* We calculate our income tax by use of a Tax Table, employing one of four columns depending on filing status. Out of curiosity, I plotted this year's "filing-jointly" column to see its implications and to see whether an equation could generate the tax-vs-income relation.
* The plot consists of about seven end-on-end straight-line segments. Their slopes are called marginal tax rates.
* Surprisingly, the plot of tax (Y) vs income (X) could be generated faithfully by the formula:
Y = Yao + Sa*X - Yao*e^(-X*ln2/H)
which makes me suspect that the line segments were positioned to mimic the smooth curve generated by this formula. The first two terms express the y intercept (Yao = -52.354k) and slope (Sa = 0.396) of the highest-income segment (Income > 450k) extrapolated to zero income, this being treated as the asymptote of the curve. The exponential term adds to the asymptote diminishingly as income rises above zero. The exponential term falls to half its intercept value at an income (H) of 120k. The natural-log of 2 is 0.693.
* The red lines in the chart illustrate the dependence of the equation's terms and their sum (calculated tax) on income in thousands. The calculated tax is almost superimposed on Tax-Table values (black diamonds and dashes).
* If every taxpayer had a calculator with the e^x function (inverse natural-log or exp function) and knew how to use it, then the tax tables could be replaced by four such formulae (one for each filing status) differing only in the constants.
* Though it is unrealistic to expect every taxpayer to calculate tax by use of the above equation, it would be rational to generate the tax tables by use of that equation rather than by a string of straight-line segments. Then the taxpayer would have the option of using the equation or the table. The result would be consistent with the purpose of today's tax table and would obviate concern with tax brackets, which often confuse people.
* The applicable marginal rate for a given income is the first derivative of the above formula:
dY/dX = Sa + Yao*(ln2/H)*e^(-X*ln2/H)
For example, at 120k income the marginal rate is 0.245, but there's no need to know it.
