Shaikh and Ragab's `Incomes of the Vast Majority': Some additions and extensions

The `Vast majority of incomes ratio (VMIR)' $R_0$ is the ratio of the average income $\mu_0$ of a poorest majority $p_0$ of the population to the overall average income $\mu$. Another measure of equality is $E_0\equiv(1-G)$ where $G$ is the Gini coefficient of inequality of the distribution. Shaikh and Ragab (2007, 2008), employing a wide variety of data sets, have shown (among other things), that when $p_0$ is 70 per cent, $R_0$ serves as a uniformly stable and excellent approximation of $E_0$. We extend the Shaikh-Ragab findings in two directions. We explore some analytics of the relationship between $R_0$ and $E_0$ for a lognormal distribution; and we derive a couple of other VMIRs and related measures of welfare which are distinct from but inspired by the Shaikh-Ragab formulations.