First, the dimension should be important for human well-being. Second, a significant fraction of the population should have low outcomes in the dimension.<sup>38</sup> Whether a particular dimension satisfies these requirements is an empirical question. Yet, there is little doubt that it is typically the case for the monetary dimension and for the health dimension.

Assume that the practitioner faces no data constraints. Hence, there is a survey that collects individual outcomes on both the monetary and the health dimensions. We show that an identification method combining information on both dimensions identifies the poor better than an identification method that ignores one of these two dimensions. This result is a straightforward implication of our theory. This result may in fact appear trivial, but there are substantial debates on whether monetary poverty should be combined with non-monetary poverty. For instance, Atkinson (2016) does not recommend the World Bank to track a multidimensional poverty measure that combine both.<sup>39</sup> The dominant practice seems to be to keep monetary poverty measures separated from non-monetary poverty measures. Notable exceptions are the MPM of the World Bank (see definition in Section 7.1) and the official multidimensional poverty measure adopted in Mexico (CONEVAL, 2010).<sup>40</sup> Formally, let x1 denote health achievements, as measured by an ordinal variable. Let x2 denote monetary achievements as measured by the total monetary value of consumption on market dimensions (food, non-food, etc). Assume at this stage that individuals cannot increase their health outcomes by consuming specialized market goods (e.g., drugs do not exist). As we explain below, this assumption is not necessary for our next result. We say that an LIM ignores dimension j ∈{1, 2} if this LIM attributes a weight w = 0. j Any LIM that ignores monetary outcomes identifies the poor worse than some LIM that accounts for both dimensions. Indeed, the LIM that ignores monetary outcomes identifies the (multidimensionally) poor simply by selecting some health deprivation cutoff z1. Provided that the two dimensions are not perfectly correlated, this LIM makes exclusion errors on some extremors. In particular, it makes exclusion errors on individuals whose health achievement is above z1 but whose monetary achievement is below the extreme monetary threshold e2. Consider the union-LIM with the same health deprivation cutoff z1 and whose monetary deprivation cutoff is equal

> 38 A dimension for which virtually no-one has low achievement cannot be a major source of well-being losses. For instance, many peasants in medieval Europe where subjected to serfdom, a condition of servitude to the local Lord. There is little doubt that being subjected to serfdom drastically reduces a person’s well-being. However, including a freedom-from-serfdom dimension would make little sense when measuring multidimensional poverty in a society where serfdom has been abolished or never existed. In contrast, such dimension would be highly relevant in a society where serfdom is widespread.

> 39 See p 170 in Atkinson (2016): “It is not proposed that the [multidimensional] indicator should include a monetary poverty dimension. [. . . ] The aim of Recommendations 18 and 19 is to provide indicators that complement the monetary indicator, and not to seek to combine the two different approaches.”

> 40Other countries that include a monetary dimension in their national multidimensional poverty measure can be found in Tables A.2 and A.3 in Unicef et al. (2021).

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