2.8 Clark-Hemming-Ulph class of poverty measures (svychu)

Clark, Hemming, and Ulph (1981)Clark, Stephen, Richard Hemming, and David Ulph. 1981. “On Indices for the Measurement of Poverty.” The Economic Journal 91 (June). https://doi.org/10.2307/2232600. proposes two classes of distribution-sensitive poverty measures. Yet, the poverty measurement literature focuses on the second class4 See Clark, Hemming, and Ulph (1981)Clark, Stephen, Richard Hemming, and David Ulph. 1981. “On Indices for the Measurement of Poverty.” The Economic Journal 91 (June). https://doi.org/10.2307/2232600. and Clark, Hemming, and Ulph (1981)Clark, Stephen, Richard Hemming, and David Ulph. 1981. “On Indices for the Measurement of Poverty.” The Economic Journal 91 (June). https://doi.org/10.2307/2232600., for instance., expressed as

\[ CHU_\alpha = \begin{cases} \frac{1}{\alpha N} \sum_{i \in U} \big[ 1-(y_i/\theta)^\alpha \big] \cdot \delta ( y_i \leqslant \theta ) , & \alpha \leqslant 1 , \alpha \neq 0 \\ 1 - \bigg( \prod_{i \in U} y_i^{\delta ( y_i \leqslant \theta )} \bigg)^{1/N} \bigg/ \theta , & \alpha = 0 \end{cases} \]

As an special case, \(CHU_0 = 1 - \exp{(-Watts)}\). The \(\alpha\) parameter defines the sensivity towards regressive income transfers among the poor, such that the lower its value, larger is the regressive transfer impact on the index. When \(\alpha \rightarrow 1\), \(CHU_1 = FGT_0 \cdot I\), a measure insensitive to regressive income transfers among the poor.