CES function, generalised mean and human poverty index: Exploring some links

The Sennian capability approach has facilitated to capture poverty in its multi-dimensional incidence and thus to raise a new aggregate poverty index - the UNDP's Human Poverty Index (HPI). The UNDP has found power mean of order > 1 as possessing some of the most desirable properties in describing the distribution of deprivation dimensions and hence as the most appropriate aggregate index of multi-dimensional deprivation. The UNDP elevates power mean of order > 1 (PM) in comparison with arithmetic mean (AM) commonly used for averaging, leaving out others. It would hence be worthwhile to look into the links among the means, both the known and the potential ones, and their strengths and weaknesses in terms of their properties in comparison with each other. The present paper is a preliminary attempt at this. We find that the means we commonly use, the AM, the geometric mean (GM) and the harmonic mean (HM), along with the PM, are special cases of the CES function. We acknowledge the possibility of an inverse CES function, and hence, that of an inverse power mean (IPM) also. Among these means, the AM is an average, typical of all the components, but its infinite elasticity of substitution renders it less desirable. To the extent that we need an average, typical of the components, we seek for one that is closer to the AM, so that this second best choice will have the minimum deviations next to the AM. And we find this basic criterion is satisfied by the IPM only. Hence, while the PM captures the multi-dimensional deprivation, its inverse, the IPM, seems to offer a multi-dimensional development index.