Notes on Gini Coe cient

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1 otes on Gini Coe cient Michael Bar July, 07 Contents Introduction Standard Gini coe cient 3 Standard Gini, Weighted Sample 4 Gini for Samples with egative Values 5 Introduction The Gini coe cient is a popular measure of inequality, widely used to measure inequality of income, wealth and progresivity of a tax system. In these notes I de ne the Gini coe cient and derive a convenient formula for the computing the Gini. The last section shows how to modify the standard Gini coe cient when the sample contains negative values. Standard Gini coe cient Suppose that we have a vector of non-negative values of some attribute (e.g. income, wealth), sorted in ascending order, Y (0 Y Y ::: Y ), with mean Y. The mean absolute di erence of all pairs is M j jy i Y j j j X (Y j Y i ) () The second formula recognizes that the absolute di erence is positive when i 6 j, or twice the number of pairs with j >. The the standard Gini coe cient for the sample is de ned San Francisco State University j

2 as the relative mean absolute di erence, i.e. one half of the mean di erence, relative to the sample mean: G M j j (Y j Y i ) () Y Y The reason why the Gini de nition uses of the mean absolute di erences is bound its maximal value at. Suppose that all the income is given to one individual, i.e. Y > 0 and Y i 0 for i ; ; :::;, and the total income is also Y. This case is referred to as the maximum inequality. There are ( ) non-zero absolute di erences, all equal to Y, and the Gini is j G j (Y j Y i ) Y Y Y otice that in this case of maximum inequality, lim! G. Thus, the division by in the de nition of Gini coe cient, equation () is a normalization of maximum inequality to in case of large population. The maximum inequality with individuals gives G 0:5, and with 00 individuals G 0:99. In contrast, if all individuals have the same value, i.e., perfect equality, then M G 0, since Y i Y j 0 8i; j. Thus, G [0; ) for any nite sample Y. roposition The Gini coe cient is unit-free, i.e. scaling the sample by a constant b > 0, does not a ect the Gini. roof. G (by ) j j (by j by i ) by i b j j (Y j Y i ) b Y i G (Y ) The above property establishes that the Gini is only a relative measure, so doubling the income of all individuals, or using di erent units (e.g. thousands of dollars instead of dollars) or di erent currencies, does not a ect the resulting Gini coe cient. Let S Y i be the total income. The income share of individual i is therefore y i Y i S. Then we can write the Gini as: G j j (Y j Y i ) Y i j j (Y j Y i ) S j X (y j y i ) j 3 Standard Gini, Weighted Sample Let f (Y i ) be the relative weight (frequency) of observation i, so that f (Y i ) 0 and f (Y i). Then, the mean absolute di erence of all pairs is M j X f (Y i ) f (Y j ) jy i Y j j f (Y i ) f (Y j ) (Y j Y i ) j j

3 The Gini coe cient is de ned as where G M Y j j f (Y i) f (Y j ) (Y j Y i ) Y (3) Y f (Y i ) Y i otice that the case of unweighted sample is a special case of (3) when f (Y i ) 8i. We now derive a computational convenient formula for computing the Gini coe cient for large samples. The cumulative weighted sum of income of the rst i individuals as S i ix f (Y j ) Y j, S 0 0, and S j f (Y j ) Y j j roposition The Gini coe cient in (3) can be computed as where s i S i S. G f (Y i) (S i + S i ) S f (Y i ) (s i + s i ) (4) roof. Rewriting the de nition of Gini (3): G ji+ f (Y h i) f (Y j ) (Y j Y i ) f (Y i) ji+ f (Y i j) Y j ji+ f (Y j) Y i S S f (Y i i) hs S i ji+ f (Y j) Y i f (Y i) hs i + i ji+ f (Y j) Y i S S otice that the third step used the fact that ji+ f (Y j) Y j S to show that f (Y i) ji+ f (Y j) Y i f (Y i) S i. S i. Finally, we need j X f (Y i ) f (Y j ) Y i j j X f (Y j ) f (Y i ) Y i j f (Y j ) S j j The formula in (4) can be written in a way, that has a geometric interpretation, as the 3

4 area under the Lorenz curve divided by the area under the equality line. In the above gure, The Gini is de ned graphically as G A A + B Since A B, the Gini is also G B B + B B Letting s i S i S be the cumulative fraction of total income earned by the rst i individuals. The formula in (4) si + s i G f (Y i ) The term f (Y i ) s i +s i represents area of a trapezoid under the Lorenz curve, between i and i. The two parallel sides are s i and s i, and the height is f (Y i ). Thus, the above sum is the total area B under the Lorenz curve. 4

5 4 Gini for Samples with egative Values In some applications, samples with negative values can lead to Gini coe cient which is greater than. Chen et al (98) develop a normalization formula, which was improved by Berebbei and Silber (985). Recently, a more appropriate normalization factor was introduced by Ra netti et al (05), which we present in this section. Generalizing Ra netti et al (05) notation to weighted samples, we de ne: (i) the total amount of the attribute (e.g. income, wealth) as T Y i; (ii) the total amount of positive attributes as T + max (0; Y i); (iii) the total amount of negative attributes, in absolute value, as T max (0; Y i) min (0; Y i). Ra netti et al (05) propose as the maximum inequality (olarization) case if we assign T to one individual, T + to another individual, and 0 to everybody else. In this case, the vector of income is ( T ; 0; 0; :::; 0; 0; T + ). The mean di erence in this (olarization) case is j " X i f (Y i ) f (Y j ) jy i Y j j T + i T + + T + + T ( ) T + ( ) T + T + T + # jy i otice that since any of the i ; ; :::; individuals can be assigned the T or T +, the above quantity does not depend on speci c sample weights, and we use f (Y i ) 8i. Thus, the Ra netti et al (05) proposed Gini is: j Y j j G RSV M ji+ f (Y i) f (Y j ) (Y j Y i ) (T + T + ) (5) To derive the computational formula, we divide the numerator and denominator of G RSV by S j f (Y j) Y j : Simplifying, gives where G RSV ji+ f (Y i) f (Y j ) (Y j G RSV (T + T + ) S G (t + t+ ) t T S, t + T + 5 S Y i ) S (6)

6 If there are no negative incomes, we have T 0, T + T and the Gini is G RSV M Y ji+ f (Y i) f (Y j ) (Y j Y i ) T ji+ f (Y i) f (Y j ) (Y j Y i ) Y i G Thus, even if there are no negative values, G RSV 6 G. In particular, G RSV G > G. Large samples, without negative values of the attribute, will produce very similar coe cients G RSV and G. In case of maximal inequality, G RSV even for nite. Without negative values, G RSV G > G, and for large samples, with no negative values, GRSV is very close to the standard (not-normalized) Gini coe cient. Recall that the standard Gini coe cient for maximum inequality, is G <. In conclusion, G RSV can always be used, in any sample, whether it contains negative values or not. References [] Berrebi, Zeev M., and Jacques Silber. "The Gini coe cient and negative income: A comment." Oxford Economic apers 37, no. 3 (985): [] Chen, Chau-an, Tien-Wang Tsaur, and Tong-Shieng Rhai. "The Gini coe cient and negative income." Oxford Economic apers 34, no. 3 (98): [3] Ra netti, Emanuela, Elena Siletti, and Achille Vernizzi. "On the Gini coe cient normalization when attributes with negative values are considered." Statistical Methods & Applications 4, no. 3 (05):