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

The ideal fit yields the parameters m = 0.0672 and b = -0.5121. The table below outlines how closely the mathematical log formula predicts actual statutory tax rates:

  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%

US federal tax brackets match a base-e natural logarithm (ln) model surprisingly well, boasting a statistical correlation R^2 of approximately 0.962.

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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%

Politicians would set the log-linear tax rate slope (the 4.7%, 15.5% or 6.7% depending on log base) and the tax rate base (50% which might vary between perhaps 45-55%).

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%

Lets see if the calculated m slope would lower taxes:

  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

It's obvious from the last summary that incomes under $100,000 would pay less under a log-linear flat tax than a 10% flat tax. Millionaires and multimillionairs would pay less than their current 37% top marginal tax rate too. Only top multimillionaires and billionaires would pay higher taxes than they do now.

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.

The Price of Happiness (2024)