Proportionality-Induced Distribution Laws
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The differential equation for Lorenz curves which leads to the one-parametric Pareto distribution, see Chap. 7, can be considered as proportionality law. This law relates the Lorenz density to the income share of certain population segments and allows a variety of variations and relaxations leading to a functional rather than differential equation. One kind of variation of the differential equation allows to raise income share to powers and other variations replace proportionality factors by proportionality functions. Many of the equations can be solved in closed form leading to a system of (new) types of one-parametric Lorenz curves.
Some of the other proportionality-induced Lorenz curves better fit empirical data than the Pareto distribution.