Data show that an increase in the Gini coefficient is associated with a falling bottom $p_{B}$% income share and an increasing top $p_{T}$ % income share where, for example $p_{B}$ = 40 and $p_{T}$ = 1. This relationship, which we call the $X$ inequality relationship, is pervasive in the sense that it is observed in many countries, including the US, the UK, France and others. The purpose of this paper is (i) to construct a Schumpeterian growth model to explain the relationship, and (ii) to identify/quantify factors behind it via calibration of the US economy. Our model gives rise to a double-Pareto distribution of income as a result of entrant and incumbent innovations. Its advantage is that it allows us to develop iso-Gini loci and iso-income share schedules in a tractable way. Using a double-Pareto distribution as an approximation of an underlying income distribution, calibration analysis reveals that a declining business dynamism and fiscal policy changes in the past decades played a significant role in generating the $X$ inequality relationship in the US.

A new approach to goodness-of-fit for Pareto distributions is introduced. Based on Euclidean distances between sample elements, the family of statistics and tests is indexed by an exponent in (0,2) on Euclidean distance. The corresponding tests are statistically consistent and have excellent performance when applied to heavy-tailed distributions. The exponent can be tailored to the particular Pareto distribution. The goodness-of-fit statistic measures all types of differences between distributions, hence it is also applicable as a minimum distance estimator. Implementation of the test statistics is developed and applied to estimation of the tail index in three well known examples of claims data, and compared with the classical EDF statistics.

Les mesures d'inégalité du revenu rassemblent deux types d'indicateurs décomposables : les indices décomposables en sous-populations et les indices décomposables en sources de revenu. Les premiers permettent de partager l'inégalité totale en une inégalité intragroupe et une inégalité intergroupe et les seconds d'attribuer à chaque facteur de revenu (revenu du travail, revenu du capital, taxes, etc.) une part de l'inégalité totale. Dans cet article, nous examinons d'une part la construction de ces techniques et d'autre part nous relatons les débats auxquels elles ont aboutis et plus particulièrement celui de la convergence vers un emploi simultané des deux types de décomposition.