# as _IDθa_ ˆ and _GDθa_ ˆ.

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

# _Proof._ See Appendix D.

We now discuss more systematically the differences between _EDθa_ ˆ and _GDθa_ ˆ. _EDθa_ ˆ and _GDθa_ ˆ rank non-stationary societies differently. The main difference between _EDθa_ ˆ and _GDθa_ ˆ comes from the way the two indices account for the number of years prematurely lost. _GDθa_ ˆ records the number of years prematurely lost over all premature deaths actually taking place in year _t_ . _EDθa_ ˆ also counts the number of years prematurely lost but, instead of being computed on the actual population pyramid, _EDθa_ ˆ uses a counterfactual population pyramid, which is the one that would prevail in a stationary society characterized by the age-specific mortality rates observed in the period.

A major implication of this difference is that _EDθa_ ˆ is more reactive to policy changes than _GDθa_ ˆ. Consider a permanent mortality shock. The population dynamics is such that a transition phase sets in during which the population pyramid slowly adjusts to the new mortality rates. This transition stops when a new stationary population pyramid is reached, typically after _a_<sup>_∗_</sup> periods. _GDθa_ ˆ records each step of this transition and therefore exhibits inertia in its response to a permanent mortality shock.<sup>20</sup> By contrast, _EDθa_ ˆ immediately refers to the new stationary population pyramid and disregards the inertia caused by these transitory demographic adjustments. We provide an illustration of this property in Appendix E.

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

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

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

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

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

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