The most popular index to measure income distribution in a country is the GINI Index. The lower the GINI index the more evenly distributed the wealth of a county among its citizens, see: https://en.wikipedia.org/wiki/Gini_coefficient
the Gini index measures the income distribution and I think I can measure income distribution with only a linear regression. I'm thinking about the Atkinson index or other appropriate coefficients!
Linear regression is perhaps too simplistic for what is a non-linear system.
There are a number of alternative indicators of wealth distribution in a country such as the Theil Index, the Atkinson index and the generalized entropy index.