Gini Index:

First, we consider Gini coefficient. A measure the has been widely used to represent the extent of inequality is the Gini coefficient attributed to Gini (1912). One way of viewing it is in terms of the Lorenz curve whereby the percentages of the population arranged from the poorest to the richest are represented on the horizontal axis and the percentages of income enjoyed by the bottom x% of the population is shown on the vertical axis. Obviously, 0 % of the population enjoys 0 % income and 100 % of the population enjoy all the income. So the Lorenz curve runs from one corner of the unit square to the diametrically opposite corner. If everyone has the same income, the Lorenz curve will be simply the diagonal, but in the absence of perfect equality, the bottom income group will enjoy a proportionately lower share of income. It is obvious that any Lorenz curve must lie below the diagonal ( except the one of complete equality which would be the diagonal ), and its slope will increasingly rise - at any rate not fall - as we move to richer and richer sections of the population.

The Gini coefficient is the ratio of the difference between the line of absolute equality ( the diagonal ) and the Lorenz curve. There are various ways of defining the Gini coefficient. Undoubtedly, one apeal of Gini coefficient lies in the fact that it is a very direct measure of income difference, taking consideration of differences between every pair of income . A bit of manipulation reveals that it is exactly one-half of the relative mean difference, which is defined as the arithmetic average of the absolute values of differences between all pairs of income.  For convenience, we use Sen’s formula for Gini coefficient. The formula is:

G = 1 +  - ( +  + … … … + ) for      … … …  

However, some authors prefer  instead of .

            It is clear from the Sen's formula that the Gini coefficient implies a welfare function which is just a weighted sum of different people's income levels with the weights being determined by the rank-order position of the person in the ranking by income level. Therefore, Sen's formula shows that the implicit welfare function underlying the Gini coefficient is a rank-order weighted sum of different person's income share. However, the welfare function must be thought to be -G, since a higher value of G shows a greater inequality, which corresponds to less welfare.

Now, from BBS income data collected from Naogaon in 1996, we had the following data-

G         = 1 +  - (20653.528 + 2*9116.266 + 3*5853.555 + 4*5238.323 + 5*5076.333 + 6*4028.331 + 7* 2626.620 + 8*1620.806 + 9*1439.137)

            = 1 +  - (171256.7)

            = 1 + 0.1111111 - 0.6838286

            = 0.4272825

See the S-plus / R codes used to calculate this result of the Gini Coefficient.

Now, from BBS expenditure data collected from Naogaon in 1996, we had the following data-

Calculating similarly for expenditure, we get-

G         = 0.443723

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