using equal weights.

We call “objective” the indices that are constructed using the joint distributions of outcomes. Objective indices first aggregate across dimensions at the individual level to ˆ generate an individual index u(di). Then, the individual indices are aggregated across individuals, typically by averaging them. This is for instance the case of the two main global multidimensional poverty indices: the global MPI of UNDP-OPHI and the World Bank’s MPM, both defined in Appendix 6.1. These indices generate their individual indices uˆ(di) following the Alkire-Foster aggregation methodology (Alkire and Foster, 2011a). This methodology makes a weighted sum of dichotomous deprivation statuses in a number of dimension-specific variables (see Section 3.2).

From a conceptual perspective, objective indices are more appealing than synthetic indices because the former account for well-being inequalities across individuals (Fleurbaey, 2009). Synthetic indices ignore that some individuals cumulate low achievements in several dimensions. Moreover, synthetic indices do not generate an individual index uˆ(di) and can therefore not be used for inter-personal well-being comparisons. However, synthetic indices can be computed even when the joint distribution is missing.

Consider then the columns in Table 1. In absence of data on preference u, the typical practice consists in using ad-hoc methods to set weights, like the popular “equal-weights” procedure.<sup>12</sup> The risk when resorting to this ad-hoc procedure is to select weights that are not plausible, which casts doubt on the added-value of the multi-dimensional indicator (Ravallion, 2012). Going beyond ad-hoc methods may help increase the confidence in well-being comparisons obtained from multidimensional indicators. This requires selecting the values of weights from some empirical source, like dedicated preference-elicitation surveys (willingness-to-pay, stated preferences, etc.), subjective well-being data or external sources in the literature. For instance, the dimension-specific statistics feeding synthetic indices are sometimes aggregated using a parametric utility function. Its parameters can be calibrated using empirical values taken from specialized literatures. For the trade-off between income and longevity, one can draw on empirical estimates for the elasticity of inter-temporal substitution and the value of a statistical life (see section 3.3).

> 12 In the case of multidimensional poverty measures, the equal-weights procedure is more nuanced than what its name may suggest. The reason is that, in practice, each dimension is captured by several dimension-specific variables. The equal-weights practice attributes equal weights between dimensions and also attributes equal weights among the variables capturing the same dimension. Hence, the weight 1 1 received by each variable is given by ndim<sup>∗</sup> nvar<sup>wherendimdenotesthenumberofdimensionsand</sup> nvar denotes the number of variables capturing the dimension. For instance, the global MPI has three dimensions: education, health and living standards. Health is captured by two variables (nutrition and child mortality) that each receives a weight of 6<sup>1=</sup> 3<sup>1∗</sup> 2<sup>1.Livingstandardsiscapturedbysixvariables</sup> that each receives a weight of 181<sup>.</sup>

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