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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, _{i}

For more than a century, Lorenz plots [_{i}

Phase transitions have been reported in connection with interest rate models [

We discuss the relation between Lorenz graphs and statistical distributions in

Once averaged over many individuals and sufficiently long intervals of time (typically, a year) for data to be statistically significant, the individual’s share of total benefit defines one out of many possible

Of course, as will become clear below, supplementary assumptions will be necessary to obtain a workable expression for the entropy.

Standard entropy maximisation with no other constraint than the obvious

Lorenz plots

The expectation value of

(

The unequal-chance inequality (UCI) region is defined by

The median, the mode and the mean coincide for symmetrical distributions. The natural boundary between classes is then the axis of symmetry, as illustrated in

Inequality thus shows up in at least two conceivable and non equivalent ways: for ECI, the poor fraction of the population equals the rich fraction,

If a phase transition is indeed at work here, we expect a law of corresponding states to apply, as it does for similar transformations in physics [

Symmetry-dependent universal behaviours. (

Data shows this maximum to occur at values

We define the

At first sight, three parameters,

If

Its continuation into the region

One recovers the approximate expression for ^{2} that becomes _{r}

Equations (5), once solved for

Conventional

The main advantage of the Gini coefficient is its conceptual simplicity, though counterbalanced by possible inaccuracies when obtained from discontinuous data. Equations (5) and (9) provide a function describing

Quantities like

Characteristic parameters of unequal-chance inequality for different types of data.

BENEFIT | ^{2} |
_{P} |
_{M} |
_{uc} |
||||
---|---|---|---|---|---|---|---|---|

Income | 0.008 | 0.58 | 0.68 | 0.16 | 0.99 | 0.35 | 0.46 | 0.16 |

Electricity consumption | 0.046 | 0.66 | 0.52 | 0.32 | 0.97 | 0.04 | 0.60 | 0.38 |

Life expectation | 0.159 | 0.76 | 0.55 | 0.52 | 0.89 | 0.10 | 0.79 | 0.68 |

Survival after cancer | 0.344 | 0.86 | 0.45 | 0.71 | 0.78 | –0.11 | –– | –– |

This work provides a model that fairly fits Lorenz curves, up to

Quite different phenomena, from income distribution to cancer rate of survival, obey the same statistical laws. The resulting description of inequality implies an apparently oversimplified two-class division of society. A more detailed analysis should provide criteria allowing recognition of existing classes, whatever their number, out of real-life distributions. This amounts to a nontrivial challenge – modelling the probability density function.

We assumed that

Let the density