---
title: Finding the ideal tax curve
date: 2025-05-16
tags: [economics, math]
description: "Having fun by finding better alternatives to those absurd piecewise-defined tax curves"
---

I recently tried to figure out income taxes and came across one of those absurd
[piecewise-defined](https://en.wikipedia.org/wiki/File:Income_Tax_Germany_2010.png)
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:

$$f(x) = ax+b$$

![](linear.png)

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](https://en.wikipedia.org/wiki/Negative_income_tax) 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:

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

![](log.png)

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:

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

![](pow.png)

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.