I've held the notion that money behaves in a log-linear way for decades, so it's nice to see that formalized finally. A river exhibits different behavior than a lake or the ocean. It's interesting that Daniel Bernoulli (famous for the Bernoulli effect) had a similar instinct in 1738. Compared to the talking heads in news and politics today, he was an intellectual giant to say the least.
I did a deep dive on finding how well tax brackets correlate with a log-based tax rate, but couldn't find much. I'll just summarize the results of my AI-assisted research:
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https://www.fidelity.com/learning-center/personal-finance/ta...
https://www.reddit.com/r/AskEconomics/comments/1iri8nf/tax_b...
By plotting the 2026 single filer tax bracket thresholds against their marginal rates, we can fit them to the classic logarithmic function:
log-linear equation for slope of line (y = m * x + b):
tax rate = m * ln(income) + b
income tax rate ln() tax rate deviation
$12,400 12% 12.10% +0.10%
$50,400 22% 21.52% -0.48%
$105,700 24% 26.50% +2.50%
$201,775 32% 30.84% -1.16%
$256,225 35% 32.44% -2.56%
$640,600 37% 38.60% +1.60%
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The general public might have a hard time understanding logarithms, so I investigated using base 2, base 10 and base e (ln) to explain them (the base doesn't affect the computed tax rate). Here are the two simplest rules of thumb for a log-based tax system:
a) base 2 log: every time your income doubles, you pay 4.7% higher taxes on the total
b) base 10 log: every time you add a 0 to the end of your income, you pay 15.5% higher taxes on the total
income tax rate taxes paid approximation
a) base 2 log:
$8,192 9.37% $768 ~10%
$16,384 14.03% $2,299 ~15%
$32,768 18.69% $6,124 ~20%
$65,536 23.35% $15,303 ~25%
$131,072 28.01% $36,713 ~30%
$262,144 32.67% $85,642 ~35%
$524,288 37.33% $195,717 ~37% (current top marginal tax rate capped above this point)
$1,048,576 41.99% $440,297 ~40% vs 37%
$2,097,152 46.65% $978,321 ~45% vs 37%
$4,194,304 51.31% $2,152,097 ~50% vs 37%
$8,388,608 55.97% $4,695,104 ~55% vs 37%
$16,777,216 60.63% $10,172,026 ~60% vs 37%
$33,554,432 65.29% $21,907,689 ~65% vs 37%
$67,108,864 69.95% $46,942,650 ~70% vs 37%
$134,217,728 74.61% $100,139,847 ~75% vs 37%
$268,435,456 79.27% $212,788,786 ~80% vs 37%
$536,870,912 83.93% $450,595,756 ~85% vs 37%
$1,073,741,824 88.59% $951,227,882 ~90% vs 37%
b) base 10 log:
$10,000 10.66% $1,066 ~10%
$100,000 26.12% $26,120 ~25%
$1,000,000 41.59% $415,900 ~40% retains current millionaire tax rate near 37%
$10,000,000 57.06% $5,706,000 ~50% at mid-millions vs 37%
$100,000,000 72.52% $72,520,000 ~75% at $100 million vs 37%
$1,000,000,000 87.99% $879,900,000 ~90% at $1 billion vs 37%
From those tables, it's easy to see how a log-linear flat tax rate would work:
a) base 2 log:
4.7% flat tax: tax rate = 4.7% * (number of doublings) - 50%
b) base 10 log:
15.5% flat tax: tax rate = 15.5% * (number of zeros) - 50%
c) base e log (for completeness):
6.7% flat tax: tax rate = 6.7% * (number of zeros) - 50%
After grokking this, we might ask why a non-logarithmic 10% flat tax wouldn't work? The answer is subtle, but it's because it wouldn't incorporate the increased buying power over expenses ratio of higher incomes, so the formula would become tax rate = 0 * (number of zeros) + 10%, making it a regressive tax that penalizes low incomes and lowers taxes on high incomes that don't need the help.
To demonstrate why a 10% flat tax would be regressive, lets calculate the log-linear tax rate that meets the current $2 trillion US tax income:
tax rate = m * ln(income) + b
calculation of m for ln(income) derived from current values:
m = (T - (b * AGI)) / (AGI * ln(u))
m = log-linear slope to solve for
T = total US tax revenue (currently about $2 trillion)
b = -50% (floor held constant as a starting point)
AGI = annual gross income of US (currently about $15 trillion)
u = center of mass income of all taxpayers with half of tax revenues above and below (currently about $250,000)
m = (2e12 - (-0.5 * 15e12)) / (15e12 * ln(250000)) = 0.05096 ~= 5%
calculation of m for base 2 log and base 10 log for completeness:
a) base 2 log:
m = (2e12 - (-0.5 * 15e12)) / (15e12 * log2(250000)) = 0.03532 ~= 3.5%
b) base 10 log:
m = (2e12 - (-0.5 * 15e12)) / (15e12 * log10(250000)) = 0.11733 ~= 12%
final tax rates to meet $2 trillion in tax revenue using base 10 log-linear tax at m = 12%:
tax rate = 12% * log10(income) - 50%
income tax rate taxes paid approximation
b) base 10 log:
$10,000 -2.00% -$200 ~0% tax floor/credit for poverty line
$100,000 10.00% $10,000 ~10% tax for working class
$1,000,000 22.00% $220,000 ~20% for millionaires (37% top marginal tax rate currently)
$10,000,000 34.00% $3,400,000 ~35% for multimillionaires
$100,000,000 46.00% $46,000,000 ~50% for top millionaire incomes vs 37%
$1,000,000,000 58.00% $580,000,000 ~60% for billionaires vs 37%
notable thresholds:
$50,000 6.39% $3,194 ~6.5% tax for median income taxpayers
$250,000 14.78% $36,938 ~15% tax for center of mass income taxpayers
After running the math, I feel that it's objectively self-evident that a log-linear tax reflects reality better than a 10% flat tax.