Drawing is easy.

Narratives are easy.

Numbers are hard.

When people post online, they are probably not putting too much time into thinking about what drawings their brain renders and what narratives they are following.

Then, we get comments in threads like this ELI5 thread which claim that progressive taxation is fair because it imposes equal disutility on those taxed. And crucially, that the reason why it is justified is because utility is logarithmic.

They are wrong.

Let's set up a function to calculate the proportion of income that should be taxed to get constant disutility under logarithmic utility, where y is income, x is non-taxed proportion, and u is the disutility. log(y * x) = log(y) - u. Then, let's solve for x with Wolfram Alpha because I can't be arsed to do it by hand.

The solution is x = e^-u. The tax, 1 - x, does not vary in y (income). Logarithmic utility therefore justifies flat taxes, the ones where the rate is the same, not progressive ones.

The intuition behind this requires going beyond "line curves right". Logarithms also have the (nice) feature of turning the difference of two logarithms into per cent changes. How a constant difference in logarithms (the disutility) leads to a constant per cent value should then be obvious.

How can you justify progressive taxation under equal disutility? Well, if you adopt a constant relative risk aversion function, just jack up the IES parameter beyond 1. (And if you take the IES parameter down to zero you can then justify head taxes.)