the Econometric Society, pages 219–231. Sullivan, D. F. (1971). A single index of mortality and morbidity. HSMHA health reports, 86(4):347. Watkins, K. (2006). Human Development Report 2006 - Beyond scarcity: Power, poverty and the global water crisis, volume 28. Whitehead, S. J. and Ali, S. (2010). Health outcomes in economic evaluation: the qaly and utilities. British medical bulletin, 96(1):5–21. World Bank (2023). Poverty and inequality platform. 23 Appendices A Notation and definition of stationary society We present here the formal notation used for the proofs. There is a discrete set of periods {. . . , t − 1, t, t + 1, . . . }. In each period, some individuals are born (at the beginning of the period) 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. 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 that an individual born in t survives to age a is given nt ( a ) by Vt (a) = nt , and the conditional probability that an individual born in t will be p t (a) poor when reaching age a is πt (a) = nt ( a ) . We denote the distribution on the set of lives that Ct implicitly defines 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 period t 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 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 (by definition of a ). The particularity of stationary societies is to have their natality, mortality and poverty constant over time, so that 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. 24 Definition 2 (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}.28 B Proof of Proposition 1 The proof is based on Lemma 1, which shows that, in a stationarity society, the poverty and mortality observed in a given period completely reflects the life profile of newborns. Lemma 1. If society St is stationary, then −1 k=0 (1 − µk ) Vt ( a ) = Π a t for all a ∈ {0, . . . , a∗ − 1}, (6) Nt = nt ∗ LEt , (7) a∗ −1 N t ∗ Ht = nt ∗ V ( a) π ( a) . (8) a=0 Proof. We first prove Eq (6). 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, we have for all a ∈ {1, . . . , a∗ − 1} that nt ( a ) Vt ( a ) = , nt −1 nt (k + 1) = Πa k=0 , nt ( k ) −1 nt−k (k + 1) = Πa k=0 , nt − k ( k ) −1 k=0 (1 − µk ). = Πa t We then prove Eq (7). As St is stationary, we have nt (a) = nt−a (a) for all a ∈ {1, . . . , a∗ − 1}. Recalling Eq (6) and Vt (a) = nt ( a ) nt , we can successively write a∗ −1 −1 k=0 (1 − µk ), Πa t LEt = a=0 a∗ −1 = Vt ( a ) , a=0 a∗ − 1 a=0 nt (a) = , nt a∗ − 1 a=0 nt − a ( a ) = , nt = Nt /nt . 28 Clearly, a constant distribution of lives is not sufficient for these equalities, one also needs a constant size of cohorts. 25 Finally, we prove Eq. (8). As St is stationary, we have pt (a) = pt−a (a) for all a ∈ {1, . . . , a∗ − 1}. Given that πt (a) = p t (a ) nt ( a ) and Vt (a) = nt ( 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 ) V t ( a ) n t = . Nt We use Lemma 1 to prove Proposition 1. The assumption that individuals only enjoy binary consumption statuses implies that Eu(ca ) = π (a)uP + (1 − π (a))uN P where uN P = u(N P ) and uP = u(P ). By a∗ −1 Eq. (6), life expectancy at birth can be written as LE = a=0 V (a). We can thus rewrite Eq. (4) as a∗ − 1 EU = uN P LE − (uN P − uP ) V ( a) π ( a) . (9) a=0 The result follows directly when substituting Eq. (7) and (8) into Eq. (9). C Calibrating values for θ uN P − uP Proposition 1 shows that θ = uN P , where uP and uN P respectively denote the Bernouilli utility of being poor and being non-poor (and the utility of being dead is normalized to zero). Consider the constant elasticity of substitution Bernouilli utility function defined as ˆ1−ϵ c 1− ϵ − c u( c ) = , (10) 1−ϵ ˆ denotes the subsistence consumption, for which u(ˆ where c c) = 0, and ϵ is the coefficient of relative risk aversion that captures the curvature of utility function u. A parametric value for θ requires defining representative consumption for the (consumption) poor and non-poor statuses such that uP = u(cpoor ) and uN P = u(cnon−poor ). Typically, cpoor and cnon−poor could respectively be defined as mean or median consumption among the poor and non-poor. ˆ Parametric values for θ are sensitive to the values selected for the parameters c and ϵ. We illustrate this by providing values for 1/θ for India in 2019 for different values of these parameters. We define poverty using the the International Poverty Line, whose value is $ 2.15 per person per day (2017 PPPs). We assume that cpoor and cnon−poor are defined as mean consumption among the poor and non-poor, which we extract from the Poverty and Inequality Platform from the World Bank. D Proof of Proposition 2 The proof builds on the framework presented in Appendix A. 26 Table 3: Parametric values for 1/θ for India in 2019 using the International Poverty Line. ˆ c ˆ c ˆ c ˆ c 0.5 0.75 1.0 1.25 ($ a day) ($ a day) ($ a day) ($ a day) ϵ 2.5 7.8 4.1 2.6 1.8 ϵ 2.0 4.7 3.0 2.1 1.6 ϵ 1.5 3.0 2.2 1.7 1.4 ϵ 1.0 2.1 1.7 1.5 1.3 Note: According to the Poverty and Inequality Platform, mean consumption in India in 2019 was $ 5.13 per person per day. For the International Poverty Line, mean consumption among the poor is $ 1.75 and mean consumption among the non-poor is $ 5.51. The utility function considered is CES. 1/θ can be interpreted as the number of years spent in poverty yielding the same well-being loss as one year of life lost. By Proposition 2 in Baland et al. (2021) we have for any stationary society St that ˆ (St ) = IDθ a GDθa ˆ (St ) = EDθ a ˆ (St ). Hence, we only need to prove that GDθ a ˆ (St ) for any stationary society St . By definition, we have that Nt = #P +#N P , Nt Ht = #P and nt−a (a) = Na , i.e., Y LLt N t Ht ˆ = GDθa + θ , Nt + Y LLt Nt + Y LLt where ˆ −2 a Y LLt = a ∗ (ˆ nt − a ( a ) ∗ µ t a − (a + 1)). a=0 As society St is stationary, Lemma 1 applies and Nt = nt LEt (Eq. (7)). Substi- ˆ proves our result, provided tuting this expression for Nt into the definition of GDθa ˆ , which remains to be shown. As society St is stationary, Lemma 1 Y LLt = nt LGEa nt −a ( a ) a−1 applies and we have nt = k=0 (1 − µt k ) (Eq. (6)). Substituting this expression for nt−a (a) into the definition of Y LLt gives: ˆ −2 a a−1 Y LLt = nt a∗ a − (a + 1)) ∗ µt (ˆ (1 − µt k ), a=0 k=0 which shows that Y Lt = nt LGEa ˆ − 1), the ˆ − (a + 1) = 0 when a = a ˆ (recall that a desired result. E ˆ and GDθa EDθa ˆ under a transitory shock ˆ and GDθ a We illustrate the 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 27 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 10. / {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 10: Response of GDθa ˆ to a permanent mortality shock in t = 0. ˆ and EDθ a The years prematurely lost are shaded. ˆ . In period 0, the actual population pyramid is not stationary Consider first GDθa 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, ˆ takes value 7/11. In period 1, the population pyramid is stationary, and and GDθa ˆ is equal to 5/8 from then on. GDθa ˆ . Even if the actual population pyramid is not stationary in We now turn to EDθa ˆ is immediately equal to 5/8 since it records premature mortality as if period 0, EDθa 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). F ˆ index Characterization of the EDθa 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.29 29 All newborns have age 0 during period t and some among these newborns may die at the end 28 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 ai < aˆ. A distribution x = (x1 , . . . , xn(x) ) specifies the age and the status in period 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: ˆ > t − b i }. X = {x ∈ ∪n∈N (Z × S )n | there is i for whom either si ̸= D or si = D and a 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 of prematurely dead individuals in distribution x, which is the number of individuals ˆ > t − bi , p(x) the number of individuals who are poor and i for whom si = D and a ˆ index is defined as: f (x) the number of alive and non-poor individuals. The IDθa d( x) p( x ) ˆ ( x) = IDθa +θ , (11) 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 individuals 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 nota- tion 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 pe- riod 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 deprivation 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 of period t. This implies that bi = t ⇒ si ̸= D . 29 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 ) + d a ( 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 index is a function P : O × N → R+ . We simplify the notation P (x, µ, a ˆ) to P (x, µ) ˆ is assumed. as a fixed value for a 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 30 ˆ in the case of a stationary pair: rates is equivalent to IDθa Deprivation axiom 1 (IDθa ˆ Equivalence). ˆ≥a There exists some θ ∈ (0, 1] and a ˆ such that for all (x, µ) ∈ O that are stationary we have P (x, µ) = IDθa ˆ ( x) . 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 = s′ ′ i , 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). ˆ index. Proposition 6 shows that these properties jointly characterize the EDθa ˆ ). Proposition 6 (Characterization of EDθa ˆ if and only if P satisfies Independence of Dead, IDθ a P = EDθa ˆ Equivalence, Replication Invariance and Independence of Birth Year. 30 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. 30 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 2) 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 ) = ′′′ ˆ (x, µ). Given that (x , µ) is stationary, we have by IDθ a EDθa ˆ Equivalence that P (x′′′ , µ) = IDθa ′′′ ˆ (x , µ) for some θ ∈ (0, 1]. As IDθ a ˆ = EDθ a ˆ = GDθ a ˆ 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. 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 = c na (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 a∗ −1 a−1 a∗ − 1 n′ ′ n′ cj j =0 (1 − ej ) and n = ca 31 such that µa = ea . Let e = j =0 ej , a = e j =0 j. a∗ − 1 Let x′ be a n′ −replication of x. Letting nx = j =0 n j ( x ) be the number of alive ′ ′ individuals in distribution x, we have that x has n ∗ n alive individuals. We have x 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 . 32 k=0 j =0 (1− e ) j 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 31 These numbers imply that a constant natality of e newborns leads to a stationary population of n′ alive individuals. ∑a ∗ − 1 ∏k − 1 cj n′ ∗nx j =0 (1 − e ) = LE , implying that e = ∑a∗ −1 ∏k−1 32 Observe that k=0 cj . j =0 (1− ej j ) k=0 31 G Proof of Proposition 3 The proof builds on the framework presented in Appendix A. We first show that LE + LGEa ˆ = a ˆ ≥ a∗ . By definition, LE and ˆ when a t ˆ only depend on the age-specific mortality vector µ . Thus, the values for LE LGEa ˆ do not depend on whether the society is stationary or not. Consider and LGEa any stationary society St whose constant mortality vector is µt . We show for this stationary society St that LE + LGEa ˆ =a ˆ ≥ a∗ . ˆ when a As society St is stationary, Lemma 1 applies and we have Nt = nt ∗ LEt (Eq. a∗ −1 (7)). As by definition Nt = a=0 nt (a), we get a∗ −1 nt ( a ) LE = . (12) a=0 nt a− 1 =0 (1 − µk ) (Eq. t As society St is stationary, Lemma 1 applies and we have Vt (a) = Πk nt−a (a)−nt−a (a+1) (6)). Using the definition of age-specific mortality rate, namely µt a = nt −a ( a ) , ˆ as we can rewrite LGEa ˆ −1 a nt−a (a) − nt−a (a + 1) ˆ ( St ) = LGEa a − (a + 1)) ∗ (ˆ ∗ Vt ( a ) . a=0 nt − a ( a ) 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 successively write ˆ −1 a nt (a) − nt (a + 1) nt (a) ˆ = LGEa a − (a + 1)) ∗ (ˆ ∗ , a=0 nt ( a ) nt ˆ −1 a ˆ −1 a nt (a) − nt (a + 1) nt (a) − nt (a + 1) = ˆ∗ a − (a + 1) ∗ , a=0 nt a=0 nt ˆ −1 a 1 = a)) − ˆ ∗ (nt (0) − nt (ˆ 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 that 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 ) a=0 nt = a=0 nt . We have shown that LGEa ˆ− ˆ = a a=0 nt , which together with Eq. (12) proves that LE + LGEa ˆ =a ˆ ≥ a∗ . ˆ when a The fact that LE + LGEa ˆ =a ˆ(1 − EDθa ˆ implies that P ALE θ = a ˆ ) because ˆ(1 − EDθa a ˆ )(1 − ED θ a ˆ ) = (LE + LGEa ˆ) = LE (1 − θH ), = P ALE θ . Thus, when aˆ ≥ a∗ , P ALEθ is a linear function of EDθa ˆ that depends negatively on EDθa ˆ . Therefore, these two indicators yields opposite ranking of any two societies ˆ (A) ≤ EDθ a A and B , i.e. P ALE θ (A) ≥ P ALE θ (B ) ⇔ EDθa ˆ (B ). 32 H ˆ and Mortality shocks and the evolution of EDθa P ALEθ ˆ , assum- We briefly contrast the impact of mortality shocks on P ALEθ and EDθa ing that these mortality shocks are independent of the poverty status. Consider a ˆ while 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 ˆ , since LGEa shock reduces EDθa ˆ is thereby reduced. It is indeed easy to check 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 ˆ , decreases: EDθa ∂LE ˆ < 0, for θH < 1. Moreover, P ALEθ is more sensitive to 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 ˆ . Again, P ALEθ improves and EDθ a and reduces LGEa ˆ decreases since both LE increases and LGE decreases. I Proof of Proposition 4 The proof builds on the framework presented in Appendix A. ˆ is Paradox-free if and only if IDθ a By Proposition 2, Mθa ˆ is Paradox-free. ˆ ≥ a∗ . The proof is ˆ is Paradox-free only if θ ≤ 1 and a First, we prove that IDθa by contradiction. ˆ ≥ 2. Consider two alternative stationary societies Assume first that θ > 1 and a A and B that both feature only one newborn i every year. The life of i is respectively A B li = (P, D) and li = (P, P, D ). Society B is obtained from A by a lifespan increment a−1)+θ (ˆ a−2)+2θ (ˆ ˆ ( A) = to the poor person i. However, we have IDθa ˆ a ˆ (B ) = and IDθa ˆ a , ˆ (A) < IDθ a which yields IDθa ˆ is not Paradox-free. ˆ (B ), which shows that IDθ a ˆ < a∗ . Consider two alternative stationary societies Assume then that θ > 0 and a A’ and B’ that both feature two newborns i and j every year. Their lives are ′ ′ A B respectively li = (P, . . . , P, D ) and li ˆ = (P, . . . , P, P, D ), where i’s lifespan is a ′ ′ A B ˆ + 1 years in society B’, while lj years in society A’ and a = lj = (N P, . . . , N P, D ), ˆ years in both societies. Society B’ is obtained from A’ by where j ’s lifespan is a ′ a) θ (ˆ a lifespan increment to the poor person i. However, we have IDθa ˆ (A ) = a ˆ+ˆ a and ′ a+1) θ (ˆ ′ ′ ˆ (B ) = IDθa a ˆ+ˆ a+1 , ˆ (A ) < IDθ a which yields IDθa ˆ is ˆ (B ), which shows that IDθ a not Paradox-free. ˆ ≥ a∗ . ˆ is Paradox-free only if θ ≤ 1 and a We have thus proven that IDθa ˆ ≥ a∗ . Take any ˆ is Paradox-free if θ ≤ 1 and a Second, we prove that IDθa two stationary societies A” and B” such that B” is obtained from A” by a lifespan increment to the poor person i. Poor person i dies prematurely in society A” because ˆ ≥ a∗ . The difference between societies A” and B” is thus that i spends an additional a year in poverty in society B”, instead of prematurely loosing that year in society A”. Thus, the following two equalities hold #P (A′′ ) + #N P (A′′ ) + #P D(A′′ ) = 33 #P (B ′′ )+#N P (B ′′ )+#P D(B ′′ ) and #P D(A′′ ) − #P D(B ′′ ) = #P (B ′′ ) − P (A′′ ) = ′′ ′′ ′′ ˆ (A ) ≥ IDθ a 1. The former equality implies that IDθa ˆ (B ) if and only if #P D (A )+ θ#P (A′′ ) ≥ #P D(B ′′ ) + θ#P (B ′′ ). This inequality is equivalent to #P D(A′′ ) − #P D(B ′′ ) ≥ θ(#P (B ′′ ) − #P (A′′ )), which further simplifies to 1 ≥ θ given the ˆ is Paradox-free. latter equality, which proves that IDθa J Proof of Proposition 5 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 ∗ (LE (B )H (B ) − LE (A)H (A)), (13) 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) ≥ LE (B )H (B ) − LE (A)H (A). It immediately follows that inequality (13) is verified for all θ ∈ (0, 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 . K ˆ comparisons Robust EDθa ˆ ≥ a∗ . However, EDθa ˆ is Paradox-free when θ ∈ (0, 1] and a EDθa ˆ no longer encap- ˆ ≥ a∗ . To ease the impossibility between paradox- sulates the minimal view when a freeness and the minimal view, we define a weaker notion of paradox-freeness. A deprivation index is minimally paradox free when the index does not record a wors- ening for increments to the lifespan of a poor person who dies prematurely, i.e., ˆ. In that case, EDθa whose lifespan is smaller than a ˆ is minimally paradox free when θ ∈ (0, 1]. ˆ ∈ N0 must respect a lower-bound a We assume that the age threshold a ˆ ∈ N0 , ˆ ≥ 0. Clearly, the value for the lower bound a ˆ ≥ a such that a ˆ influences the set ˆ. Proposition 7 of comparisons that are robust to the values selected for θ and a ˆ is robust for all θ ∈ (0, 1] provides the conditions under which the ranking by EDθa ˆ≥a and all a ˆ. ˆ ). Proposition 7 (Robust comparisons with EDθa ˆ (B ) for all θ ≤ 1 and ˆ (A) ≥ EDθ a (i) For any two societies A and B we have EDθa 34 ˆ≥a all a ˆ if and only if a (A) ≥ ED0ˆ ED0ˆ ˆ≥a a (B ) for all a ˆ, and a (A) ≥ ED1ˆ ED1ˆ ˆ≥a a (B ) for all a ˆ (generalized Condition C1) ˆ (A) ≥ EDθ a ˆ ≥ 2, there exist societies A and B for which EDθa (ii) For any a ˆ (B ) for ˆ≥a all θ ≤ 1 and all a ˆ even though H (A) < H (B ). These societies are such that LE (A) < LE (B ). Proof. See Appendix L for the straightforward proof. We illustrate Proposition 7 in Figure 11.33 The vertical axis represents the share of pairs of societies for which H and LE provide identical (at the top) or opposite rankings (at the bottom). By definition, rankings by H and LE are insensitive to the ˆ considered. The horizontal axis represents all possible values of a age threshold a ˆ, the lower bound on the age threshold. ˆ provides robust rank- The left panel describes the share of pairs for which EDθa ˆ. Lower values of a ings as a function of a ˆ imply a fall in the share of cases that EDθa ˆ ˆ over which EDθa can rank robustly. Indeed, a larger age interval of values of a ˆ has to be computed implies a larger number of comparisons for ED. As a result, the num- ber of pairs for which it can provide the same ranking for all age thresholds falls.34 ˆ provides the same ranking as Second, if H and LE provide the same ranking, EDθa H when aˆ = a∗ . Finally, as discussed above, when H and LE disagree, a larger value ˆ implies that the share of cases for which H provides an robustly wrong ranking of a gets larger. ˆ, the share of pairs of societies for which The right panel reports, for all values of a ˆ provide robust rankings. Since P ALEθ does not depend on the P ALEθ and EDθa age threshold, it is able to rank a larger set of comparisons. As shown in Proposition ˆ = a∗ , the two indices are equivalent. 3, when a L Proof of Proposition 7 ˆ≥a We first prove the following: for any a ˆ and any two societies A and B, we have ˆ (B ) for all θ ∈ (0, 1] if and only if ˆ (A) ≥ EDθ a EDθa a (A) ≥ ED0ˆ ED0ˆ a (A) ≥ ED1ˆ a (B ) and ED1ˆ a (B ). a ( A) < We start with the “only if” part. Assume on the contrary that ED0ˆ a (B ) or ED1ˆ ED0ˆ a (B ). This implies that EDθ a a (A) < ED1ˆ ˆ (B ) for ˆ (A) < EDθ a ˆ (A) ≥ EDθ a some θ ∈ (0, 1] and therefore we cannot have EDθa ˆ (B ) for all θ ∈ (0, 1]. 33 All graphs that follow are constructed using a lower bound on a ˆ equal to 1. Indeed, for θ = 0 and aˆ = 0, EDθa ˆ is equal to zero for all societies and cannot therefore deliver robust comparisons. 34 It is not a sufficient condition that the rankings by H and LE are identical for the ranking by EDθa ˆ to be robust. The reason is that, when a ˆ < a∗ , LE no longer contains all the relevant information on mortality: for instance, two societies can share the same life expectancy at birth but one with several deaths occurring below a ˆ while the other has all deaths occurring above aˆ. 35 ˆ comparisons as a function of a Figure 11: Share of robust EDθa ˆ. % pairs of societies A and B % pairs of societies A and B 100 100 * * same ranking * * with H & LE * * * * opposite rankings with H & LE 0 * * ˆ a * * ˆ a 0 1 a∗ 1 a∗ ˆ rank robustly PALEθ & EDθa * ˆ ranks robustly EDθa * ˆ do not rank robustly PALEθ & EDθa ˆ does not rank robustly EDθa only PALEθ ranks robustly Reading: Left: The smaller the lower-bound a ˆ, the lower the share of societies pairs robustly ranked by EDθa ˆ . Right: The higher the lower-bound aˆ, the higher the share of societies pairs robustly ranked by both EDθa ˆ and P ALEθ . 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 ) + LGEa ˆ (B ) LE (B ) ∗ H (B ) LE (A) ∗ H (A) θ − for all θ ∈ (0, 1]. LE (B ) + LGEa ˆ (B ) LE (A) + LGEaˆ ( A) (14) a (A) ≥ ED0ˆ As ED0ˆ a (B ), the left hand side of Eq. (14) is non-negative. As a (A) ≥ ED1ˆ ED1ˆ a (B ), Eq. (14) holds for θ = 1. As a result, inequality (14) holds for all θ ∈ (0, 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 − 1. Therefore, society A is ˆ ( A) = a such that ˆ −1 a ( A) = 1 − ˆ≥a a 0 .6 a ( A) = • ED0ˆ ˆ a and ED1ˆ ˆ a for all a ˆ. Society B is such that H (B ) = 0.5 and all its individuals die at the maximal age a∗ − 1, which implies that LE (B ) = a∗ and • LGEa ˆ ∈ {2, . . . , a∗ }, ˆ (B ) = 0 if a • LGEa ˆ > a∗ . ˆ − a∗ if a ˆ (B ) = a Therefore, society B is such that • ED0ˆ a (B ) = 0 and ED1ˆ ˆ ∈ {2, . . . , a∗ }, a (B ) = 0.5 for all a 36 ˆ −a∗ 0 .5 a ∗ • ED0ˆ a (B ) = a aˆ a (B ) = 1 − and ED1ˆ a ˆ ˆ > a∗ . for all a ˆ (B ) for all θ ∈ (0, 1] and all a ˆ (A) ≥ EDθ a By statement (i), we get EDθa ˆ ≥ a ˆ if a (A) ≥ ED0ˆ we have ED0ˆ a (A) ≥ ED1ˆ a (B ) and ED1ˆ ˆ≥a a (B ) for all a ˆ. Recalling a (A) ≥ ED0ˆ ˆ ≥ 2, one can then easily check that we have ED0ˆ that a a (B ) and a (A) ≥ ED1ˆ ED1ˆ ˆ > a∗ . ˆ ∈ {2, . . . , a∗ } and for all a a (B ) both for all a M Descriptive statistics Table 4 lists all the countries present in our data set as well as the 2019 values of the main variables of interest. Table 4: Countries used in the dataset and descriptive statistcs Country H LE LGE70 LE*H P ALE1 ED1,70 % Years Years Years Years % Albania 0 77 3 0 77 4 Algeria 0 75 4 0 75 6 Angola 32 64 11 21 43 42 Armenia 1 75 4 1 74 6 Azerbaijan 0 70 6 0 70 8 Bangladesh 11 74 6 8 65 18 Belarus 0 73 5 0 73 6 Belize 18 73 6 13 60 24 Benin 19 63 12 12 51 32 Bhutan 0 72 6 0 72 8 Bolivia 2 71 6 1 70 10 BosniaandHerzegovina 0 76 3 0 76 4 Botswana 13 61 13 8 53 28 Brazil 5 75 5 4 71 12 Bulgaria 1 72 5 1 72 7 BurkinaFaso 31 61 14 19 42 44 Burundi 72 63 12 45 18 76 CaboVerde 3 73 5 2 71 9 Cameroon 23 62 13 14 48 36 CentralAfricanRepublic 65 51 21 33 18 75 Chad 31 59 15 18 41 45 China 0 77 3 0 77 4 Colombia 5 79 4 4 75 10 Comoros 18 68 9 12 56 27 CostaRica 1 79 3 1 78 5 CotedIvoire 11 63 12 7 56 25 Djibouti 18 66 10 12 54 29 DominicanRepublic 1 72 7 1 72 9 Ecuador 4 75 5 3 73 9 ElSalvador 1 75 6 1 74 8 Eswatini 34 57 16 19 38 48 Ethiopia 18 68 9 12 56 27 37 ...continued from previous page Country H LE LGE70 LE*H P ALE1 ED1,70 % Years Years Years Years % Fiji 1 68 8 1 67 11 Gabon 2 67 9 2 65 14 GambiaThe 15 66 10 10 56 26 Georgia 5 72 5 4 69 11 Ghana 22 65 10 14 51 33 Guatemala 7 72 7 5 67 15 Guinea 13 60 14 8 52 30 GuineaBissau 21 60 14 13 47 36 Guyana 6 66 9 4 62 18 Haiti 25 63 12 16 47 37 Honduras 13 71 6 9 62 19 India 12 70 7 8 62 20 Indonesia 4 71 6 3 67 12 Iraq 0 72 5 0 72 7 Jamaica 1 75 5 1 74 7 Jordan 0 77 3 0 77 4 Kazakhstan 0 71 6 0 71 8 Kenya 32 66 10 21 45 41 Kiribati 2 60 13 1 59 19 KyrgyzRepublic 1 73 5 0 72 8 Lebanon 0 76 4 0 76 5 Lesotho 34 51 21 18 33 54 Liberia 31 65 11 20 45 40 Madagascar 79 65 11 51 14 82 Malawi 69 64 12 44 20 74 Malaysia 0 74 4 0 74 5 Maldives 0 78 3 0 78 4 Mali 15 61 15 9 52 31 MarshallIslands 1 65 10 1 64 14 Mauritania 5 70 8 4 66 14 Mauritius 0 74 5 0 74 6 Mexico 3 75 5 2 73 9 Moldova 0 73 5 0 73 7 Mongolia 1 67 8 0 67 11 Montenegro 3 75 3 2 73 7 Morocco 1 72 5 1 72 8 Mozambique 71 57 16 41 17 78 Myanmar 1 68 8 1 68 12 Namibia 17 64 11 11 53 29 Nepal 3 70 7 2 68 11 Nicaragua 4 74 4 3 71 9 Niger 50 61 14 31 31 59 Nigeria 31 63 13 20 44 43 38 ...continued from previous page Country H LE LGE70 LE*H P ALE1 ED1,70 % Years Years Years Years % NorthMacedonia 3 74 4 2 72 7 Pakistan 5 65 11 3 62 19 PapuaNewGuinea 31 64 11 19 44 41 Paraguay 1 75 5 1 75 7 Peru 3 79 4 2 77 7 Philippines 5 71 7 3 68 13 Romania 2 75 4 2 73 7 RussianFederation 0 72 6 0 72 8 Rwanda 44 68 9 30 38 50 Samoa 1 70 7 1 69 10 SaoTomeandPrincipe 15 70 6 10 60 22 Senegal 9 67 9 6 61 20 Serbia 0 75 3 0 75 4 SierraLeone 25 61 14 15 46 39 SolomonIslands 25 58 14 14 44 39 SouthAfrica 21 64 12 13 51 33 SriLanka 1 76 4 1 76 6 StLucia 5 74 5 4 71 11 Sudan 23 69 8 16 53 31 SyrianArabRepublic 69 73 5 50 23 71 TaiwanChina 0 79 3 0 79 4 Tajikistan 4 69 7 3 66 12 Tanzania 43 66 10 29 38 51 Thailand 0 77 4 0 77 6 Togo 28 64 11 18 46 39 Tonga 1 72 6 1 71 9 Tunisia 0 77 3 0 77 4 Turkmenistan 1 70 7 1 69 10 Tuvalu 0 67 8 0 67 11 Uganda 42 65 11 27 38 50 Ukraine 0 69 7 0 69 10 UnitedArabEmirates 0 73 5 0 73 6 Uzbekistan 28 68 7 19 49 35 Vanuatu 9 65 10 6 59 21 Vietnam 1 74 5 1 73 7 Zambia 61 62 12 38 24 67 Zimbabwe 40 60 14 24 36 51 N Building PALE and ED in practice How should a practitionner build our different indices with available data ? Table 5 presents the different step required to build our PALE and ED indices, the data source as well as their 2019 value. 39 Table 5: PALE and Expected Deprivation in the developing world in 1990 and ˆ = 70. 2019, with a Unit 1990 2019 Computation Value Value Life Expectancy (LE) Years 62.2 71.0 Source: GBD (2019) Poverty Headcount (H) % 48.8 9.8 Source: Poverty and Inequality Platform LE*H Years 30.4 7.0 LE* H P ALE1 Years 31.9 64.0 LE-θH*LE Life Gap Expectancy70 Years 13.0 7.2 See Section 3.2 LGEa LE ∗H Expected % 57.6 18.1 ˆ LE +LGEaˆ + θ LE +LGEaˆ Deprivation1,70 O Ambiguous countries’ trajectories In Figure 12, 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 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 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. 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) compared to the situation prevailing in the same country five years earlier.35 35 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. 40 Figure 12: Resolution of ambiguous countries’ trajectories, 1990-2019 30 Non Poverty Headcount Growth −15 0−30 15 −8 −4 0 4 8 Life Expectancy Growth Unambiguous cases Unsolved ambiguous cases Solved ambiguous cases zero growth−PALE Reading: Each dot represents a country-year. Countries located in the southwest (northeast) quadrant are worse (better) off than they were 5 years earlier. Countries’ evolution located in the other quadrants can not be unambiguously assessed with a dashboard approach. Countries’ trajectories located between the zero growth-PALE curve and the zero non poverty headcount growth line can be unambiguously assessed with PALE. Note: for readibility, the graph only shows the points situated between a growth rate of +/- 30% in non poverty headcount and of +/-8% in life expectancy. These are 85% of all observations. 41