Tax Brackets III

 

Do You Hear The People Sing

Do you hear the people sing?
Singing the song of angry men?
It is the music of the people
Who will not be slaves again!
When the beating of your heart
Echoes the beating of the drums
There is a life about to start
When tomorrow comes

Fool me once shame on me.  Fool me twice shame on you.

In an ideal world, the income tax debate would go like this. What is the lowest acceptable income tax rate. The Bible might call this a tithe, 10%. If that is the lowest rate, then according to the laws of normal distributions, the mode/mean/median rate should be 20%, and the highest rate should be 40%. The Tax Cut and Jobs Act, TCJA, sets the lowest tax rate at 10% and the highest tax rate at 37%, which is not exactly normal, but is arguably close enough. It also confuses marginal tax rates and effective tax rates, but that will be discussed later.

The question is how to transition from a rate of 10% to a rate of 37% in a fair manner. One suggestion might be to pick an arbitrary amount and transition at that income. However establishing that transition income would be contentious politically, but mathematically it should be such that the median tax rate is paid by the median income.  Since it is a tax on income, not a tax on individuals, if income is not normally distributed among individuals, then arguably the transition should follow the distribution of income not people. The distribution of income is not normal (a statistical term, not an ethical term). If incomes were distributed normally, then the mean would be equal to the median would be equal to the mode, the most common value. According to the US Census for income, the mean is definitely not the median. One of the most common non-normal distributions is the exponential distribution. Without debating why incomes are not normally distributed, the transition from the lowest tax rate to the highest tax rate should arguably follow this existing distribution of income. (Interestingly, a flat tax also implies that everyone has the same income. To be mathematically consistent, those that argue for a flat tax must also be also communists who argue that every individual should have the same income.)

Any non-linear distribution can be approximated by a series of straight lines of EQUAL intervals, which is how calculus works. There are seven brackets in the TCJA. Assuming that the top bracket is the full amount, and the first bracket is the lowest amount, then there are five brackets in which to make the transition (e.g. the total tax paid in the highest income in the sixth of seven brackets should have an effective rate of 37%, the highest rate). The effective tax rate at any income x is 10% + 27%*(1-e^-λx), where 1/λ is the mean income according to the exponential distribution. This can be approximated by a series of straight lines, tax brackets, as:

a=λ*10%= 10% λ
b=λ*(10% + (1-e^-.2)) = 28.1% λ
c=λ*(10% + (1-e^-.2) + (1- e^-.4)) = 61.1% λ
d=λ*(10% + (1-e^-.2) + (1-e^-.4) + (1-e^-.6)) = 106.2% λ
e=λ*(10% + (1-e^-.2) + (1-e^-.4) + (1-e^-.6) + (1-e^-.8)) = 161.3% λ
f= λ*(10% + (1-e^-.2) + (1-e^-.4) + (1-e^-.6) + (1-e^-.8) + (1-e^-.1) ) = 224.5%  λ

This means that the tax brackets are a simple problem given the highest tax rate, 37%, the lowest tax rate, 10%, and the number of brackets, in this case seven. The only variable in the table above that needs to be specified is the mean income. This should be computed for each tax filing using the tax status (e.g. Single, Married Filing Jointly, Married Filing Separately, Head Of Household, etc.). As an example, the US Census specifies the mean household income as $102,316 in 2021 according to Income in the United States: 2021: Current Population Reports. With this amount, and the actual percentages substituted in place of the exponential equations, the table would be: 

Bracket	Lower income	Upper Income	Fixed Tax	Marginal Tax Rate  (on amount above lowest income in bracket)	Lowest Effective Tax	Highest Effective Tax
1	$0	$102,316	$0	10%	10%	10%
2	$102,316	$204,632	$10,232	18.13%	10%	14%
3	$204,632	$306,948	$28,751	32.97%	14%	20%
4	$306,948	$409,264	$62,515	45.12%	20%	27%
5	$409,264	$511,580	$108,660	55.07%	27%	32%
6	$511,580	$613,896	$165,036	63.21%	32%	37%
7	$613,896	No Limit	$229,699	37%	37%	37%

 The 2022 TCJA Tax rates for Married Filing Jointly or Qualifying Widow (Widower) are: 

Bracket	Lower income	Upper Income	Fixed Tax	Marginal Tax Rate  (on amount above lowest income in bracket)	Lowest Effective Tax	Highest Effective Tax
1	$0	$20,550	$0	10%	10%	10.0%
2	$20,550	$83,550	$2,055	12%	10%	11.5%
3	$83,550	$178,150	$9,615	22%	11.5%	17.1%
4	$178,150	$340,100	$2,055	24%	17.1%	20.4%
5	$340,100	$431,900	$9,615	32%	20.4%	22.8%
6	$431,900	$647,850	$2,055	35%	22.8%	26.9%
7	$647,850	No Limit	$9,615	37%	26.9%	35.7%*

*The highest effective tax rate in the highest tax bracket is not reached until the income is infinite. This is the effective tax rate at an income of $5,000,000.

 As can be seen for the TCJA tax brackets, the brackets are not equal AND the effective tax rates are confused with the marginal tax rates. This second error is like confusing speed and acceleration, since both are first and second derivatives.  The combination of these two errors results in a tax code that does not transition between the lowest rate and the highest rate according to the distribution of income. As can be seen in the figure below, this means  the incomes below ~$300,000 are currently paying effective tax rates that should be charged only at higher incomes. The basic errors were made beginning with the “Reagan” Tax Cuts of 1981 and have resulted in an over taxation of the poor and middle class and an under taxation of the rich for over 40 years. If the intent were to transition from the lowest tax rate to a maximum tax rate according to the distribution of income, a math professor would give these tax codes a failing grade.