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Thanks, that makes sense. I guess that's why flat income tax rates are considered "unfair" since they are modeled on a linear expectation of utility.

cperciva 11y ago
Not really. Completely flat percentage taxes (e.g., pay 10% of every dollar you earn) reduce everybody's utility by the same amount if you have a utility = log(dollars) model. Actual "flat taxes" (e.g., pay 10% of every dollar you earn after the first $10k) have a larger impact on the utility of high income earners (at high incomes you approach a change of log(0.9), while at low incomes the impact of the tax on utility reaches zero). This is however an argument for poll taxes (e.g., "every adult must pay $1000") being unfair, as well as sales taxes on inferior goods (warning: technical term) if the regressive effects aren't compensated by a quasi-fixed-dollar tax rebate.
- tunesmith 11y ago
  I don't understand this. If the utility of money is roughly logarithmic, then that would imply the most fair tax bracketing would be a logarithmic curve. An example would be the "logarithmic flat tax" I see mentioned sometimes - figure out how many times above the poverty rate you make, log-10 it, and multiply by some flat constant (they usually recommend 9 or 10). If you plug in a middle-class earner's revenue to that formula, their effective tax rate is far lower than it would be under any flat tax I've seen mentioned. Flat tax in comparison is very regressive.
  - evanpw 11y ago
    Logarithmic utility of money implies that the last X% of your income is worth the same amount of utility, regardless of what your income is: log(x) - log[(1 - r)x] = log(x) - log(1 - r) - log(x) = -log(1 - r). But "does the same harm to every person" is not necessarily the right definition of "fair" in the context of taxes, though.
    - tunesmith 11y ago
      Can you expand on that? What is x, what is r? And you are saying that the last 5% of a millionaire's income has the same utility as the last 5% of a different person's income that makes 25k/year?

cperciva 11y ago

Not really. Completely flat percentage taxes (e.g., pay 10% of every dollar you earn) reduce everybody's utility by the same amount if you have a utility = log(dollars) model. Actual "flat taxes" (e.g., pay 10% of every dollar you earn after the first $10k) have a larger impact on the utility of high income earners (at high incomes you approach a change of log(0.9), while at low incomes the impact of the tax on utility reaches zero). This is however an argument for poll taxes (e.g., "every adult must pay $1000") being unfair, as well as sales taxes on inferior goods (warning: technical term) if the regressive effects aren't compensated by a quasi-fixed-dollar tax rebate.

tunesmith 11y ago
I don't understand this. If the utility of money is roughly logarithmic, then that would imply the most fair tax bracketing would be a logarithmic curve. An example would be the "logarithmic flat tax" I see mentioned sometimes - figure out how many times above the poverty rate you make, log-10 it, and multiply by some flat constant (they usually recommend 9 or 10). If you plug in a middle-class earner's revenue to that formula, their effective tax rate is far lower than it would be under any flat tax I've seen mentioned. Flat tax in comparison is very regressive.
- evanpw 11y ago
  Logarithmic utility of money implies that the last X% of your income is worth the same amount of utility, regardless of what your income is: log(x) - log[(1 - r)x] = log(x) - log(1 - r) - log(x) = -log(1 - r). But "does the same harm to every person" is not necessarily the right definition of "fair" in the context of taxes, though.
  - tunesmith 11y ago
    Can you expand on that? What is x, what is r? And you are saying that the last 5% of a millionaire's income has the same utility as the last 5% of a different person's income that makes 25k/year?