Inequality Dynamics: A Note on the Time-wise Decompositions of Entropy-based Measures of Inequality with a Special Focus on Theil Measures

This paper discusses the dynamic properties of generalized entropy measures of inequality, deriving an explicit expression linking changes in inequality with changes in the growth income. This establishes a time-wise decomposition for the family of generalized entropy measures of inequality, showing that the rate of change in the distribution of income can be divided into two parts. One is purely a function of the levels and rates of change of macroeconomic variables; a second part depends exclusively on the micro nature of the distribution process. We then focus on a subset of generalized entropy inequality indexes, focusing on Theil measures. By assuming that the distribution process follows a specific rule, we further simplify the time-wise decomposition formula, establishing a direct relationship between the rate of income growth and the rate of change in inequality as measured by Theil measures. A specific application of this formula to the simulation of an economy with a constant average positive growth rate produces behavior consistent with the Kuznets hypothesis.