A nice estimation of Gini index and power Pen's parade

Under the assumption that an economic variable y such as income, is a power function of its rank among n individuals, we provide the coefficient of variation and the Gini index as functions of the power degree of the Pen's parade. In fact, knowing the power degree of the Pen's parade function, we obtain the coefficient of variation and a sharp analytical simple way to compute Gini index in terms of the power degree. Reciprocally, knowing, the coefficient of variation of a said variable, we can also provide the power degree of the corresponding Pen's parade, hence deducting its shape. Several examples are given to illustrate our methods. Special attention is given to countries' Gini indices of the World development indicators report 2009 (WDI 2009).