God Plays Dice

Here's a question of some practical importance: let's say that I make 2,000ドル in each of the first six months of the year, and 4,000ドル in each of the second six months. Will I have more or less tax withheld than if I make 3,000ドル every month?

(This is inspired by the fact that I taught this summer, and was paid for doing so, but because of the way my contract is written, my pay for the academic year is spread out over twelve months. As a result I got extra-large paychecks over the summer, partially for the work I was doing in the summer and partially for work that I had already done in the previous academic year or will be doing in the next academic year.)

For those not familiar with the US tax system: if your net pay is X you don't get a check for X, but for some smaller amount, because various taxes are withheld. Chief among these is the federal income tax. Now, the federal income tax is not a flat tax, but a progressive tax -- if your income is higher then you pay a larger percentage of your income in tax. Tax returns have to be filled out on a yearly basis, but most people get paid more often than yearly. So the amount of tax withheld is determined, based on the amount of the paycheck and the period that the paycheck is for, in such a way that the total amount of tax withheld is somewhere near the amount that you're expected to owe. (Most Americans actually end up overpaying through this system, and get a small refund back at tax-filing time.)

So say that if you make 36x per year, then your taxes will be f(36x). Then you'd expect that if you make 3x in a given month, you will have f(36x)/12 withheld, for a total of f(36x) over the course of the year.. If instead you make 2x in each of six months and 4x in each of six months, then in each of the months in which you make 2x tax will be withheld as if you make 24x per year, and in each month in which you make 4x tax will be withheld as if you make 48x per year. So total withholding will be

6f(24x)/12 + 6f(48x)/12

or, simplifying, [f(24x) + f(48x)]/2. Call this T'. Is this less than or greater than f(36x), which we'll call T?

We can easily see that T' ≥ T if and only if

f(48x)-f(36x) ≥ f(36x) - f(24x).

That is, T' ≥ T if and only if the amount of extra tax owed when you go from 36,000ドル to 48,000ドル is more than the extra amount owed when you go from 24,000ドル to 36,000ドル. But since marginal tax rates are increasing -- since the tax is progressive -- this is true.

More generally, given progressive taxation, withholding is smallest for a given annual income if that income is spread out exactly evenly throughout the year. This is a consequence of Jensen's inequality. The more unevenly spread out the earnings are, the more money will be withheld.

In reality this is slightly more complicated because there are tax brackets, which are reflected in the withholding formulas, so f is actually piecewise linear (see page 36 of this IRS publication). For example, a single person paid between 883ドル and 3,050ドル per month (after subtracting withholding allowances) will have 70ドル.80 + .15(x-883ドル) withheld from a paycheck of x; a single person paid between 3,050ドル and 7,142ドル will have 395ドル.85 + .25(x-3050ドル) withheld. (Note that putting 3,050ドル into either of these formulas gives 395ドル.85; the amount withheld is a continuous function of the amount earned.) So if every paycheck is under 3,050,ドル or if every paycheck is over 3,050,ドル then the amount withheld ends up being the same no matter how the pay is distributed. But a person who makes, say, 3,000ドル in each of two months will have 388ドル.35 withheld from each, for a total of 766ドル.70; a person who makes 2,000ドル in one month and 4,000ドル in another will have 238ドル.35 withheld from the first and 633ドル.35 withheld from the second, for a total of 871ドル.70 withheld.

I had known all this intuitively before this afternoon but I'd never bothered to actually write down why it is...

This all applies to people who make varying amounts in differing pay periods from a single job. People who have multiple jobs can be burnt in the withholding process because we have progressive taxation; if you make x in each of two jobs you have less withheld than if you make 2x in a single job. If that's you, be careful.

(I'm not an accountant. None of this should be taken as financial advice.)