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Pseudo-flat tax formula:
Assume utility is logarithmic in income, and the goal is to set the experienced tax burden to be constant.
Then, we have the formula that the average tax rate, where is a parameter controlling the experienced tax burden and is the break-even point, is as follows:
is the input income, and is the average tax rate.
x is the initial income, and I forgot to cancel it. Good point.
Turns out, it's far simpler than I had it as.
Fans of math exercises related to toy models of taxation might enjoy this old post of mine.