Abstract
A very common criticism of globalisation is that it has led to an increase in inequality. To check the validity of the criticism, one would need an inequality index. Presumably, inequality here means inequality in the distribution of well-being. However, well-being is a multidimensional concept. Hence, we need a multidimensional inequality index. Moreover, since in the unidimensional context, the most widely used inequality index is the Gini index, and it is natural to want to use a multidimensional Gini index (MGI) for the purpose. However, while the unidimensional Gini is uniquely defined, an MGI is not. The existing literature contains various suggestions as to how an MGI can be defined. Unfortunately, none of the suggested MGIs seems to have all the properties that one would, on intuitive grounds, expect such an index to possess. Moreover, some of the suggested MGIs are mathematically quite complex. In this paper, we first mathematically characterise a specific MGI and then show that it possesses all the intuitively expected properties. It is also seen to be very simple and transparent in nature. It is hoped that while this paper is theoretical, the derived index can be used by empirical researchers to investigate the effect of globalisation on inequality somewhat more satisfactorily than has hitherto been the case.
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Notes
- 1.
In fact, so far as inequality indices are concerned, there is an embarrassment of riches because there are an infinite number of such indices. See Sen (1997).
- 2.
Actually, in place of what we have called IAP, Huntington (1927) used a condition which is slightly weaker, requiring only that, for any K in K, m(K) = m(L) if L is obtained by replacing each of the first two members of K by their average. It is easily seen, however, that in the presence of PI the weaker version is equivalent to IAP as stated in the text.
- 3.
Attempts at characterisation of various types of averaging functions have by now a history of more than a hundred years. The first such attempt was by Schimmack (1909) who proved that a number of conditions on the averaging function uniquely identify the arithmetic mean. The independence of Schimmack’s postulates was proved by Beetle (1915). For two reasons, however, we have preferred to use the Lemma which, as mentioned in the text, is due to Huntington (1927). First, the earlier literature which was in the German language may not be accessible to all readers. Secondly (and more importantly), Huntington’s postulates were somewhat different from those used by Schimmack. In fact, he gave several different sets of postulates all of which characterise the arithmetic mean (and, therefore, all of which are logically equivalent to each other). The Lemma stated in the text is based on one of his sets of postulates. Our choice of this particular set was motivated by the fact that all the postulates in the set, including the one that we called AC, seem to be natural requirements of the average in our specific context. (Huntington, however, did not use the names and the abbreviations given by us to the conditions in the text.) The curious reader will also find in Huntington’s paper several alternative characterisations of each of three other types of average: the geometric mean, the harmonic mean and the root-mean-square.
- 4.
It may be noted that what Gajdos and Weymark (2005) seek to extend to the multidimensional context is not exactly the classical unidimensional Gini index but a set of indices (called “generalised Gini indices”) that includes the classical Gini. When q = 1, the multidimensional Gajdos-Weymark class reduces to the unidimensioal “generalised Gini” class.
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Banerjee, A.K. (2019). Measuring Multidimensional Inequality: A Gini Index. In: Chakrabarti, G., Sen, C. (eds) The Globalization Conundrum—Dark Clouds behind the Silver Lining. Springer, Singapore. https://doi.org/10.1007/978-981-13-1727-9_5
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