Consider the attached charts, which illustrate that low-inequality Country A can have both a lower average income and higher average log income than high-inequality Country B. This sparked my interest in determining how to integrate inequality data with income data in such a way as to address the way in which higher inequality can suppress living standards (even if average income is unchanged).
I believe that one way to do this would be by giving the % difference in individual income (between Countries A and B) the same weight at any given income percentile. To do this, we can plot log(Y=individual income) against X=income percentile and compare the area underneath the graph for Country 1 vs Country 2 (as in Graph 2). Or we can compare average log(individual income) for Country 1 vs Country 2. Unfortunately, this takes a long time using raw income distribution data from the World Bank (which it uses to calculate Gini).
Therefore, I would like to know the best way to approximate average log(individual income) from average individual income and the Gini coefficient?
Perhaps by way of ranking the countries by
(average individual income).((1-Gini)^a), where a is either a constant or a function of Gini,
and determining which a results in the most similar ranking order as when ranking by
average log(individual income)?
Should this be the case I would like to know how to determine a, via iteration or otherwise.
NB: I am not experienced at all in economics, so please suggest to me how I could make my question more clear. If it is clear and there is no need for me to keep on editing/improving it let me know as well. Reference 1 suggests that average individual income (GDP per capita) can be adjusted for inequality by using the expression (GNI per capita PPP).(1-Gini), but countries ranked by (GNI per capita PPP).(1-Gini) are not in exactly the same order as countries ranked by average log(individual income). So a=1 is incorrect.