The Gini decomposition formula was developed by Fei et al. (1978) and Pyatt et al. (1980). It can be written as follows:
where G(Y) is the Gini ratio of total income, Yi is income from income source i, si is share of income source i, R(Y,Yi) equals the rank correlation ratio, and G(Yi) equals the Gini ratio of income source i. R(Y,Yi) is the rank correlation ratio expressed as:
where r(Y) is the ranking of households in terms of total income and r(Yi) is ranking of income source i.
Moreover, Alderman and Gracia (1993) elucidated the decomposition of the Gini coefficient with the following expression:
∑sigi = 1 where gi is the relative concentration coefficient of income source i in overall inequality.
To determine whether income source i increases or decreases inequality, we compare the relative concentration coefficient gi to unity.