QUADRATIC PEN's PARADES: SOME NEW RESULTS AND AN APPLICATION

In this paper we derive several new results with respect to the Gini index for a quadratic Pen’s parade, building upon recent results by Mussard et al. (2011). To ensure that our results are relevant to many applied settings we impose only the parameter restrictions that guarantee the desirable property of convexity of the quadratic parade, which makes it in accord with the nature of many observed income distributions. We also account for the fact that the correlation between income and its rank is a function of the parameters of the parade. It turns out that for sufficiently large samples the reference value of the Gini index for a quadratic Pen’s parade is 1/2. Whether the Gini index is equal to, less than or greater than 1/2 in these large sample situations depends on whether or not the quadratic parade passes through, above or below the origin, respectively. The problem of fitting a quadratic Pen’s parade to observed income distributions is discussed and illustrated using Canadian data.