Abstract
We propose a model for Lorenz curves. It provides two-parameter fits to data on incomes, electric consumption, life expectation and rate of survival after cancer. Graphs result from the condition of maximum entropy and from the symmetry of statistical distributions. Differences in populations composing a binary system (poor and rich, young and old, etc.) bring about chance inequality. Symmetrical distributions insure equality of chances, generate Gini coefficients Gi £ ⅓, and imply that nobody gets more than twice the per capita benefit. Graphs generated by different symmetric distributions, but having the same Gini coefficient, intersect an even number of times. The change toward asymmetric distributions follows the pattern set by second-order phase transitions in physics, in particular universality: Lorenz plots reduce to a single universal curve after normalisation and scaling. The order parameter is the difference between cumulated benefit fractions for equal and unequal chances. The model also introduces new parameters: a cohesion range describing the extent of apparent equality in an unequal society, a poor-rich asymmetry parameter, and a new Gini-like indicator that measures unequal-chance inequality and admits a theoretical expression in closed form.
Highlights
For more than a century, Lorenz plots [1] have been used to describe inequalities in economic, social or biological systems
We discussed [3,4] the relevancy for the characterisation of inequality of a notion originated in thermodynamics and statistical physics-entropy
We show that symmetry determines whether the number of such intersections, if any, is even or odd
Summary
For more than a century, Lorenz plots [1] have been used to describe inequalities in economic, social or biological systems. Predictive features of the resulting model will be checked against data on incomes, electricity consumption, life expectation and (male) cancer-specific rate of survival. Different substances may feature dissimilar data, but suitable normalisation of relevant quantities results in a single universal curve on each side of the transition, describing the properties of all such substances simultaneously. This is the signature of a phase transition and is known as the law of corresponding states for liquid-gas transitions, and universality in the general case. We show here that it holds for quite different economic and demographic data, and we obtain a mathematical expression for it It allows the calculation of Lorenz plots and thereby their possible intersections. An appendix proves two lemmas on relationships between the symmetry of the statistical distribution and the resulting graphs
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