increase proportionally more at the top than at the bottom of the distribution (and the mean of the well-being indicator would increase). Nothing can be stated a priori regarding inequality under scenario (iii). If rent were a growing (decreasing) share of individuals’ welfare, inequality would increase (decrease). Yet, if there is no linear relation between the original welfare distribution and rents, the distributional impact of imputed rents cannot be predicted ex-ante.<sup>5</sup> For example, if rents are higher for some specific subset of the population, such as individuals living in urban dwellings, individuals belonging to that subgroup may also present higher rent shares relative other subpopulations _._ The effect of adding rents on inequality would not be straightforward in this case.

The distributional impact of imputed rents on poverty is assessed analyzing changes in poverty incidence, depth and severity. Recall that poverty measurement involves three steps: defining a welfare indicator (usually consumption or income), setting a poverty line, and finally aggregating all information into a single poverty indicator. Recall, moreover, that the poverty line is usually defined as a function of the welfare indicator itself. This holds true not just in case of a relative poverty line, e.g. 60 percent of median income, but also in case of an absolute poverty line. An absolute poverty line, e.g. based on the cost of basic needs approach, is generally computed in two steps, first by identifying a food poverty line and then by adding to it a non-food poverty line. Seeking simplicity, we can think of the non-food component as determined residually by inflating the food poverty line with the ratio of non-food consumption to food consumption for the reference population.<sup>6</sup> Therefore, if the welfare distribution changes, it should be good practice, whenever possible, to re-estimate the poverty line accordingly. Trivially, if rents are included in the welfare aggregate, but the poverty line is left unchanged, poverty will decrease. When both the poverty line and the original welfare distribution increase as a consequence of including rents in the welfare aggregate, the effect on poverty becomes unpredictable, unless the poverty line is relative, and rents are modeled following scenario (ii). In this case, given that the poverty indices used in this analysis satisfy scale invariance, poverty will not change.

The third indicator of well-being taken into account in this paper is shared prosperity. In 2013, the World Bank adopted two new targets: ending extreme poverty by 2030 and promoting shared prosperity. The emphasis on reducing extreme poverty is in line with the commitment to eliminate extreme poverty enshrined in the first target of the Sustainable Development Goals, which were adopted by the United Nations in 2015 (United Nations 2015). Shared prosperity focuses on the less well-off more broadly. It is defined as the annualized growth rate of the average welfare of the bottom 40 percent of the population between two years and , where stands for the number of years between two data surveys. yy yy+ ττ Let us define the average welfare aggregate at timeττ , for the bottom 40 percent of the population as<sup>,</sup> Monitoring shared prosperity is challenging, as it requires building comparable welfare aggregates over time. For our purposes, the three scenarios we introduced before are here increased to nine, as summarized in Table 1, depending on the prevalent rent-imputation scenario in each one of the observed years.

> 5 Blades (2009), for example, shows that the share of housing on total consumption is positively correlated to GDP per-capita. However, there is still no evidence that the share of housing on total consumption is correlated to the distribution of the welfare aggregate in developing countries. More evidence on this is needed, preferably using large samples of countries.

> 6 Specifically, the lower level of the total poverty line is defined based on those households whose total consumption is close to the food poverty line. On the other hand, the upper level of the poverty line happens when households whose total _food_ consumption is close to the food poverty line (Ravallion 1994). In the following we assume that the reference population for the estimation of the food poverty line does not vary under different definitions of the consumption aggregate, namely, including or not the housing component.

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