20 These and a host of analogous manipulations generate a set of counterfactual income distributions, �<sup>�</sup> ��� . For each one, counterfactual poverty and inequality measures can be computed, and compared to the initial and final distributions actually observed. As an illustration, the results for just such a set of comparisons – for the distributional change between 1976 and 1996 in Brazil – are presented in Table 1, which is taken from Ferreira and Barros (2005), one of the case studies in Bourguignon, Ferreira and Lustig (2005). The columns of Table 1 list the mean, four inequality, and three poverty measures (for two different poverty lines) for each actual and counterfactual distribution. The first two rows are for the actual distributions observed in 1976 and 1996. Each of the subsequent rows is for a counterfactual distribution, denoted by the Greek letters corresponding to the estimated coefficients that are imported from 1996 into 1976 in each case.

But just as the counterfactual distributions can be used to compute scalar summary measures of inequality and poverty, they can also be used to construct quantile functions, and to compute the GIC decomposition in equation (13). Four such “full distribution” decompositions are depicted, for the same study of Brazil between 1976 and 1996, in Figure 2.

In this figure, the solid dark line is an approximation of the actual growth incidence curve between 1976 and 1996.<sup>22</sup> Each of the other curves, identified by a set of Greek letters, corresponds to the counterfactual GIC obtained from comparing a counterfactual income distribution with the actual 1976 distribution. In each case, the counterfactual income distribution was constructed by importing a set of parameters (denoted by the corresponding Greek letters) from a model estimated in 1996, to the 1976 model, and micro‐simulating the resulting changes. So in Figure 2, for example, the line denoted “αs and βs” gives the counterfactual growth incidence attributable only to changes in the returns in all Mincer equations. The next line, which also includes γs, combines those changes in returns with changes in occupational choices (estimated using multinomial logits). The line that also includes ���� further incorporates behavioral changes associated with fertility choices. Finally, the line that includes “ ����, ���� , αs, βs, and γs”, adds the changes in educational attainment, from a multinomial logit that estimates the conditional distribution of years of schooling on a set of exogenous variables such as race, age, gender, and region of residence. Critically, when the GIC represents a _combination_ of effects, as in each of these cases, they are made internally consistent. For example, if a person in the 1976 sample is ascribed a new level of education by the simulation arising from importing the ���� parameters, then that new level of education carries through to the person’s new occupational choice, decision on fertility, and earnings determination.

In the Brazil case study which I have used to illustrate this approach, the decompositions summarized in Table 1 and in Figure 2 suggest that three main economic forces combine to explain changes in income levels, poverty and inequality – i.e. in Brazil’s “poverty‐growth‐inequality triangle” – between 1976 and 1996. The first is a shift in the occupational structure towards greater unemployment and informality, and towards fewer hours worked within the informal sector by less educated workers. This effect was responsible for the pronounced income losses in the first decile of the distribution, and the

> 22 The writing preceded the definition of GICs, and the authors calculated the log differences in incomes for each percentile, which approximates their growth rates.