Distribution of Income

 

Good Morning Judge

I filed my income tax return, thought I'd saved some dough
I cheated just a little bit, I knew they'd never know
I got some money back this year, like I always do
They'll have to catch me before I pay Internal Revenue!

When you file, will there be the same flat tax rate in all income brackets?

A flat tax on income is socialist, if socialism is defined as everyone having the same income.  That is not my definition of socialism, but it is the definition of many.

The degree of the variation is a statistical measure called the..... variance.  If everyone had the same income, then the variance of income would be zero.  A flat tax rate, the same percentage of tax on everyone, also has a variance of zero.  If there is a distribution of income, then if there should also be a distribution of taxes on that income, then its variance could not be zero.

A uniform normal distribution is one in which the variance is 1.0, for example the familiar bell‑shaped curve. The phrase “Flatten the curve” became familiar during the COVID-19 pandemic.  It  means that the normal distribution of COVID-19 cases requiring hospitalization remained the same,  but its variance increased.  A normal distribution is one in which the median equals the median (where these values are also equal to a seldom used statistic, the mode). Flattening the curve is still a normal distribution but it was no longer a uniform normal distribution. 

A normal distribution requires the consideration of negative values.  It is common not to report negative incomes ( negative incomes could include donations, debt, etc. but those are not often considered to be income).  A distribution which does not allow negative values is the exponential distribution.  In this case the median does not equal the median, but the mean is a constant factor of the median.  In an exponential distribution, the mean is always 1.44 times the median.  The variance of an exponential distribution is by definition equal to its mean.

An exponential function is NOT an exponential distribution.

An exponential function is

a*e ^-bx

An exponential distribution is the special case where a=b, which is traditionally expressed using  γ, where the exponential distribution probability function is

γ*e^-γx

In an exponential distribution  the mean is 1/γ, the variance is also 1/γ , and the median is ln(2)⁄γ, or 0.69 * mean, or its inverse, mean = 1.44 * median. The variance of all exponential functions can be described.  It is at a minimum when the mean is the variance, which is the exponential distribution.

Any large collection of independent objects can be expected to follow an exponential function when their cumulative values are reported.  Individual incomes in the United States appear to follow an exponential function.  Prior to 1980, the distribution of incomes as reported by the US Census was highly correlated with the exponential distribution. These incomes were not equal.  If the taxes on this income followed the same distribution, it also could not be equal unless all incomes were also equal.  Allowing for a distribution of taxes, not a flat tax rate, considers this distribution of income.  A flat tax rate considers no variation in the distribution of income.  If individual incomes are not equal, then the distribution of taxes, which is what the tax rate is, should also not be equal.

Incomes in the United States prior to 1980 correlated well with an exponential distribution, especially at the higher incomes, and less well for lower incomes.  After 1980, the incomes follow an exponential function, but not an exponential distribution.  The variance in incomes are much larger than the mean income.  The variance in incomes increased between 1980 and 1990, and between 1990 and 2000. There was a set-back between 2000 and  2010, ( it is almost as if there was a recession during this period😏). However between 2010 and 2019 the increase in variance continued the previous trend.  The distribution of incomes have bent, i.e. the variance from an exponential distribution has increased. It has not yet broken yet, but how long is it healthy to continue this trend?