Measuring Inequality - DARP

1.3. INEQUALITY MEASUREMENT, JUSTICE AND POVERTY 9even if a collection of such scales (I 1 and I 2 ) cannot be found, we might be ableto agree on an inequality ranking. This is a situation where –although you maynot be able to order or to sort the income distributions uniquely (most equalat the bottom, most unequal at the top) – you nevertheless …nd that you canarrange them in a pattern that enables you to get a fairly useful picture of whatis going on. To get the idea, have a look at Figure 1.2. We might …nd that overa period of time the complex changes in the relevant income distribution canbe represented schematically as in the league table illustrated there: you cansay that inequality went down from 1980 to 1985, and went up from 1985 toeither 1990 or 1992; but you cannot say whether inequality went up or down inthe early nineties. Although this method of looking at inequality is not decisivein terms of every possible comparison of distributions, it could still providevaluable information.Numerical RepresentationWhat interpretation should be placed on the phrase “numerical representation”in the de…nition of an inequality measure? The answer to this depends onwhether we are interested in just the ordering properties of an inequality measureor in the actual size of the index and of changes in the index.I 1 I 2 I 3 I 4A :10 :13 :24 :12B :25 :26 :60 :16C :30 :34 :72 :20D :40 :10 :96 :22Table 1.1: Four inequality scalesTo see this, look at the following example. Imagine four di¤erent social statesA; B; C; D, and four rival inequality measures I 1 , I 2 , I 3 , I 4 . The …rst column inTable 1.1 gives the values of the …rst measure, I 1 , realised in each of the foursituations. Are any of the other candidates equivalent to I 1 ? Notice that I 3has a strong claim in this regard. Not only does it rank A; B; C; D in the sameorder, it also shows that the percentage change in inequality in going from onestate to another is the same as if we use the I 1 scale. If this is true for all socialstates, we will call I 1 and I 3 cardinally equivalent. More formally, I 1 and I 3 arecardinally equivalent if one scale can be obtained from the other multiplying bya positive constant and adding or subtracting another constant. In the abovecase, we multiply I 1 by 2:4 and add on zero to get I 3 . Now consider I 4 : itranks the four states A to D in the same order as I 1 , but it does not give thesame percentage di¤erences (compare the gaps between A and B and betweenB and C). So I 1 and I 4 are certainly not cardinally equivalent. However, if itis true that I 1 and I 4 always rank any set of social states in the same order,we will say that the two scales are ordinally equivalent. 1 Obviously cardinal1 A mathematical note: I 1 and I 4 are ordinally equivalent if one may be written as a