The Atkinson measure.

A seco n d critical p ro b lem in u sin g an d in te rp re tin g G ini coefficients, referred to as transfer sensitivity, w as d em onstrated b y A tkinson (1970) w ho

showed that changes in the Gini coefficient are sensitive to where transfers accrue. Atkinson noted that the Gini coefficient usually attaches more weight to transfers affecting middle income earners. Kakwani (1986) more specifically notes that it is transfers around the modal classes to which the Gini coefficient is most sensitive. These views do not necessarily conflict, it is a matter of recognising whether the middle income range coincides with the modal income class.

To illustrate this point, consider the income distributions shown in Table 6.2. Y° represents an initial income distribution which is concentrated around the median income. In Y1, a transfer of 10 units occurs between two individuals (B and A) who are initially separated by an income gap of 25 units, at the lower end of the distribution. As a result of this transfer, the Gini coefficient falls from 0.4363 to 0.4336, implying a reduction in inequality of 0.6%

Table 6JL Income redistributions around the median.

Distr. A B _Q D E F £ H I Gini Y° 5 30 40 45 50 65 180 200 205 0.4363 Yl 15 20 40 45 50 65 180 200 205 0.4336 Y2 5 30 50 45 50 55 180 200 205 0.4295 Y3 5 30 40 45 50 65 190 200 195 ] 0.4322

In Y2 a transfer of 10 units is again effected across a similar income gap, but this time in the middle of the distribution (between individuals F and C). As a result of this transfer, the Gini coefficient falls to 0.4295, representing a reduction in inequality of 1.6%. Thus for a transfer of a similar magnitude of that in Y1, in Y2 there is a much greater fall in the Gini coefficient and a red istrib u tio n estim ated w hich is m ore than double th at of the redistribution estimated for the transfer in the lower income range. In Y3 a similar transfer takes place at the upper end of the income distribution, betw een individuals I and G. The resultant Gini coefficient is 0.4322, representing a redistribution of 0.9%.

To overcome this bias Atkinson (1970) proposed the use of an inequality aversion param eter 'S', which w ould reflect the relative sensitivity to transfers at different income levels. As £ rises, more weight is attached to transfers at the lower end of the distribution and less weight to transfers at the top. The limiting cases for this measure are where £ —> 00 which takes account of transfers only to the very lowest income group; and at the other

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extreme, where £ = 0, which is completely insensitive to transfers and ranks distributions solely according to aggregate income.

Atkinson’s approach can be best illustrated by the following example. We start with two individuals A and B, where B has an income four times greater than A. If we are inequality averse we would approve of a redistribution of $1 from B to A with no net loss of income (ie the redistribution process is "cost-less.") In reality, the transfer process is in fact not cost-less, so that we may still approve of a transfer even if it is going to take more than $1 from B in order to give $1 to A. How much more we are prepared to take from B will be reflected in the weighting given to £. Let us suppose that we are prepared to take up to ($4)e from B in order to give $1 to A. So th a t:

£ = 0 implies a transfer of $1 from B to provide $1 to A; £ = 0.5 implies a transfer of $2 from B to provide $1 to A; £ = 1 implies a transfer of $4 from B to provide $1 to A; and £ = 2 implies a transfer of $16 from B to provide $1 to A.

If we return to the problem raised earlier in Section 6.1.1, in relation to the preferred ranking of distributions B and D, Table 6.3 shows that for higher values of £ (ie attaching greater weight to the lower end of the income distribution), the Atkinson measure reverses the inequality ranking of the Gini coefficient, so that distribution D is to be preferred to distribution B.

Table 6 3 . Atkinson's inequality aversion ranking of income distributions.

Atkinson parameter Distribution B Distribution D

0.5 0.0571 0.0655 1.0 0.1167 0.1231 1.5 0.1760 0.1714 2.0 0.2315 0.2105 5.0 0.4324 0.3222 Gini coefficient 0.256 0.272

An even more marked result is demonstrated for distributions Y° to Y3 in Tables 6.4 and 6.5, where for all values of £ > 0 the given transfer reduces inequality most when it occurs at the lower end of the distribution (ie Y1) and reduces inequality least at the upper end (ie Y3). For values of £ > 2, the transfer at the upper end of the distribution (Y3) is given such a low weighting that the impact on inequality is considered negligible and thus the inequality index is identical to the original distribution Y°.

_T ab le^ 4^ A ^ d n ^ n ^ in eq u ^ i^ av e^ io n ^ m d exford is^ b u tio n s^ ^ toY ^ . Atkinson parameter yO _{Y1 Y2 Y3} 0.5 0.1808 0.1674 0.1786 0.1805 1.0 0.3730 0.3228 0.3691 0.3728 1.5 0.5576 0.4484 0.5539 0.5574 2.0 0.7019 0.5399 0.6999 0.7019 5.0 0.9050 0.7359 0.9050 0.9050 Gini coefficient 0.4364 0.4336 0.4295 0.4322

Table 6.5. Rank of distributions YO to Y3 based on the Atkinson index.

Distrib'n Gini _{0 5 LÜ L5 Z&}

Y° 4 4 4 4 =3 =2

Y1 3 1 1 1 1 1

Y2 1 2 2 2 2 =2

Y3 2 3 3 3 =3 =2