# A Logical Tax Rate

The approach of February can only mean one thing: its tax season! I get inordinately excited for this task to the continued puzzlement of my wife. Every year I download the forms, build a new Excel spreadsheet with all the bells and whistles, and crunch numbers. It is extremely cathartic to go through all the paperwork yourself and mail a stack of documents to some poor bureaucrat in Kansas City.

The primary benefit of doing your own taxes is that you really learn about your finances. Giving your W-2 and 1099s to an accountant keeps the entire process a black box, while simply entering numbers into a program like TurboTax doesn’t really give a complete picture of everything that’s going on and things you might be leaving on the table. But going through line-by-line, reading instructions, and researching arcane terms means you understand what these numbers mean and what your money actually means.

It can also help keep you from making mistakes. Think you’re getting the benefit from donations? Not if you’re taking the standard deduction every year. Can you get money back from installing energy efficient windows without itemizing deduction? Yes, because that’s a credit, which is a whole different thing. The rules may seem needlessly Byzantine, but they all follow the same rhythm once you start digging-in and comprehending the biggest bill you pay every year.

Tax knowledge is also important for understanding various policy debates in the news. Recently, there was a lot of discussion over SALT deductions and their caps. Does that affect you? Will that ever effect you? Do you support the status quo or a policy revision? You won’t know unless you actually go through and see how it presents itself on your tax return.

More people doing their own taxes would mean fewer people with fundamental misunderstandings about how they work. Consider income tax brackets. They are a consequence of a progressive marginal system. The concept best works with an example. The average household in the United States makes $79,900 a year in income. A married couple filing jointly would pay a 10 percent tax on their income up to $20,550, and a 12 percent tax on income between $20,551 and $83,550 for a total tax bill of $9,177.

An incorrect interpretation of tax brackets is unfortunately common. The wrong way to think of them is that by having income within the 12 percent bracket, all the income would be taxed at that rate. If that were so, the bill would increase to $9,588. Such a misunderstanding is bad, because it leads people to not make extra money. For instance, if a raise would put you in a higher marginal tax bracket, some people will turn it down because they falsely believe they will be subject to higher taxes that negate the new income. A better understanding of your own taxes as a whole will only help to avoid that and similar mistakes.

This isn’t to say that tax brackets are a perfect system. They certainly aren’t. Each bracket introduces three variables, the lower limit, the upper limit, and the tax rate. With seven brackets currently, that means policymakers have to tune 14 separate variables to have the desired societal outcome, with each needing to be updated yearly to at least keep up with inflation. We may never be able to fit everyone’s tax return on a note card, but we can make this part better.

Instead of marginal brackets, let’s simplify things and make the amount of tax you pay (*T*_{F}) be a simple function of your taxable income (*I*) multiplied by a tax rate dependent on your income (*R*). Equation 1 summarizes the functional form.

The question is then what form the tax rate function should be. For context, let’s see what the current real tax rate is as a function of taxable income. Figure 1 shows this information for married couple filing jointly on a logarithmic scale to help highlight the overall shape. We can see that income taxed up to $10,000 is taxed at a flat rate of ten percent, with higher rates slowing creeping up over the middle income brackets and then increasing once you get above $100k before tapping off for the very wealthy. This kind of trend is known as a sigmoid or S-shape curve. As such we, can easily approximate its current shape with many fewer variables than our tax brackets.

I’ve written out the framework of our new continuous function in equation 2 below. We’re down to four variables now instead of the 14 used to define our brackets.* R*_{0} and *R*_{m} are the minimum and maximum tax rates across the entire range of incomes, while *H* is the income that pays half the difference between them. The final variable, *m*, determines how sharp of a transition it is from *R*_{0} to *R*_{m}. These allow us to create a very good simulacrum of the actual 2021 tax rates, as shown in the following figure. Some of the behavior toward the middle will be different, either plus or minus a few fractions of a percent, but on the whole will provide the same result.

The criticism that would likely be levied against this new method is that it is too complicated of a calculation for the average American do make. This may be correct, but it ignores the fact that the current tax instructions include a tax table with all the numbers up to $100,000 printed out and that it is trivial to implement on a website. So long as the math is hidden, no one will even know the difference. And in return, we have a much simpler system to amend and update every year. At least in one small way.