We consider now the implication of the escaping-poverty property when the value of median income _<u>y</u>_ varies. The key observation is that self-centered individuals escape welfare poverty when their income surpasses the subsistence income, even if median income is simultaneously increased. For instance, in Figure 2, the self-centered individual 1 is absolutely income poor in distribution **y** , and thus welfare poor. In contrast, individual 1 is only-relatively income poor in distribution **y**<sup>_′′_</sup> , and thus not welfare poor. Hence, the escaping-poverty property is satisfied only if any only-relatively income poor individual is attributed a strictly smaller poverty score than any absolutely income poor individual, even if they live in societies with different values of median income. The view that only-relatively income poor individuals should always be attributed a smaller poverty score than absolutely income poor individuals seems to be largely shared, as survey evidence suggests (Decerf and Ferrando, 2022). We have thus shown that the escaping-poverty property provides a preference-based foundation for this view.

This second implication of the escaping-poverty property is violated by the societal headcount ratio but satisfied by the hierarchical headcount ratio. Index _HS_ violates it because _HS_ attributes a poverty score equal to one to all income poor individuals. Index _HHS_ satisfies it because _HHS_ attributes a poverty score equal to one to absolutely income poor individuals and a smaller poverty score to only-relatively income poor individuals.

We conclude this section with three remarks. First, index _HHS_ need not always satisfy the escaping-poverty property. Indeed, _HHS_ violates that property if some absolutely income poor individuals are not welfare poor.<sup>16</sup> Thus, even if index _HHS_ comes closer than index _HS_ to satisfying the escaping-poverty property, it does not fully do so.<sup>17</sup> Unfortunately, this is the price to pay for considering additive indices that do not treat individuals differently based on their sensitivity to relative income. No index defined by Equation (2) fully satisfies the escapingpoverty property.<sup>18</sup> Importantly, the interpersonal comparisons across different

> 16This could, for instance, be the case for an individual with preference _uσ_  ̄ when the value of median income is smaller than the reference median income _<u>y</u>_<sup>_~~z~~_</sup> . Such individual could escape welfare poverty without changing her absolute income poverty status, while index _HHS_ attributes a poverty score equal to one to all absolutely income poor individuals.

> 17As we show in online Appendix S3, _HHS_ satisfies a weak version of the escaping-poverty property that _HS_ violates.

> 18Even the expected fraction of poor EH( **y** ) violates the second implication of the escapingpoverty property when some absolutely income poor individuals are not welfare poor under some preference _u_<sup>_σ_</sup> . Indeed, one can then find two bundles, ( _y,_ _<u>y</u>_ <u>)</u> and ( _y_<sup>_′_</sup> _,_ _<u>y</u>_<sup>_~~′~~_</sup> ), with _y < za < y_<sup>_′_</sup> and _<u>y</u> <_ _<u>y</u>_<sup>_~~z~~_</sup> _<_ _<u>y</u>_<sup>_~~′~~_</sup> such that a self-centered individual is welfare poor in ( _y,_ _<u>y</u>_ <u>)</u> but not in ( _y_<sup>_′_</sup> _,_ _<u>y</u>_<sup>_~~′~~_</sup> ), whereas an individual with preference _u_<sup>_σ_</sup> is welfare poor in ( _y_<sup>_′_</sup> _,_ _<u>y</u>_<sup>_~~′~~_</sup> ) but not in ( _y,_ _<u>y</u>_ <u>).</u>

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