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Abstract
The Human Development Index, which is multidimensional by construction, is criticized on the ground that it is insensitive to any form of inequality across persons. Inequality in the multidimensional context can take two distinct forms. The first pertains to the spread of the distribution across persons, analogous to unidimensional inequality. The second, in contrast, deals with interactions among dimensions. The second form of inequality is important as dimensional interactions may alter individual level evaluations as well as overall inequality. Recently proposed indices have incorporated only the first form of inequality, but not the second. It is an important omission. This paper proposes a two‐parameter class of Human Development Indices that reflects sensitivity to both forms of inequality. It is revealed how consideration of interactions among dimensions affects policy recommendations. Finally, the indices are applied to the year 2000 Mexican census data to contrast the present approach with the existing approaches.
Acknowledgements
All opinion and errors in this paper are mine. I am thankful to Professor James Foster for his valuable comments, to Luis Felipe Lopez‐Calva for providing access to the year 2000 Mexican census data, and to the UNDP/RBLAC for financial support. This article is based on a background paper for the 2009 Latin America and the Caribbean Regional Human Development Report. I also appreciate the valuable comments from the two unknown referees that helped to improve this paper.
Notes
1 A region can be a country, a state, or even a society.
2 For the purpose of this paper, we assume the terms correlation and association as synonymous. The term association is broader; whereas the term correlation implies Pearson’s product moment correlation, in general.
3 For detail calculation please see the technical note on page 394 of the Human development Report, 2006.
4 A permutation matrix is a square matrix where each row and column have exactly one element equal to one, while the other elements are equal to zero. An identity matrix is a special type of permutation matrix.
5 The subgroup consistency axiom that we discuss here is ‘population subgroup consistency’ and not ‘dimension subgroup consistency’ (see Foster and Shorrocks, Citation1991).
6 Note that the concept of path independence in the single‐dimensional context is different from this concept of multidimensional path independence.
7 This type of transfer is also called progressive transfer.
8 To see the relation between these two definitions in the single‐dimensional and the multidimensional context, see Weymark (Citation2006).
9 A bistochastic matrix is a non‐negative square matrix whose row sum and column sum are both equal to one. Thus, a permutation matrix is always a bistochastic matrix, by definition, but the reverse is not true.
10 If there are two distributions with the same mean and the same population size, and one is less dispersed than the other, then the former is more equal than the latter, yielding higher human development. These two distributions need not necessarily be obtained from each other resulting from direct transfers.
11 This approach has been applied by Bourguignon (Citation1999), Tsui (Citation1999, 2002), and Alkire and Foster (Citation2007). Bourguignon and Chakravarty (Citation2003) used an almost similar concept called correlation‐increasing switch in the two‐dimensional context.
12 The idea behind the theorem is analogous to Proposition 1 and Proposition 2 in Pattanaik et al. (Citation2007). Note that the Non‐Invariance axiom in Pattanaik et al. (Citation2007) is not identical to the concept of association increasing transfer that motivates axioms SDIA and SIIA.
13 For α = 0 and β = 0, the proposed indices take the corresponding geometric mean forms.
14 The above formulation can be generalized to consider unequal weights.
15 Generalized mean µγ(x) is both concave and quasi‐concave for γ ≤ 1. Similarly, µγ(x) is both strictly concave and strictly quasi‐concave for γ ≤ 1.
16 For detailed proofs, see Seth (Citation2009).
17 The original welfare index by Bourguignon (Citation1999) was defined using any arbitrary weights on dimensions. However, here we consider the equal weight version of the index without loss of generality.
18 Following the methodology of the UNDP, we take the logarithm of income and, accordingly, restrict the lower bound of income to a positive value.
19 This value is equivalent to US$40 000 that is applied by the UNDP as an upper limit of per‐capita‐GDP. We use a deflator from the Human Development Report, 2002.
20 For the households having no person between 6 and 24 years of age, the literacy rate receives a weight equal to one; whereas for the households having no person older than 14 years of age, the enrolment rate receives a weight equal to one.
21 Child mortality is measured as the number of children not surviving per 1000 births.