blog

commit

7a6c12d77196964392fd5e8129081ce7ad3f3c49

parent

e0c381f950f282144c8b7537211a0b15ee3b38e5

Author

Tobias Bengfort <tobias.bengfort@posteo.de>

Date

2025-05-24 18:10

tweak tax curves

Diffstat

1 files changed, 15 insertions, 9 deletions

diff --git a/_content/posts/2025-05-16-tax-curves/index.md b/_content/posts/2025-05-16-tax-curves/index.md

@@ -16,7 +16,10 @@ We will look at functions where $x$ is the income before taxes and $f(x)$ is
   16    16 the income after taxes.
   17    17 
   18    18 I am a big fan of folding some form of social security into the tax system, so
   19    -1 $f(0) = b$ will be the level of basic income everyone should have.
   -1    19 $f(0) = b$ will be the level of basic income everyone should have. This would
   -1    20 of course not replace other mechanisms of social redistribution. It would just
   -1    21 provide a baseline. I find it weird that we use one system above a certain
   -1    22 income and a completely different one below that.
   20    23 
   21    24 All functions will have a single additional parameter $a$ that can be used to
   22    25 adjust the system to the current economic situation, basically raising or
@@ -30,6 +33,11 @@ $f(x) = ax+b$
   30    33 
   31    34 ![](./linear.png)
   32    35 
   -1    36 The marginal tax rate is what you have to pay on any additional income you
   -1    37 make. In mathematical terms, it is $1 - f'(x)$. The average tax rate, on the
   -1    38 other hand, is the total fraction of your income that you have to pay, i.e. $1
   -1    39 - f(x) / x$.
   -1    40 
   33    41 With this system, the marginal tax rate is constant, but the average tax rate
   34    42 starts out [negative](https://en.wikipedia.org/wiki/Negative_income_tax) and
   35    43 then quickly grows towards the marginal tax rate.
@@ -37,19 +45,17 @@ then quickly grows towards the marginal tax rate.
   37    45 ## Logarithmic
   38    46 
   39    47 Most real-world economies have progressive tax rates. That means: the higher
   40    -1 your income, the higher your marginal tax rate. The classic mathematical
   41    -1 function that grows to infinity but slows down while doing so is the logarithm:
   -1    48 your income, the higher your marginal tax rate. Or: your income after tax still
   -1    49 grows, but it grows slower. The classic mathematical function that grows to
   -1    50 infinity but slows down while doing so is the logarithm:
   42    51 
   43    52 $f(x) = a \ln(\frac{x}{a} + 1) + b$
   44    53 
   45    54 ![](log.png)
   46    55 
   47    -1 Now the marginal tax rate starts at 0% and grows towards 100%.
   48    -1 
   49    -1 The marginal tax rate is what you have to pay on any additional income you
   50    -1 make. The common believe is that a high marginal tax rate will disincentivize
   51    -1 people from earning more (= being more productive). That is why I wanted to
   52    -1 start with 0%.
   -1    56 Now the marginal tax rate starts at 0% and grows towards 100%. The common
   -1    57 believe is that a high marginal tax rate will disincentivize people from
   -1    58 earning more (= being more productive). That is why I wanted to start with 0%.
   53    59 
   54    60 On the other end of the spectrum, I don't see any reason to stop the
   55    61 progression at some arbitrary value. If a single person has some astronomical