Finding the ideal tax curve

I recently tried to figure out income taxes and came across one of those absurd piecewise-defined tax curves. So I thought it might be a fun activity to figure out better options for tax curves, without any claim to real-world merit.

Basics

We will look at functions where $x$ is the income before taxes and $f(x)$ is the income after taxes.

I am a big fan of folding some form of social security into the tax system, so $f(0) = b$ will be the level of basic income everyone should have. This would of course not replace other mechanisms of social redistribution. It would just provide a baseline. I find it weird that we use one system above a certain income and a completely different one below that.

All functions will have a single additional parameter $a$ that can be used to adjust the system to the current economic situation, basically raising or lowering taxes.

Linear

The simplest model is a flat tax rate with basic income:

$f(x) = ax+b$

The marginal tax rate is what you have to pay on any additional income you make. In mathematical terms, it is $1 - f'(x)$. The average tax rate, on the other hand, is the total fraction of your income that you have to pay, i.e. $1 - f(x) / x$.

With this system, the marginal tax rate is constant, but the average tax rate starts out negative and then quickly grows towards the marginal tax rate.

Logarithmic

Most real-world economies have progressive tax rates. That means: the higher your income, the higher your marginal tax rate. Or: your income after tax still grows, but it grows slower. The classic mathematical function that grows to infinity but slows down while doing so is the logarithm:

$f(x) = a \ln(\frac{x}{a} + 1) + b$

Now the marginal tax rate starts at 0% and grows towards 100%. The common believe is that a high marginal tax rate will disincentivize people from earning more (= being more productive). That is why I wanted to start with 0%.

On the other end of the spectrum, I don't see any reason to stop the progression at some arbitrary value. If a single person has some astronomical income, sure, let them pay 99.9% marginal tax. They will still have much more money than everyone else.

Power

Another type of functions that go up to infinity but slow down on the way are power functions like this one:

$f(x) = (\frac{x}{ab} + 1)^a b$

Like with the logarithm, the marginal tax rate starts at 0% and grows towards 100%. However, it doesn't grow as quickly, which puts it somewhere in between the logarithmic and linear versions. So this feels the most balanced.

I also like that $a$ has an intuitive interpretation (how curved the curve is). So if I were ever going to be minister of finance, this would probably be the version that my advisors would have to talk me out of.