Measuring Inequality - DARP

1.3. INEQUALITY MEASUREMENT, JUSTICE AND POVERTY 7it by assuming that a given whole is to be shared as a number of equal orunequal parts. For some descriptions of inequality this assumption is irrelevant.However, since the size of the cake as well as its distribution is very importantin social welfare theory, we shall consider the relationship between inequalityand total income in Chapter 3 (particularly page 47), and examine the practicalimplications of a growing –or dwindling –cake in Chapter 5 (see page 139.)1.3 Inequality measurement, justice and povertySo what is meant by an inequality measure? In order to introduce this devicewhich serves as the third “ingredient”mentioned previously, let us try a simplede…nition which roughly summarises the common usage of the term: a scalar numerical representation of the interpersonal di¤erences in incomewithin a given population.Now let us take this bland statement apart.Scalar InequalityThe use of the word “scalar”implies that all the di¤erent features of inequalityare compressed into a single number –or a single point on a scale. Appealingarguments can be produced against the contraction of information involved inthis aggregation procedure. Should we don this one-dimensional straitjacketwhen surely our brains are well-developed enough to cope with more than onenumber at a time? There are three points in reply here.Firstly, if we want a multi-number representation of inequality, we can easilyarrange this by using a variety of indices each capturing a di¤erent characteristicof the social state, and each possessing attractive properties as a yardstick ofinequality in its own right. We shall see some practical examples (in Chapters3 and 5) where we do exactly that.Secondly, however, we often want to answer a question like “has inequalityincreased or decreased?” with a straight “yes” or “no.” But if we make theconcept of inequality multi-dimensional we greatly increase the possibility ofcoming up with ambiguous answers. For example, suppose we represent inequalityby two numbers, each describing a di¤erent aspect of inequality of thesame “income”attribute. We may depict this as a point such as B in Figure 1.1,which reveals that there is an amount I 1 of type-1 inequality, and I 2 of type-2inequality. Obviously all points like C represent states of society that are moreunequal than B and points such as A represent less unequal states. But it ismuch harder to compare B and D or to compare B and E. If we attempt toresolve this di¢ culty, we will …nd that we are e¤ectively using a single-numberrepresentation of inequality after all.Third, multi-number representations of income distributions may well havetheir place alongside a standard scalar inequality measure. As we shall see inlater chapters, even if a single agreed number scale (I 1 or I 2 ) is unavailable, or