A note on the maximum value of the Kakwani index

The Kakwani index computes the departure from proportionality of a progressive income tax by measuring the difference between the concentration coefficient for tax liabilities and the Gini coefficient for pre-tax incomes. In case of maximum progression, that is a situation in which only one taxpayer faces the overall tax burden, the index reaches its theoretical maximum value, given by 1 minus the Gini coefficient for pre-tax incomes. We argue that this phenomenon can happen in one special case that is not satisfied in real-world personal income taxes. As a matter of fact, the overall tax revenue of a real-world personal income tax cannot be eventually paid only by the richest taxpayer. Therefore, the maximum concentration coefficient for taxes cannot be equal to 1, and, consequently, the maximum value of the Kakwani index cannot be 1 minus the Gini coefficient for pre-tax incomes, as generally described in the related literature. According to different hypotheses, we give evidence of this phenomenon by employing the Italian personal income tax