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Premature mortality and poverty measurement in an OLG economy

Abstract

Following Kanbur and Mukherjee (Bull Econ Res 59(4):339–359 2007), a solution to the “missing poor” problem (i.e., selection bias in poverty measures due to income-differentiated mortality) consists in computing hypothetical poverty rates while assigning a fictitious income to the prematurely dead. However, in a dynamic general equilibrium economy, doing “as if” the prematurely dead were still alive is likely to affect wages, output and capital accumulation, with an uncertain effect on poverty. We develop a three-period OLG model with income-differentiated mortality and compare actual poverty rates with hypothetical poverty rates that would have prevailed if everyone faced the survival conditions of the top income class. Including the prematurely dead has an ambiguous impact on poverty, since it affects income distribution through capital dilution, composition effects, and horizon effects. Our results are illustrated by quantifying the impact of income-differentiated mortality on poverty measures for France (1820–2010).

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Notes

  1. On this correlation, see Duleep (1986), Backlund et al. (1999), Deaton and Paxson (1998), Deaton (2003), Salm (2007) and Belloni et al. (2013). Note that the direction of causality has been much studied in the recent years, some studies finding no causal link from income to mortality (see Snyder and Evans 2006; Evans and Moore 2011; Ahammer et al. 2015).

  2. The selection bias arises because of the income/mortality correlation, independently from any causality issue. To see this, suppose that an exogenous bus strike arises on the day of a classroom test, which prevents some students from attending the test. The average mark at the test (computed on students attending) suffers from a selection bias as soon as there is a correlation between potential marks and being victim of the bus strike, implying that the average mark would have been different provided no bus strike took place. This is true even if there is no causal link between being subject to a bus strike and potential marks.

  3. That case occurs in reality. Lefebvre et al. (2017) show that, although poverty beyond age 60 is higher in Portugal than in Estonia, once we correct for the missing poor problem (by computing the poverty rates that would have prevailed if all income groups had faced the survival conditions of the richest), old-age poverty is higher in Estonia than in Portugal.

  4. Lefebvre et al. (2013) propose to take as a fictitious income the income equivalent to death, i.e. the income that would make a person indifferent between, on the one hand, remaining alive with that income level, and, on the other hand, dying.

  5. Note that we cannot make the same claim for measures of relative poverty.

  6. See, on the empirical side, studies by Carter et al. (2007), Adhvaryu and Beegle (2012) and Ardington et al. (2014).

  7. Another difference lies in the fact that Chakraborty (2004) considers the possibility of endogenous survival conditions (depending on public health spendings proportional to the wage), whereas we keep here survival conditions as exogenous (but also related to income).

  8. Our reliance on a three-class model is made for analytical tractability. Section 6 develops a numerical example with 10 income classes.

  9. This assumption amounts to assume that, independently from the survival conditions faced by the members of the previous cohort, we know for sure that, in the following cohort, the same proportions of the three types will prevail in terms of innate human capital. To see this, take the extreme case where all low productivity individuals die before becoming adults and parents. We then suppose that, among the children of two remaining types, there will be the same proportion of high and low productivity as in the previous cohort.

  10. Note that, while assuming a perfect rank correlation between innate human capital and survival probabilities consists of a reasonable assumption, the empirical literature focusing on some particular causes of death can, in some cases, show some occurrences of skill-biased mortality, which contradict that assumption. For instance, Cogneau and Grimm (2008) show that people with relatively high schooling levels are more exposed to the risk of being hit by AIDS because they have more sexual partners.

  11. Alternatively, we could have assumed that all individuals save a positive amount, with saving propensities varying with the survival probability. This would not have affected our results.

  12. The next section will consider the impact of adding prematurely dead persons (i.e. missing persons) on the measurement of relative poverty (with a varying poverty line).

  13. See the Appendix for the presentation of all cases, including the ones under either no one is in poverty before the adjustment, or everyone is in poverty before the adjustment.

  14. Here again, we focus on the case where yy1 < ω and yy3 > ω.

  15. We focus on the case where, yo1 < ω and yo3 > ω. We also assume, without loss of generality, that \(\hat {y}_{o1}<\hat {y}_{o2},\hat {y}_{o3}\).

  16. Here again, we focus on the case where yo1 < ω and yo3 > ω.

  17. See the Proof of Proposition 7 in the Appendix.

  18. Without loss of generality, we focus on the case where, before the adjustment, the income group 1 lies in poverty (i.e. yy1 < yym), whereas the income group 3 does not lie in poverty (i.e. yym < yy3).

  19. Here again, we assume yy1 < yym and yym < yy3.

  20. Here again, we assume yo1 < yom and yom < yo3.

  21. Here again, we assume yo1 < yom and yom < yo3.

  22. For simplicity, we assume here that all income groups save some fraction of their income for the old age, and we also abstract here from public pensions (i.e. we fix 𝜃 = 0).

  23. If the group with the highest occurrence is the very poor, and, then, the second income group, it follows that the second class represents 1/2 of the occurrence of the first (i.e. \(n_{2}=\frac {n_{1}}{2}\)), and so on for other groups. Hence, normalizing the sum of ni to 1, we thus have \( n_{1}+\frac {n_{1}}{2}+\frac {n_{1}}{3}+...= 1\), which yields n1 = 0.34, n2 = 0.17, n3 = 0.11, etc, up to n10 = 0.034.

  24. The income/longevity gradient is less steep nowadays. Blanplain (2016) shows that white collar men (resp. women) in France live 6 years (resp. 3 years) more than blue collar men (resp. women).

  25. The reason why the hypothetical GDP pattern is close to the actual one lies in the fact that, under our calibration, the major engine of growth lies in the growth in TFP.

  26. It is assumed here that individuals enjoy the same earnings during the entire periods considered, which are of length 25 years.

  27. In the Appendix, we use a poverty line equal to 40% of the median income.

  28. The intuition is that selection through fertility is cumulative, and could thus be, in theory, much stronger than selection through mortality. In terms of the inclusion of “missing persons,” we would have to take into account not only the missing persons, but their missing children, and their missing grandchildren, etc.

  29. See Boucekkine and Laffargue (2010) on the intricate distributional effects of mortality shocks when health expenditures are private.

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Acknowledgments

The authors would like to thank A. Cigno and two anonymous referees for their helpful suggestions and comments on this paper.

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Correspondence to Gregory Ponthiere.

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Appendix

Appendix

A.1 Proof of Proposition 1

The capital accumulation equation is:

$$k_{t + 1}=\frac{\phi_{2}n_{2}\frac{\pi_{2}}{1+\pi_{2}}\lambda_{2}(1-\alpha )\left( 1-\theta \right) Bk_{t}^{\alpha }+\phi_{3}n_{3}\frac{\pi_{3}}{ 1+\pi_{3}}\lambda_{3}(1-\alpha )\left( 1-\theta \right) Bk_{t}^{\alpha }}{ gA} $$

This can be rewritten as:

$$k_{t + 1}=\frac{(1-\alpha )\left( 1-\theta \right) Bk_{t}^{\alpha }}{gA}\left( \frac{\phi_{2}\pi_{2}n_{2}\lambda_{2}}{1+\pi_{2}}+\frac{\phi_{3}\pi _{3}n_{3}\lambda_{3}}{1+\pi_{3}}\right) \equiv \varphi \left( k_{t}\right) $$

The existence of a stationary equilibrium amounts to search for fixed point of \(\varphi \left (k_{t}\right ) \). We have \(\varphi \left (0\right ) = 0\), so that 0 is a stationary equilibrium. Note that we have:

$$\begin{array}{@{}rcl@{}} \varphi^{\prime }\left( k_{t}\right) &=&\frac{(1-\alpha )\alpha \left( 1-\theta \right) Bk_{t}^{\alpha -1}}{gA}\left( \frac{\phi_{2}\pi _{2}n_{2}\lambda_{2}}{1+\pi_{2}}+\frac{\phi_{3}\pi_{3}n_{3}\lambda_{3}}{ 1+\pi_{3}}\right) >0 \\ \lim_{k_{t}\rightarrow 0}\varphi^{\prime }\left( k_{t}\right) &=&+\infty \end{array} $$

We thus know that the transition function \(\varphi \left (k_{t}\right ) \) is above the 45 line in the \(\left (k_{t},k_{t + 1}\right ) \) space when kt tends to 0. We also have:

$$\lim_{k_{t}\rightarrow +\infty }\frac{\varphi \left( k_{t}\right) }{k_{t}}= \frac{(1-\alpha )\left( 1-\theta \right) Bk_{t}^{\alpha -1}}{gA}\left( \frac{ \phi_{2}\pi_{2}n_{2}\lambda_{2}}{1+\pi_{2}}+\frac{\phi_{3}\pi_{3}n_{3}\lambda_{3}}{1+\pi_{3}}\right) = 0 $$

Hence, when kt tends to \(+\infty \), the \(\varphi \left (k_{t}\right ) \) function lies below the 45 line. Hence, given that \(\varphi \left (k_{t}\right ) \) is above the 45 line in the \(\left (k_{t},k_{t + 1}\right ) \) space when kt tends to 0, and lies below the 45 line in the \(\left (k_{t},k_{t + 1}\right ) \) when kt tends to \(+\infty \), it must be the case, by continuity, that \(\varphi \left (k_{t}\right ) \) crosses the 45 line at least once for some kt > 0.

Finally, given that

$$\varphi^{\prime \prime }\left( k_{t}\right) =\frac{(1-\alpha )\alpha \left( \alpha -1\right) \left( 1-\theta \right) Bk_{t}^{\alpha -2}}{gA}\left( \frac{ \phi_{2}\pi_{2}n_{2}\lambda_{2}}{1+\pi_{2}}+\frac{\phi_{3}\pi_{3}n_{3}\lambda_{3}}{1+\pi_{3}}\right) <0 $$

we know for sure that this intersection is unique. This intersection is achieved when:

$$k^{\ast }=\left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{i}\pi_{i}n_{i}\lambda_{i}}{1+\pi_{i}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{gA}\right]^{\frac{1}{1-\alpha }} $$

Regarding stability, note that

$$\varphi^{\prime }\left( 0\right) =\frac{(1-\alpha )\alpha \left( 1-\theta \right) B0^{\alpha -1}}{gA}\left( \frac{\phi_{2}\pi_{2}n_{2}\lambda_{2}}{ 1+\pi_{2}}+\frac{\phi_{3}\pi_{3}n_{3}\lambda_{3}}{1+\pi_{3}}\right) >1 $$

so that 0 is clearly unstable.

On the contrary, we have

$$\varphi^{\prime }\left( k^{\ast }\right) =\frac{(1-\alpha )\alpha \left( 1-\theta \right) B\left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{i}\pi _{i}n_{i}\lambda_{i}}{1+\pi_{i}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{gA}\right]^{\frac{\alpha -1}{1-\alpha }}\left( \sum\limits_{i = 2,3}\frac{\phi_{i}n_{i}\pi_{i}\lambda_{i}}{1+\pi_{i}} \right) }{gA}=\alpha <1 $$

Hence, k is locally stable.

A.2 Proof of Proposition 3

At the stationary equilibrium, we have three income levels at the young age (net of taxes):

$$\begin{array}{@{}rcl@{}} y_{y1} &\equiv &\lambda_{1}B(1-\alpha )\left( 1-\theta \right) \left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{i}n_{i}\pi_{i}\lambda_{i}}{1+\pi _{i}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g{\Sigma} n_{i}\phi_{i}\lambda_{i}}\right]^{\frac{\alpha }{1-\alpha }} \\ y_{y2} &\equiv &\lambda_{2}B(1-\alpha )\left( 1-\theta \right) \left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{i}n_{i}\pi_{i}\lambda_{i}}{1+\pi_{i}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g{\Sigma} n_{i}\phi_{i}\lambda_{i}}\right]^{\frac{\alpha }{1-\alpha }}=\frac{\lambda_{2}}{ \lambda_{1}}y_{y1} \\ y_{y3} &\equiv &\lambda_{3}B(1-\alpha )\left( 1-\theta \right) \left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{i}n_{i}\pi_{i}\lambda_{i}}{1+\pi_{i}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g{\Sigma} n_{i}\phi _{i}\lambda_{i}}\right]^{\frac{\alpha }{1-\alpha }}=\frac{\lambda_{3}}{ \lambda_{1}}y_{y1} \end{array} $$

Hence, we have that the poverty rate Py at the young age can take, in theory, four distinct values, depending on the structural parameters of our economy:

$$P_{y}=\left\{ \begin{array}{l} 0\text{ when }y_{y1}\geq \omega \\ \frac{\phi_{1}n_{1}}{{\Sigma} \phi_{i}n_{i}}\text{ when }y_{y1}<\omega \leq \frac{\lambda_{2}}{\lambda_{1}}y_{y1} \\ \frac{\phi_{1}n_{1}+\phi_{2}n_{2}}{{\Sigma} \phi_{i}n_{i}}\text{ when } \frac{\lambda_{2}}{\lambda_{1}}y_{y1}<\omega \leq \frac{\lambda_{3}}{ \lambda_{1}}y_{y1} \\ 1\text{ when }\frac{\lambda_{3}}{\lambda_{1}}y_{y1}<\omega \end{array} \right. $$

Thus, for a given poverty line ω, whether a more or less large proportion of young adults lies in poverty depends on technological parameters (B,α, and λi), on the cohort growth rate 1 − g, on the distribution of skills (ni), and on survival conditions (ϕi and πi).

Consider now the hypothetical case where all types of individuals benefit from the survival conditions of the richest (i.e. ,ϕ3 and π3). In that hypothetical case, the three income levels at the young age are:

$$\begin{array}{@{}rcl@{}} \hat{y}_{y1} &\equiv &\lambda_{1}B(1-\alpha )\left( 1-\theta \right) \left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{3}n_{i}\pi_{3}\lambda_{i}}{1+\pi _{3}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g\phi_{3}{\Sigma} n_{j}\lambda_{j}}\right]^{\frac{\alpha }{1-\alpha }} \\ \hat{y}_{y2} &\equiv &\lambda_{2}B(1-\alpha )\left( 1-\theta \right) \left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{3}n_{i}\pi_{3}\lambda_{i}}{1+\pi _{3}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g\phi_{3}{\Sigma} n_{j}\lambda_{j}}\right]^{\frac{\alpha }{1-\alpha }}=\frac{\lambda_{2}}{ \lambda_{1}}\hat{y}_{y1} \\ \hat{y}_{y3} &\equiv &\lambda_{3}B(1-\alpha )\left( 1-\theta \right) \left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{3}n_{i}\pi_{3}\lambda_{i}}{1+\pi _{3}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g\phi_{3}{\Sigma} n_{j}\lambda_{j}}\right]^{\frac{\alpha }{1-\alpha }}=\frac{\lambda_{3}}{ \lambda_{1}}\hat{y}_{y1} \end{array} $$

The adjusted poverty rate at the young age can take the following values:

$$\hat{P}_{y}=\left\{ \begin{array}{l} 0\text{ when }\hat{y}_{y1}\geq \omega \\ \frac{n_{1}}{{\Sigma} n_{i}}\text{ when }\hat{y}_{y1}<\omega \leq \frac{ \lambda_{2}}{\lambda_{1}}\hat{y}_{y1} \\ \frac{n_{1}+n_{2}}{{\Sigma} n_{i}}\text{ when }\frac{\lambda_{2}}{\lambda_{1} }\hat{y}_{y1}<\omega \leq \frac{\lambda_{3}}{\lambda_{1}}\hat{y}_{y1} \\ 1\text{ when }\frac{\lambda_{3}}{\lambda_{1}}\hat{y}_{y1}<\omega \end{array} \right. $$

Quite interestingly, the income levels at the young age at the actual stationary equilibrium and at the hypothetical one can be compared as follows. Let us first compare yy1 with \(\hat {y}_{y1}\). We have:

$$\begin{array}{@{}rcl@{}} \hat{y}_{y1} &=&y_{y1}\frac{\lambda_{1}B(1-\alpha )\left( 1-\theta \right) \left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{3}\pi_{3}n_{i}\lambda_{i}}{ 1+\pi_{3}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g\phi_{3}{\Sigma} n_{j}\lambda_{j}}\right]^{\frac{\alpha }{1-\alpha }}}{\lambda _{1}B(1-\alpha )\left( 1-\theta \right) \left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{i}\pi_{i}n_{i}\lambda_{i}}{1+\pi_{i}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g{\Sigma} n_{j}\phi_{j}\lambda_{j}}\right]^{ \frac{\alpha }{1-\alpha }}} \\ &=&y_{y1}\left[ \frac{\sum\limits_{i = 2,3}\left( \frac{\phi_{3}\pi_{3}n_{i}\lambda_{i}}{1+\pi_{3}}\right) \frac{1}{\phi_{3}{\Sigma} n_{j}\lambda_{j}}}{\sum\limits_{i = 2,3}\left( \frac{\phi_{i}\pi_{i}n_{i}\lambda_{i}}{1+\pi_{i}}\right) \frac{1}{{\Sigma} n_{j}\phi _{j}\lambda_{j}}}\right]^{\frac{\alpha }{1-\alpha }} \end{array} $$

Hence, we have \(y_{y1}\gtrless \hat {y}_{y1}\) when:

$${\Delta} \equiv \frac{\sum\limits_{i = 2,3}\left( \frac{\phi_{3}n_{i}\pi_{3}\lambda_{i}}{1+\pi_{3}}\right) \frac{1}{\phi_{3}{\Sigma} n_{j}\lambda _{j}}}{\sum\limits_{i = 2,3}\left( \frac{\phi_{i}n_{i}\pi_{i}\lambda_{i}}{ 1+\pi_{i}}\right) \frac{1}{{\Sigma} n_{j}\phi_{j}\lambda_{j}}}\lessgtr 1 $$

Note that, when \(y_{y1}>\hat {y}_{y1}\), we have:

$$\begin{array}{@{}rcl@{}} y_{y2} &=&\frac{\lambda_{2}}{\lambda_{1}}y_{y1}>\hat{y}_{y2}=\frac{\lambda_{2}}{\lambda_{1}}\hat{y}_{y1}=\frac{\lambda_{2}}{\lambda_{1}}{\Delta}^{ \frac{\alpha }{1-\alpha }}y_{y1} \\ y_{y3} &=&\frac{\lambda_{3}}{\lambda_{1}}y_{y1}>\hat{y}_{y3}=\frac{\lambda _{3}}{\lambda_{1}}\hat{y}_{y1}=\frac{\lambda_{3}}{\lambda_{1}}{\Delta}^{\frac{\alpha }{1-\alpha }}y_{y1} \end{array} $$

and inverted inequalities when \(y_{y1}<\hat {y}_{y1}\).

Based on those inequalities, Proposition 3 can be proved by considering the different possible impacts, in terms of poverty measurement, of multiplying the three income levels yyi by \({\Delta }^{\frac {\alpha }{1-\alpha }}\). We first start by considering all possible cases a priori. The conditions on income levels were first presented in terms of the common component yy1:

$$\hat{y}_{y1}<\omega \Leftrightarrow y_{y1}{\Delta}^{\frac{\alpha }{1-\alpha } }<\omega $$

Then, in a second stage, we deleted the impossible cases, i.e., cases for which conditions on inequalities before the adjustment and after the adjustment contradict themselves. For instance, when considering the case where \(y_{y1}<\omega \leq \frac {\lambda _{2}}{\lambda _{1}}y_{y1}\), a priori four cases could arise:

  • if \(y_{y1}{\Delta }^{\frac {\alpha }{1-\alpha }}\geq \omega \), then \( P_{y}=\frac {\phi _{1}n_{1}}{{\Sigma } n_{i}\phi _{i}}>\hat {P}_{y}= 0;\)

  • if \(y_{y1}{\Delta }^{\frac {\alpha }{1-\alpha }}<\omega \leq \frac { \lambda _{2}}{\lambda _{1}}y_{y1}{\Delta }^{\frac {\alpha }{1-\alpha }}\), then \( P_{y}=\frac {\phi _{1}n_{1}}{{\Sigma } n_{i}\phi _{i}}<\hat {P}_{y}=\frac {n_{1}}{ {\Sigma } n_{i}};\)

  • if \(\frac {\lambda _{2}}{\lambda _{1}}y_{y1}{\Delta }^{\frac {\alpha }{ 1-\alpha }}<\omega \leq \frac {\lambda _{3}}{\lambda _{1}}y_{y1}{\Delta }^{ \frac {\alpha }{1-\alpha }}\), then \(P_{y}=\frac {\phi _{1}n_{1}}{{\Sigma } n_{i}\phi _{i}}<\hat {P}_{y}=\frac {n_{1}+n_{2}}{{\Sigma } n_{i}};\)

  • if \(\frac {\lambda _{3}}{\lambda _{1}}y_{y1}{\Delta }^{\frac {\alpha }{ 1-\alpha }}<\omega \), then \(P_{y}=\frac {\phi _{1}n_{1}}{{\Sigma } n_{i}\phi _{i} }<\hat {P}_{y}= 1;\)

But clearly, given that Δ < 1, it cannot be the case that both yy1 < ω and \(y_{y1}{\Delta }^{\frac {\alpha }{1-\alpha }}\geq \omega \), so that impossible case was deleted. Similar cancellations were made for other cases, when there is a conflict between conditions.

Finally, note that, when Δ < 1, so that \(y_{\text {yi}}>\hat {y}_{yi}\forall i = 1,2,3\), the extreme cases not treated in Proposition 3 are:

  • If yy1ω

    • if \(y_{y1}{\Delta }^{\frac {\alpha }{1-\alpha }}\geq \omega \), then \( P_{y}=\hat {P}_{y}= 0\);

    • if \(y_{y1}{\Delta }^{\frac {\alpha }{1-\alpha }}<\omega \leq \frac { \lambda _{2}}{\lambda _{1}}y_{y1}{\Delta }^{\frac {\alpha }{1-\alpha }}\), then \( P_{y}= 0<\hat {P}_{y}=\frac {n_{1}}{{\Sigma } n_{i}}\);

    • if \(\frac {\lambda _{2}}{\lambda _{1}}y_{y1}{\Delta }^{\frac {\alpha }{ 1-\alpha }}<\omega \leq \frac {\lambda _{3}}{\lambda _{1}}y_{y1}{\Delta }^{ \frac {\alpha }{1-\alpha }}\), then \(P_{y}= 0<\hat {P}_{y}=\frac {n_{1}+n_{2}}{ {\Sigma } n_{i}}\);

    • if \(\frac {\lambda _{3}}{\lambda _{1}}y_{y1}{\Delta }^{\frac {\alpha }{ 1-\alpha }}<\omega \), then \(P_{y}= 0<\hat {P}_{y}= 1\);

  • If \(\frac {\lambda _{3}}{\lambda _{1}}y_{y1}<\omega \), then \(P_{y}=\hat { P}_{y}= 1\).

A.3 Proof of Proposition 5

We have, at the stable stationary equilibrium:

$$\begin{array}{@{}rcl@{}} y_{o1} &=&\left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{i}n_{i}\pi_{i}\lambda_{i}}{1+\pi_{i}}\right) {\Omega} \right]^{\frac{\alpha }{ 1-\alpha }}\frac{\theta B(1-\alpha )\left( {\Sigma} n_{i}\phi_{i}\lambda_{i}\right) }{n_{1}\phi_{1}\pi_{1}} \\ y_{oi} &=&\left[ \sum\limits_{j = 2,3}\left( \frac{\phi_{j}n_{j}\pi_{j}}{ 1+\pi_{j}}\lambda_{j}\right) {\Omega} \right]^{\frac{2\alpha -1}{1-\alpha } }\left( \frac{\lambda_{i}(1-\alpha )\left( 1-\theta \right) B^{2}\alpha }{ 1+\pi_{i}}\right) \text{ for }i = 2,3 \end{array} $$

Hence, we have

$$y_{o2}=\frac{\lambda_{2}}{1+\pi_{2}}{\Xi} y_{o1}\text{ and }y_{o3}=\frac{ \lambda_{3}}{1+\pi_{3}}{\Xi} y_{o1} $$

where

$${\Xi} \equiv \frac{gn_{1}\phi_{1}\pi_{1}\alpha }{\theta (1-\alpha )\left( \frac{\phi_{2}n_{2}\pi_{2}\lambda_{2}}{1+\pi_{2}}+\frac{\phi _{3}n_{3}\pi_{3}\lambda_{3}}{1+\pi_{3}}\right) } $$

Given this, the old-age poverty Po measure takes the following values:

$$P_{o}=\left\{ \begin{array}{l} 0\text{ when }y_{o1}\geq \omega \\ \frac{\phi_{1}\pi_{1}n_{1}}{{\Sigma} \phi_{i}\pi_{i}n_{i}}\text{ when } y_{o1}<\omega \leq \frac{\lambda_{2}}{\left( 1+\pi_{2}\right) }{\Xi} y_{o1} \\ \frac{\phi_{1}\pi_{1}n_{1}+\phi_{2}\pi_{2}n_{2}}{{\Sigma} \phi_{i}\pi_{i}n_{i}}\text{ when }\frac{\lambda_{2}}{\left( 1+\pi_{2}\right) }{\Xi} y_{o1}<\omega \leq \frac{\lambda_{3}}{\left( 1+\pi_{3}\right) }{\Xi} y_{o1} \\ 1\text{ when }\frac{\lambda_{3}}{\left( 1+\pi_{3}\right) }{\Xi} y_{o1}<\omega \end{array} \right. $$

Let us now compare those poverty measures with those that would have prevailed in the absence of income-differentiated mortality. For that purpose, we now assume that all types of individuals benefit from the survival conditions of the richest (i.e., ϕ3 and π3). In that hypothetical case, the three income levels at the old age are:

$$\begin{array}{@{}rcl@{}} \hat{y}_{o1} &\equiv &\left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{3}n_{i}\pi_{3}\lambda_{i}}{1+\pi_{3}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g\phi_{3}{\Sigma} n_{j}\lambda_{j}}\right]^{\frac{ \alpha }{1-\alpha }}\frac{\theta B(1-\alpha )\left( {\Sigma} n_{j}\lambda_{j}\right) }{n_{1}\pi_{3}} \\ \hat{y}_{o2} &\equiv &\left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{3}n_{i}\pi_{3}\lambda_{i}}{1+\pi_{3}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g\phi_{3}{\Sigma} n_{j}\lambda_{j}}\right]^{\frac{ 2\alpha -1}{1-\alpha }}\frac{\lambda_{2}(1-\alpha )\left( 1-\theta \right) BB\alpha }{1+\pi_{3}} \\ &=&\frac{n_{1}\pi_{3}\lambda_{2}B\alpha g\phi_{3}}{\theta B(1-\alpha )\left( 1+\pi_{3}\right) \left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{3}n_{i}\pi_{3}\lambda_{i}}{1+\pi_{3}}\right) \right] }\hat{y}_{o1} \\ \hat{y}_{o3} &\equiv &\left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{3}n_{i}\pi_{3}\lambda_{i}}{1+\pi_{3}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g\phi_{3}{\Sigma} n_{j}\lambda_{j}}\right]^{\frac{ 2\alpha -1}{1-\alpha }}\frac{\lambda_{3}(1-\alpha )\left( 1-\theta \right) BB\alpha }{1+\pi_{3}}=\frac{\lambda_{3}}{\lambda_{2}}\hat{y}_{o2} \\ &=&\frac{\lambda_{3}}{\lambda_{2}}\frac{n_{1}\pi_{3}\lambda_{2}B\alpha g\phi_{3}}{\theta B(1-\alpha )\left( 1+\pi_{3}\right) \left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{3}n_{i}\pi_{3}\lambda_{i}}{1+\pi_{3}}\right) \right] }\hat{y}_{o1} \end{array} $$

The adjusted old-age poverty \(\hat {P}_{o}\) measure takes the following values:

$$\hat{P}_{o}=\left\{ \begin{array}{l} 0\text{ when }\hat{y}_{o1}\geq \omega \\ \frac{n_{1}}{{\Sigma} n_{i}}\text{ when }\hat{y}_{o1}<\omega \leq \hat{y}_{o1} \frac{n_{1}\pi_{3}\lambda_{2}B\alpha g\phi_{3}}{\theta B(1-\alpha )\left( 1+\pi_{3}\right) \left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{3}n_{i}\pi_{3}\lambda_{i}}{1+\pi_{3}}\right) \right] } \\ \frac{n_{1}+n_{2}}{{\Sigma} n_{i}}\text{ when }\left. \left\{ \begin{array}{l} \hat{y}_{o1}\frac{n_{1}\pi_{3}\lambda_{2}B\alpha g\phi_{3}}{\theta B(1-\alpha )\left( 1+\pi_{3}\right) \left[ \sum\limits_{i = 2,3}\left( \frac{ \phi_{3}n_{i}\pi_{3}\lambda_{i}}{1+\pi_{3}}\right) \right] } \\ <\omega \leq \hat{y}_{o1}\frac{n_{1}\pi_{3}\lambda_{3}B\alpha g\phi_{3}}{ \theta B(1-\alpha )\left( 1+\pi_{3}\right) \left[ \sum\limits_{i = 2,3} \left( \frac{\phi_{3}n_{i}\pi_{3}\lambda_{i}}{1+\pi_{3}}\right) \right] } \end{array} \right. \right. \\ 1\text{ when }\hat{y}_{o1}\frac{n_{1}\pi_{3}\lambda_{3}B\alpha g\phi_{3}}{ \theta B(1-\alpha )\left( 1+\pi_{3}\right) \left[ \sum\limits_{i = 2,3} \left( \frac{\phi_{3}n_{i}\pi_{3}\lambda_{i}}{1+\pi_{3}}\right) \right] } <\omega \end{array} \right. $$

When comparing yo1 with \(\hat {y}_{o1}\), we have:

$$\begin{array}{@{}rcl@{}} \hat{y}_{o1} &\equiv &\left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{3}n_{i}\pi_{3}\lambda_{i}}{1+\pi_{3}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g\phi_{3}{\Sigma} n_{j}\lambda_{j}}\right]^{\frac{ \alpha }{1-\alpha }}\frac{\theta B(1-\alpha )\phi_{3}\left( {\Sigma} n_{j}\lambda_{j}\right) }{n_{1}\phi_{3}\pi_{3}} \\ y_{o1} &\equiv &\left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{i}n_{i}\pi _{i}\lambda_{i}}{1+\pi_{i}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g{\Sigma} n_{j}\lambda_{j}\phi_{j}}\right]^{\frac{\alpha }{ 1-\alpha }}\frac{\theta B(1-\alpha )\left( {\Sigma} n_{j}\phi_{j}\lambda_{j}\right) }{n_{1}\phi_{1}\pi_{1}} \end{array} $$

Hence,

$$\hat{y}_{o1}=y_{o1}{\Delta}^{\frac{\alpha }{1-\alpha }}\frac{\phi_{1}\pi_{1}\left( {\Sigma} n_{j}\lambda_{j}\right) }{\pi_{3}\left( {\Sigma} n_{i}\phi_{j}\lambda_{j}\right) } $$

We also have:

$$\begin{array}{@{}rcl@{}} \hat{y}_{o2} &=&\left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{3}n_{i}\pi_{3}\lambda_{i}}{1+\pi_{3}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g\phi_{3}{\Sigma} n_{j}\lambda_{j}}\right]^{\frac{2\alpha -1}{ 1-\alpha }}\frac{\lambda_{2}\left( \frac{(1-\alpha )\left( 1-\theta \right) B}{g\phi_{3}{\Sigma} n_{j}\lambda_{j}}\right) gB\alpha \phi_{3}{\Sigma} n_{j}\lambda_{j}}{1+\pi_{3}} \\ y_{o2} &=&\left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{i}n_{i}\pi_{i}\lambda_{i}}{1+\pi_{i}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g{\Sigma} n_{j}\lambda_{j}\phi_{j}}\right]^{\frac{2\alpha -1}{ 1-\alpha }}\frac{\lambda_{2}\left( \frac{(1-\alpha )\left( 1-\theta \right) B}{g{\Sigma} n_{j}\lambda_{j}\phi_{j}}\right) gB\alpha {\Sigma} n_{j}\phi_{j}\lambda_{j}}{1+\pi_{2}} \end{array} $$

Hence,

$$\hat{y}_{o2}=y_{o2}{\Delta}^{\frac{2\alpha -1}{1-\alpha }}\frac{1+\pi_{2}}{ 1+\pi_{3}} $$

Hence, given that:

$$y_{o2}=\frac{n_{1}\phi_{1}\pi_{1}\left( \lambda_{2}g{\Sigma} n_{i}\lambda_{i}\phi_{i}\left( \frac{(1-\alpha )\left( 1-\theta \right) B}{g{\Sigma} n_{i}\lambda_{i}\phi_{i}}\right) B\alpha \right) }{\theta B(1-\alpha )\left( {\Sigma} n_{i}\phi_{i}\lambda_{i}\right) \left( 1+\pi_{2}\right) \sum\limits_{i = 2,3}\left( \frac{\phi_{i}n_{i}\pi_{i}\lambda_{i}}{1+\pi_{i}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g{\Sigma} n_{j}\lambda_{j}\phi_{j}}}y_{o1} $$

we also have:

$$\hat{y}_{o2}=y_{o1}{\Delta}^{\frac{2\alpha -1}{1-\alpha }}\frac{n_{1}\phi_{1}\pi_{1}\lambda_{2}\alpha g}{\theta (1+\pi_{3})(1-\alpha )\sum\limits_{i = 2,3}\left( \frac{\phi_{i}n_{i}\pi_{i}\lambda_{i}}{1+\pi_{i}}\right) } $$

We also have:

$$\begin{array}{@{}rcl@{}} y_{o3} &=&\left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{i}n_{i}\pi_{i}\lambda_{i}}{1+\pi_{i}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g{\Sigma} n_{j}\lambda_{j}\phi_{j}}\right]^{\frac{2\alpha -1}{ 1-\alpha }}\frac{\lambda_{3}g{\Sigma} n_{j}\lambda_{j}\phi_{j}\left( \frac{ (1-\alpha )\left( 1-\theta \right) B}{g{\Sigma} n_{j}\lambda_{j}\phi_{j}} \right) B\alpha }{1+\pi_{3}} \\ \hat{y}_{o3} &=&\left[ \sum\limits_{i = 2,3}\left( \frac{\phi_{3}n_{i}\pi_{3}\lambda_{i}}{1+\pi_{3}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g\phi_{3}{\Sigma} n_{j}\lambda_{j}}\right]^{\frac{2\alpha -1}{ 1-\alpha }}\frac{\lambda_{3}g\phi_{3}{\Sigma} n_{j}\lambda_{j}\left( \frac{ (1-\alpha )\left( 1-\theta \right) B}{g\phi_{3}{\Sigma} n_{j}\lambda_{j}} \right) B\alpha }{1+\pi_{3}} \end{array} $$

Hence,

$$\hat{y}_{o3}={\Delta}^{\frac{2\alpha -1}{1-\alpha }}y_{o3} $$

Note also that

$$\begin{array}{@{}rcl@{}} \hat{y}_{o3} &=&\frac{\lambda_{3}}{\lambda_{2}}\hat{y}_{o2}=\frac{\lambda_{3}}{\lambda_{2}}y_{o1}{\Delta}^{\frac{2\alpha -1}{1-\alpha }}\frac{ n_{1}\phi_{1}\pi_{1}\left( \lambda_{2}\left( ((1-\alpha )\left( 1-\theta \right) B\right) B\alpha \right) \frac{\left( 1+\pi_{2}\right) }{(1+\pi _{3})}}{\theta B(1-\alpha )\left( {\Sigma} n_{j}\phi_{j}\lambda_{j}\right) \left( 1+\pi_{2}\right) \sum\limits_{i = 2,3}\left( \frac{\phi_{i}n_{i}\pi _{i}\lambda_{i}}{1+\pi_{i}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g{\Sigma} n_{j}\lambda_{j}\phi_{j}}} \\ &=&\frac{\lambda_{3}}{\lambda_{2}}y_{o1}{\Delta}^{\frac{2\alpha -1}{ 1-\alpha }}\frac{n_{1}\phi_{1}\pi_{1}\lambda_{2}\alpha g}{\theta (1+\pi _{3})(1-\alpha )\sum\limits_{i = 2,3}\left( \frac{\phi_{i}n_{i}\pi_{i}\lambda_{i}}{1+\pi_{i}}\right) } \end{array} $$

In the following, we denote

$${\Psi} \equiv \frac{n_{1}\phi_{1}\pi_{1}\lambda_{2}\alpha g}{\theta (1+\pi_{3})(1-\alpha )\left( \frac{\phi_{2}n_{2}\pi_{2}\lambda_{2}}{1+\pi_{2}}+ \frac{\phi_{3}n_{3}\pi_{3}\lambda_{3}}{1+\pi_{3}}\right) }, $$

to have

$$\begin{array}{@{}rcl@{}} \hat{y}_{o1} &=&y_{o1}{\Delta}^{\frac{\alpha }{1-\alpha }}\frac{\phi_{1}\pi_{1}\left( {\Sigma} n_{i}\lambda_{i}\right) }{\pi_{3}\left( {\Sigma} n_{i}\phi_{i}\lambda_{i}\right) } \\ \hat{y}_{o2} &=&y_{o1}{\Delta}^{\frac{2\alpha -1}{1-\alpha }}{\Psi} \\ \hat{y}_{o3} &=&\frac{\lambda_{3}}{\lambda_{2}}y_{o1}{\Delta}^{\frac{ 2\alpha -1}{1-\alpha }}{\Psi} \end{array} $$

where

$${\Psi} \equiv \frac{n_{1}\phi_{1}\pi_{1}\lambda_{2}\alpha g}{\theta (1+\pi_{3})(1-\alpha )\left( \frac{\phi_{2}n_{2}\pi_{2}\lambda_{2}}{1+\pi_{2}}+ \frac{\phi_{3}n_{3}\pi_{3}\lambda_{3}}{1+\pi_{3}}\right) } $$

Based on those rewritings, we can now complete the proof of Proposition 5, by considering the different possible impacts, in terms of poverty measurement, of multiplying the three income levels at the old age by the corresponding adjustment factors. For that purpose, we suppose, without loss of generality, that \(\hat {y}_{o1}<\hat {y}_{o2},\hat {y}_{o3}\). We first start by considering all possible cases a priori. The conditions on income levels were first presented in common terms as:

$$\hat{y}_{o1}<\omega \iff y_{o1}{\Delta}^{\frac{\alpha }{1-\alpha }}\frac{\phi_{1}\pi_{1}\left( {\Sigma} n_{i}\lambda_{i}\right) }{\pi_{3}\left( {\Sigma} n_{i}\phi_{i}\lambda_{i}\right) }<\omega $$

We also deleted cases for which the conditions cannot be satisfied. For instance, in the case \(\frac {\lambda _{2}}{1+\pi _{2}}{\Xi } y_{o1}<\omega \leq \frac {\lambda _{3}}{1+\pi _{3}}{\Xi } y_{o1}\), a priori four cases could arise:

  • if \(y_{o1}{\Delta }^{\frac {\alpha }{1-\alpha }}\frac {\phi _{1}\pi _{1}\left ({\Sigma } n_{i}\lambda _{i}\right ) }{\pi _{3}\left ({\Sigma } n_{i}\phi _{i}\lambda _{i}\right ) }\geq \omega \), then \(P_{o}=\frac {\phi _{1}\pi _{1}n_{1}+\phi _{2}\pi _{2}n_{2}}{{\Sigma } \pi _{i}\phi _{i}n_{i}}>\hat {P} _{o}= 0;\)

  • if \(y_{o1}{\Delta }^{\frac {\alpha }{1-\alpha }}\frac {\phi _{1}\pi _{1}\left ({\Sigma } n_{i}\lambda _{i}\right ) }{\pi _{3}\left ({\Sigma } n_{i}\phi _{i}\lambda _{i}\right ) }<\omega \leq y_{o1}{\Delta }^{\frac {2\alpha -1}{ 1-\alpha }}{\Psi } \), then \(P_{o}=\frac {\phi _{1}\pi _{1}n_{1}+\phi _{2}\pi _{2}n_{2}}{{\Sigma } \pi _{i}\phi _{i}n_{i}}\lessgtr \hat {P}_{o}=\frac {n_{1}}{ {\Sigma } n_{i}};\)

  • if \(y_{o1}{\Delta }^{\frac {2\alpha -1}{1-\alpha }}{\Psi } <\omega \leq \frac {\lambda _{3}}{\lambda _{2}}y_{o1}{\Delta }^{\frac {2\alpha -1}{1-\alpha } }{\Psi } \), then \(P_{o}=\frac {\phi _{1}\pi _{1}n_{1}+\phi _{2}\pi _{2}n_{2}}{ {\Sigma } \pi _{i}\phi _{i}n_{i}}<\hat {P}_{o}=\frac {n_{1}+n_{2}}{{\Sigma } n_{i}};\)

  • if \(\frac {\lambda _{3}}{\lambda _{2}}y_{o1}{\Delta }^{\frac {2\alpha -1}{ 1-\alpha }}{\Psi } <\omega \), then \(P_{o}=\frac {\phi _{1}\pi _{1}n_{1}+\phi _{2}\pi _{2}n_{2}}{{\Sigma } \pi _{i}\phi _{i}n_{i}}<\hat {P}_{o}= 1;\)

But it is clear that, given \(\frac {\lambda _{2}}{1+\pi _{2}}{\Xi } y_{o1}<\omega \leq \frac {\lambda _{3}}{1+\pi _{3}}{\Xi } y_{o1}\), the first two cases cannot arise. These were thus deleted. Similar deletions were made for other logically impossible cases.

The extreme cases not treated in Proposition 5 are:

  • If yo1ω

    • if \(y_{o1}{\Delta }^{\frac {\alpha }{1-\alpha }}\frac {\phi _{1}\pi _{1}\left ({\Sigma } n_{i}\lambda _{i}\right ) }{\pi _{3}\left ({\Sigma } n_{i}\phi _{i}\lambda _{i}\right ) }\geq \omega \), then \(P_{o}=\hat {P}_{o}= 0\);

    • if \(y_{o1}{\Delta }^{\frac {\alpha }{1-\alpha }}\frac {\phi _{1}\pi _{1}\left ({\Sigma } n_{i}\lambda _{i}\right ) }{\pi _{3}\left ({\Sigma } n_{i}\phi _{i}\lambda _{i}\right ) }<\omega \leq y_{o1}{\Delta }^{\frac {2\alpha -1}{ 1-\alpha }}{\Psi } \), then \(P_{o}= 0<\hat {P}_{o}=\frac {n_{1}}{{\Sigma } n_{i}}\);

    • if \(y_{o1}{\Delta }^{\frac {2\alpha -1}{1-\alpha }}{\Psi } <\omega \leq \frac {\lambda _{3}}{\lambda _{2}}y_{o1}{\Delta }^{\frac {2\alpha -1}{1-\alpha } }{\Psi } \), then \(P_{o}= 0<\hat {P}_{o}=\frac {n_{1}+n_{2}}{{\Sigma } n_{i}}\);

    • if \(\frac {\lambda _{3}}{\lambda _{2}}y_{o1}{\Delta }^{\frac {2\alpha -1}{ 1-\alpha }}{\Psi } <\omega \), then \(P_{o}= 0<\hat {P}_{o}= 1\);

  • If \(\frac {\pi _{3}\lambda _{3}}{1+\pi _{3}}{\Xi } y_{o1}<\omega \), then \( P_{o}=\hat {P}_{o}= 1\).

A.4 Proof of Proposition 7

The relative poverty rate can take various levels, depending on the level of the median income yym:

$$p_{y}=\left\{ \begin{array}{l} 0\text{ if }y_{\text{ym}}=y_{y1} \\ 0\text{ if }y_{\text{ym}}=y_{y2}\text{ and }y_{y1}\geq \text{xy}_{y2} \\ \frac{\phi_{1}n_{1}}{{\Sigma} \phi_{i}n_{i}}\text{ if }y_{\text{ym}}=y_{y2}\text{ and }y_{y1}<\text{xy}_{y2} \\ 0\text{ if }y_{\text{ym}}=y_{y3}\text{ and }y_{y1}\geq \text{xy}_{y3} \\ \frac{\phi_{1}n_{1}}{{\Sigma} \phi_{i}n_{i}}\text{ if }y_{\text{ym}}=y_{y3}\text{ and }y_{y1}<\text{xy}_{y3}\leq y_{y2} \\ \frac{\phi_{1}n_{1}+\phi_{2}n_{2}}{{\Sigma} \phi_{i}n_{i}}\text{ if } y_{\text{ym}}=y_{y3}\text{ and }y_{y1}<y_{y2}\leq \text{xy}_{y3} \end{array} \right. $$

When considering the economy after the adjustment, we have also several possible cases, depending on where the median (adjusted) income \(\hat {y} _{\text {ym}} \) lies:

$$\hat{p}_{y}=\left\{ \begin{array}{l} 0\text{ if }\hat{y}_{\text{ym}}=\hat{y}_{y1} \\ 0\text{ if }\hat{y}_{\text{ym}}=\hat{y}_{y2}\text{ and }\hat{y}_{y1}\geq \text{x}\hat{\text{y}} _{y2} \\ \frac{n_{1}}{{\Sigma} n_{i}}\text{ if }\hat{y}_{\text{ym}}=\hat{y}_{y2}\text{ and } \hat{y}_{y1}<\text{x}\hat{\text{y}}_{y2} \\ 0\text{ if }\hat{y}_{\text{ym}}=\hat{y}_{y3}\text{ and }\hat{y}_{y1}\geq \text{x}\hat{\text{y}} _{y3} \\ \frac{n_{1}}{{\Sigma} n_{i}}\text{ if }\hat{y}_{\text{ym}}=\hat{y}_{y3}\text{ and } \hat{y}_{y1}<\text{x}\hat{\text{y}}_{y3}\leq \hat{y}_{y2} \\ \frac{n_{1}+n_{2}}{{\Sigma} n_{i}}\text{ if }\hat{y}_{\text{ym}}=\hat{y}_{y3}\text{ and }\hat{y}_{y1}<\hat{y}_{y2}\leq \text{x}\hat{\text{y}}_{y3} \end{array} \right. $$

The comparison of py and \(\hat {p}_{y}\) can be carried out by reminding that the income levels of the three classes are related as follows:

$$\begin{array}{@{}rcl@{}} y_{y2} &=&\frac{\lambda_{2}}{\lambda_{1}}y_{y1};y_{y3}=\frac{\lambda_{3}}{ \lambda_{1}}y_{y1} \\ \hat{y}_{y1} &=&y_{y1}{\Delta}^{\frac{\alpha }{1-\alpha }};\hat{y}_{y2}=\frac{ \lambda_{2}}{\lambda_{1}}y_{y1}{\Delta}^{\frac{\alpha }{1-\alpha }};\hat{y} _{y3}=\frac{\lambda_{3}}{\lambda_{1}}y_{y1}{\Delta}^{\frac{\alpha }{ 1-\alpha }} \end{array} $$

The proof of Proposition 7 follows from merely considering the different cases regarding the initial poverty line and initial extent of poverty, and regarding the new poverty line and the impact of adding missing persons on the income levels. We first start by considering all possible cases a priori. The conditions on income levels were first presented in common terms as:

$$\hat{y}_{y1}<x\hat{y}_{y2}\iff y_{y1}{\Delta}^{\frac{\alpha }{1-\alpha }}<x \frac{\lambda_{2}}{\lambda_{1}}y_{y1}{\Delta}^{\frac{\alpha }{1-\alpha } }\iff 1<x\frac{\lambda_{2}}{\lambda_{1}} $$

Then, in a second stage, we deleted the impossible cases, i.e., cases for which conditions on inequalities before the adjustment and after the adjustment contradict themselves. For instance, when considering the case where yym = yy2 and \(1<x\frac {\lambda _{2}}{\lambda _{1}}\), a priori three cases could arise:

  • if \(\hat {y}_{\text {ym}}=\hat {y}_{y1}\), then \(p_{y}=\hat {p}_{y}= 0\);

  • if \(\hat {y}_{\text {ym}}=\hat {y}_{y2}\) and \(1\geq x\frac {\lambda _{2}}{\lambda _{1}}\), then \(p_{y}=\hat {p}_{y}= 0\);

  • if \(\hat {y}_{\text {ym}}=\hat {y}_{y2}\) and \(1<x\frac {\lambda _{2}}{\lambda _{1} }\), then \(p_{y}<\hat {p}_{y}=\frac {n_{1}}{{\Sigma } n_{i}};\)

But the second case cannot arise, since we postulated that \(1<x\frac {\lambda _{2}}{\lambda _{1}}\). Hence, that case was deleted. Similar deletion were made for other cases.

Finally, note that the extreme cases not treated in Proposition 7 are as follows:

  • If yym = yy1, then \(\hat {y}_{\text {ym}}=\hat {y}_{y1}\), so that \(p_{y}= \hat {p}_{y}= 0\);

  • If yym = yy2 and \(1\geq x\frac {\lambda _{2}}{\lambda _{1}}\),

    • if \(\hat {y}_{\text {ym}}=\hat {y}_{y1}\), then \(p_{y}=\hat {p}_{y}= 0\);

    • if \(\hat {y}_{\text {ym}}=\hat {y}_{y2}\) and given \(1\geq x\frac {\lambda _{2}}{ \lambda _{1}}\), then \(p_{y}=\hat {p}_{y}= 0\);

  • If yym = yy3 and \(1\geq x\frac {\lambda _{3}}{\lambda _{1}}\),

    • if \(\hat {y}_{\text {ym}}=\hat {y}_{y1}\), then \(p_{y}=\hat {p}_{y}= 0\);

    • if \(\hat {y}_{\text {ym}}=\hat {y}_{y2}\) and given \(1\geq x\frac {\lambda _{2}}{ \lambda _{1}}\), then \(p_{y}=\hat {p}_{y}= 0\);

    • if \(\hat {y}_{\text {ym}}=\hat {y}_{y3}\) and \(1\geq x\frac {\lambda _{3}}{\lambda _{1}}\), then \(p_{y}=\hat {p}_{y}= 0\).

A.5 Proof of Proposition 9

The relative poverty rate at the old age before the adjustment can take the following values:

$$p_{o}=\left\{ \begin{array}{l} 0\text{ if }y_{\text{om}}=y_{o1} \\ 0\text{ if }y_{\text{om}}=y_{o2}\text{ and }y_{o1}\geq \text{xy}_{o2} \\ \frac{\pi_{1}\phi_{1}n_{1}}{{\Sigma} \phi_{i}\pi_{i}n_{i}}\text{ if } y_{\text{om}}=y_{o2}\text{ and }y_{o1}<\text{xy}_{o2} \\ 0\text{ if }y_{\text{om}}=y_{o3}\text{ and }y_{o1}\geq \text{xy}_{o3} \\ \frac{\pi_{1}\phi_{1}n_{1}}{{\Sigma} \phi_{i}\pi_{i}n_{i}}\text{ if } y_{\text{om}}=y_{o3}\text{ and }y_{o1}<\text{xy}_{o3}\leq y_{o2} \\ \frac{\pi_{1}\phi_{1}n_{1}+\pi_{2}\phi_{2}n_{2}}{{\Sigma} \phi_{i}\pi_{i}n_{i}}\text{ if }y_{\text{om}}=y_{y3}\text{ and }y_{o1}<y_{o2}\leq \text{xy}_{o3} \end{array} \right. $$

When considering the economy after the adjustment, we have also several possible cases, depending on where the median (adjusted) income \(\hat {y} _{\text {om}} \) lies:

$$\hat{p}_{o}=\left\{ \begin{array}{l} 0\text{ if }\hat{y}_{\text{yo}}=\hat{y}_{o1} \\ 0\text{ if }\hat{y}_{\text{om}}=\hat{y}_{o2}\text{ and }\hat{y}_{o1}\geq \text{x}\hat{\text{y}} _{o2} \\ \frac{n_{1}}{{\Sigma} n_{i}}\text{ if }\hat{y}_{\text{om}}=\hat{y}_{o2}\text{ and } \hat{y}_{o1}<\text{x}\hat{\text{y}}_{o2} \\ 0\text{ if }\hat{y}_{\text{om}}=\hat{y}_{o3}\text{ and }\hat{y}_{o1}\geq \text{x}\hat{\text{y}} _{o3} \\ \frac{n_{1}}{{\Sigma} n_{i}}\text{ if }\hat{y}_{\text{om}}=\hat{y}_{o3}\text{ and } \hat{y}_{o1}<\text{x}\hat{\text{y}}_{o3}\leq \hat{y}_{o2} \\ \frac{n_{1}+n_{2}}{{\Sigma} n_{i}}\text{ if }\hat{y}_{\text{om}}=\hat{y}_{o3}\text{ and }\hat{y}_{o1}<\hat{y}_{o2}\leq \text{x}\hat{\text{y}}_{o3} \end{array} \right. $$

The comparison of po and \(\hat {p}_{o}\) can be carried out by first reminding that:

$$\begin{array}{@{}rcl@{}} y_{o2} &=&\frac{\lambda_{2}}{1+\pi_{2}}{\Xi} y_{o1};y_{o3}=\frac{\lambda_{3} }{1+\pi_{3}}{\Xi} y_{o1} \\ \hat{y}_{o1} &=&y_{o1}{\Delta}^{\frac{\alpha }{1-\alpha }}\frac{\phi_{1}\pi_{1}\left( {\Sigma} n_{i}\lambda_{i}\right) }{\pi_{3}\left( {\Sigma} n_{i}\phi_{i}\lambda_{i}\right) } \\ \hat{y}_{o2} &=&y_{o1}{\Delta}^{\frac{2\alpha -1}{1-\alpha }}{\Psi} \\ \hat{y}_{o3} &=&\frac{\lambda_{3}}{\lambda_{2}}y_{o1}{\Delta}^{\frac{ 2\alpha -1}{1-\alpha }}{\Psi} \end{array} $$

The proof of Proposition 9 follows from merely considering the different cases regarding the initial poverty line and initial extent of poverty, and regarding the new poverty line and the impact of adding missing persons on the income levels. We first start by considering all possible cases a priori. The conditions on income levels were first presented as follows:

$$\hat{y}_{o1}\geq \text{x}\hat{\text{y}}_{o2}\iff {\Delta}^{\frac{\alpha }{1-\alpha }}\frac{ \phi_{1}\pi_{1}\left( {\Sigma} n_{i}\lambda_{i}\right) }{\pi_{3}\left( {\Sigma} n_{i}\phi_{i}\lambda_{i}\right) }\geq x{\Delta}^{\frac{2\alpha -1}{ 1-\alpha }}{\Psi} $$

Further simplifications were made by replacing for Δ and Ψ, to obtain:

$$\begin{array}{@{}rcl@{}} \frac{\phi_{1}\pi_{1}\left( {\Sigma} n_{j}\lambda_{j}\right) }{\pi_{3}\left( {\Sigma} n_{j}\phi_{j}\lambda_{j}\right) } &\geq &x\frac{ n_{1}\phi_{1}\pi_{1}\lambda_{2}\left( 1-\theta \right) B\alpha \left( \frac{\sum\limits_{i = 2,3}\left( \frac{\phi_{i}n_{i}\pi_{i}\lambda_{i}}{ 1+\pi_{i}}\right) \frac{1}{{\Sigma} n_{j}\phi_{j}\lambda_{j}}}{ \sum\limits_{i = 2,3}\left( \frac{\phi_{3}n_{i}\pi_{3}\lambda_{i}}{1+\pi_{3}}\right) \frac{1}{\phi_{3}{\Sigma} n_{j}\lambda_{j}}}\right) }{\theta (1+\pi_{3})\left( {\Sigma} n_{j}\phi_{j}\lambda_{j}\right) \sum\limits_{i = 2,3}\left( \frac{\phi_{i}n_{i}\pi_{i}\lambda_{i}}{1+\pi_{i}}\right) \frac{(1-\alpha )\left( 1-\theta \right) B}{g{\Sigma} n_{j}\lambda_{j}\phi_{j}}} \\ \frac{\theta (1-\alpha )}{g} &\geq &\frac{xn_{1}\lambda_{2}\alpha }{ n_{2}\lambda_{2}+n_{3}\lambda_{3}} \end{array} $$

We then checked to see whether there was any contradiction between the different conditions so as to delete logically impossible cases.

Finally, note that the extreme cases not treated in Proposition 9 are as follows:

  • If yom = yo1, then \(\hat {y}_{\text {om}}=\hat {y}_{o1}\), so that \(p_{o}= \hat {p}_{o}= 0\);

  • If yom = yo2 and \(1\geq x\frac {\pi _{2}\lambda _{2}}{\left (1+\pi _{2}\right ) }{\Xi } \),

    • if \(\hat {y}_{\text {om}}=\hat {y}_{o1}\), then \(p_{o}=\hat {p}_{o}= 0\);

    • if \(\hat {y}_{\text {om}}=\hat {y}_{o2}\) and \(\frac {\theta (1-\alpha )}{\pi _{3}g }\geq \frac {xn_{1}\lambda _{2}\alpha }{\left (n_{2}\lambda _{2}+n_{3}\lambda _{3}\right ) }\), then \(p_{o}=\hat {p}_{o}= 0\);

    • if \(\hat {y}_{\text {om}}=\hat {y}_{o2}\) and \(\frac {\theta (1-\alpha )}{\pi _{3}g }<\frac {xn_{1}\lambda _{2}\alpha }{\left (n_{2}\lambda _{2}+n_{3}\lambda _{3}\right ) }\leq \frac {n_{1}\lambda _{3}\alpha }{\left (n_{2}\lambda _{2}+n_{3}\lambda _{3}\right ) }\), then \(p_{o}= 0<\hat {p}_{o}=\frac {n_{1}}{ {\Sigma } n_{i}}\);

  • If yom = yo3 and \(1\geq x\frac {\pi _{3}\lambda _{3}}{\left (1+\pi _{3}\right ) }{\Xi } \),

    • if \(\hat {y}_{\text {om}}=\hat {y}_{o1}\), then \(p_{o}=\hat {p}_{o}= 0\);

    • if \(\hat {y}_{\text {om}}=\hat {y}_{o2}\) and if \(\frac {\theta (1-\alpha )}{\pi _{3}g}\geq \frac {\text {xn}_{1}\lambda _{2}\alpha }{\left (n_{2}\lambda _{2}+n_{3}\lambda _{3}\right ) }\), then \(p_{o}=\hat {p}_{o}= 0;\)

    • if \(\hat {y}_{\text {om}}=\hat {y}_{o2}\) and if \(\frac {\theta (1-\alpha )}{\pi _{3}g}<\frac {\text {xn}_{1}\lambda _{2}\alpha }{\left (n_{2}\lambda _{2}+n_{3}\lambda _{3}\right ) }\leq \frac {n_{1}\lambda _{3}\alpha }{\left (n_{2}\lambda _{2}+n_{3}\lambda _{3}\right ) }\), then \(p_{o}= 0<\hat {p}_{o}= \frac {n_{1}}{{\Sigma } n_{i}};\)

    • if \(\hat {y}_{\text {om}}=\hat {y}_{o3}\) and \(\frac {\theta (1-\alpha )}{\pi _{3}g }\geq \frac {\text {xn}_{1}\lambda _{3}\alpha }{\left (n_{2}\lambda _{2}+n_{3}\lambda _{3}\right ) }\), then \(p_{o}=\hat {p}_{o}= 0;\)

    • if \(\hat {y}_{\text {om}}=\hat {y}_{o3}\) and \(\frac {\theta (1-\alpha )}{\pi _{3}g }<\frac {\text {xn}_{1}\lambda _{3}\alpha }{\left (n_{2}\lambda _{2}+n_{3}\lambda _{3}\right ) }\leq \frac {n_{1}\lambda _{2}\alpha }{\left (n_{2}\lambda _{2}+n_{3}\lambda _{3}\right ) }\), then \(p_{o}= 0<\hat {p}_{o}=\frac {n_{1}}{ {\Sigma } n_{i}};\)

    • if \(\hat {y}_{\text {om}}=\hat {y}_{o3}\) and \(\frac {\theta (1-\alpha )}{\pi _{3}g }<\frac {n_{1}\lambda _{2}\alpha }{\left (n_{2}\lambda _{2}+n_{3}\lambda _{3}\right ) }<\frac {\text {xn}_{1}\lambda _{3}\alpha }{\left (n_{2}\lambda _{2}+n_{3}\lambda _{3}\right ) }\), then \(p_{o}= 0<\hat {p}_{o}=\frac {n_{1}+n_{2} }{{\Sigma } n_{i}}\).

A.6 Calibration

In order to fit the data for population size from the Human Mortality Database (2016), the parameter gt is calibrated as follows.

The skills parameters λit are calibrated in such a way as to be as close as possible to the headcount poverty rates in Ravallion (2016) over 1820–1995.

Table 2 Calibration of survival probabilities
Table 3 Calibration of cohort growth rate
Table 4 Calibration of parameters λit
Table 5 Calibration of TFP parameter Bt

Starting from an initial k1820 fixed to 100, we can, given the calibrated values for parameters ni, λit, ϕit, πit, α, and gt, find a level of the TFP parameter Bt for each period that allows us to perfectly fit the GDP per capita pattern for each period under study. Those values are provided in the following table.

Together, the calibrated values for all production parameters allow us to obtain a perfect fit of the GDP figures in the Maddison Project (2013) (see Fig. 1) and a reasonable proxy for the headcount poverty rates in Ravallion (2016) (see Fig. 2). The reason why our fit is not perfect in the latter case is uniquely due to our reliance on a 10 income class model.

A.7 Robustness

Fig. 8
figure 8

Comparison of actual and hypothetical absolute poverty rates for the young (poverty line = $6 a day)

Fig. 9
figure 9

Comparison of actual and hypothetical absolute poverty rates for the old (poverty line = $6 a day)

Fig. 10
figure 10

Comparison of actual and hypothetical relative poverty rates for the young (poverty line = 40% of median income)

Fig. 11
figure 11

Comparison of actual and hypothetical relative poverty rates for the old (poverty line = 40% of median income)

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Lefèbvre, M., Pestieau, P. & Ponthiere, G. Premature mortality and poverty measurement in an OLG economy. J Popul Econ 32, 621–664 (2019). https://doi.org/10.1007/s00148-018-0688-x

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Keywords

  • Income-differentiated mortality
  • Poverty measures
  • Missing poor
  • OLG models
  • Capital accumulation

JEL Classification

  • E13
  • E21
  • I32