Decomposition of Zenga's inequality index by sources

Zenga (2007) has proposed an inequality index based on the ratio between the arithmetic means of lower and upper group. In the case of the sum Y = X1 + : : : + Xj + : : : + Xc we obtain the additive contribution of each Xj to the inequality index I of the total Y . The decomposition is obtained using the property that the arithmetic mean of a sum is equal to the sum of the arithmetic means of the components. Moreover, we show that I = I + D, where I is the inequality index of the total in the situation in which among the variates Xj there is uniform ranking and D 0 measures the diminution of the synthetic inequality from the uniform ranking matrix to its data matrix. We have also obtained the contribution of each Xj to the index D. The theoretical results are applied to the 2008 Bank of Italy sample survey on household income and wealth.