Suppose everyone's utility is given by $\ln{\left(wage \cdot labor\right)} - f(labor)$, where $f$ represents the utility gained from free time. Suppose, also, that we are considering only flat taxes, but that we are allowed to tax different demographics at different rates. I'm using the word "demographic" to mean a group of people that you can't choose to be in.

These two assumptions imply (among other things) that demographics should be taxed at different rates if you want to maximize social utility [show]. In particular, they imply that each demographic should be taxed at a rate so that the average income in each demographic is equal.

For instance, imagine group A is taxed at 25% and has an average wage of \$20 hour. Then if group B has an average wage of \$25 hour, they should be taxed at 40% - this way both demographics have an average after-tax wage of \$15 hour.

I find this mathematical result non-obvious because it implies the distribution within each demographic doesn't matter - at least in this simple case.

Of course, the point this makes is rather academic. It's possible utility isn't logarithmic. Perhaps preferences for money and leisure vary both within and between demographics. And maybe, just maybe, the real world has non-flat tax systems.

Flat Tax Regression

However, the real world also contains pretty enormous income gaps between some demographics. These gaps might be caused by discrimination, differences in ability, or differing preferences, but it's unlikely that 100% of each gap is caused by the third. These gaps include race "title": "Racial wage gap in the United States", sex "title": "Gender pay gap in the United States", height "title": "The optimal taxation of height: A case study of utilitarian income redistribution", brain size [citation needed], and parental income [citation needed] - though we might want to ditch this last one as distortionary.