the reduction in the headcount in comparison to a distribution-neutral scenario. This is true for 23 countries, mostly in Sub-Saharan Africa, where the 2011 poverty rate is above 40%. As explained above, increasing the growth of the bottom 40% reduces the growth rate of the top 60% compared with a distribution-neutral scenario. When people at the poverty line are in the top 60% and well above 40%, this may lead to a slower reduction of the poverty rate than otherwise would have been the case. However, it is important to bear in mind that would still reduce the poverty gap and leads to faster poverty reduction in the long term. mm= 2% Take for example the case of Nigeria, which has around 60% poverty in 2011. Up until 2018, implies a slower rate of poverty reduction compared with (the distribution-neutral scenario). mm= 2% However, poverty drops sharply once the poverty rate falls to below 40%. With , poverty falls mm= 0% fast until it reaches 40% (in 2020). This poverty reduction comes entirely from the top 60%, who in this mm= −2% scenario are growing faster than the mean. This special case highlights the point that in the medium term, poverty reduction is slower with a shared prosperity premium compared with the distributionneutral growth scenario.

Figure 11 offers a more systematic assessment of how the change in the headcount rate depends on the initial headcount. This figure is drawn for the change in the first year (thus abstracting from re-ranking) and the zero-growth scenario (to abstract from differences in growth rates across countries).<sup>18</sup> It is clear that the initial level of poverty matters for the poverty impact of boosting shared prosperity. With an initial headcount below 40%, the amount of poverty reduction increases with the initial headcount. This comes from the fact the density at the poverty line is greater when the line is at the 40th percentile than when it is at a lower percentile.<sup>19</sup> Thus in the former case, the same amount of growth moves more people out of poverty leading to a faster decline in the headcount rate. This effect is stronger, the higher . Above the 40% threshold, poverty may increase because the top 60% are losing income in a scenario where the mean does not grow.<sup>20</sup> As a result, some of them would drop below the poverty line.<sup>21</sup> Again, mm the effect is stronger for higher values of . The precise effect depends on the country-specific shape and density of the distribution around the 40<sup>th</sup> percentile. mm

> 18 Another reason for focusing on one-year changes is that over time, the higher the poverty headcount, the higher the potential total reduction in poverty simply because poverty cannot be reduced below zero.

> 19 See the discussion in World Bank (2015) and especially Figure 1.4 in that report. Bourguignon (2003) offers a more formal treatment. 20 This is similar to what we observed for Nigeria when growth in the mean was positive not zero. In that case we saw a slower reduction in poverty, instead of an increase as in the zero-growth scenario.

> 21 Of course, the net effect would depend on how many of the bottom 40% become rich enough to cross the poverty line. From the figure, it seems that in most cases with a headcount above 60% there is a net increase in poverty. This is because the mode of the distribution is to the right of the 40% cut-off.

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