where is the household per capita income in logarithmic scale at the initial time point. The resulting coefficients from equation (2) and the average of the independent variables are used to solve the income equation, and therefore to obtain the amount of income associated to each probability.

The resulting incomes look quite similar for all three countries; non-poor individuals with a ten percent probability of falling into poverty have income levels at $8.5 a day in Chile, $9.7 in Mexico, and $9.6 a day in Peru, all of them measured using the 2005 PPP (see figure 4).<sup>6</sup>

**Figure 4: Daily income by probabilities of falling into poverty; Chile, Mexico and Peru** _Household per capita income at 2005 PPP_

<!-- Start of picture text --> 20.0<br>18.0<br>16.0<br>14.0<br>12.0<br>10.0 9.7<br>9.6<br>8.0 8.5<br>6.0<br>4.0<br>2.0<br>0.10<br>0.0<br>.00-.02 .04-.06 .08-.10 .10-.12 .14-.16 .18-.20 .22-.24 .26-.28 .30-.40 .50-.60<br>Probabilities of fall into poverty<br>Chile Mexico Peru<br>Per capita income a day (at 2005 PPP)<br><!-- End of picture text --> _Source_ : Author‘s calculations based on data from CASEN, MxFLS and ENAHO panel databases.

These figures result from the complete specifications of the models shown in annex (table A4); however, they are robust to different specifications (see table 4). As the middle class, ideally, should consist of those households facing a very low risk of falling into poverty over time, we define the income associated to that probability as the lower threshold that depicts the lower bound of the middle class.<sup>7</sup> While there is a strong association between income and vulnerability, it remains nevertheless difficult to anchor a threshold solely to vulnerability, since the curves do not suggest structural behavioral changes. In the case of the upper threshold we define it as $50 in PPP terms, being it an income amount that lies in the upper tail of the income distribution of all three countries.

> 6 Estimated as the average of the independent variables for an estimated probability range between 0.09 and 0.11. Tables A1-A3 in the annex show the coefficients and standard errors from equations (1) and (2) for Chile, Mexico, and Peru, respectively, while Figure A1 shows the correlation between the estimated probabilities from equation (1) and the fitted values from equation (2).

> 7 In order to not classify as middle class a number of lower class households if any of these incomes is used, we establish for strict comparability purposes a more demanding criterion of $10 a day. Of course, nothing comes without a trade-off. The use of the proposed threshold of $10 implies, conversely, that a number of middle class households in each country will be considered as lower class. For instance, if middle class in Mexico includes those households with per capita income at or above $9.7 a day, hence a 1.3 percent of these households will be lower class under the criterion of $10 (in Chile 8.2 percent of households are between $8.5 and $10, while in Peru 1.3 percent are between $9.6 and $10). However, we consider more reasonable to include a lower proportion of middle class households as part of the lower class, instead of the opposite situation.

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