Colombia, and Costa Rica. Both reach a similar lower bound estimate of $10 per person per day in 2005 PPPs.

The middle-class upper threshold does not follow a thorough definition or computation as the lower bound. Birdsall et al. (2011) show that varying the upper threshold from $50 to $100 a day would move the percentile of the LAC’s elite from the top 2.2 percent to the top 0.5 percent. Thus, both define the upper threshold as $50 per person per day in 2005 PPP which is equivalent to the top 2.2 percent of the population.

More recently, when 2011 PPPs were published, a very simple approach was implemented without applying previous methodologies.<sup>8</sup> Firstly, using each country's 2005 PPP conversion factor, the vulnerable and middle-class lines were converted to local currency units at 2005 prices. Secondly, these values were deflated to 2011 prices using each country's Consumer Price Index (CPI) and converted back to US dollars using their corresponding 2011 PPP conversion factors. Finally, a simple average of the resulting lines was estimated to obtain a regional value. By rounding to the closest unit, the lower and upper middle-class lines in 2011 PPP for LAC were then set at $13 and $70 a day, respectively.

The proposed approach estimates the LAC lower middle-class bound (vulnerability line) in two steps. First, it builds a two-year synthetic panel for each country using every pair of available cross-section years between 2010 and 2019. Then, the vulnerability line is defined as the median per capita household income over households who were not poor in the initial year and became poor in the final year, i.e., a household in moves into poverty if and , where is the international poverty line of 6.85 a day in purchasing power parities (PPP) 2017. ii2 ii1 ii jj = 2 yy < zz yy� > zz zz There are a few reasons why it might be useful to define the lower bound of the middle class as the median of the income distribution of people who have recently moved out of poverty. First, the median income represents the midpoint of the income distribution, so it is less affected by extreme values that could skew the results. This can help to ensure that the middle-class thresholds are fair and representative of the income needs of a typical household.

A second advantage of using the median income as the lower bound of the middle class (upper bound of the vulnerable group) is that it highlights the fact that the vulnerable population is not static, but rather constantly changing. Additionally, using a more reliable statistic like the median allows for a more precise estimate of the minimum income needed to be considered middle class. It is important to note that it does not imply that the probability-based approach (Lopez-Calva & Ortiz-Juarez, 2014) fails to fulfill these conditions; both methods acknowledge the dynamic nature of the vulnerable population.

Finally, employing the median income as the lower bound of the middle class can contribute to making the middle-class thresholds more responsive to economic changes and shifts in income distribution. As the economy grows and more people rise out of poverty, the middle-class thresholds can be adjusted accordingly to reflect these changes. This responsiveness allows for a more inclusive definition of both the vulnerable and middle-class populations, preventing the exclusion of individuals who are making

> 8 World Bank (2021).

8