Since, as noted, taxes are due in tomorrow, I thought I’d mention something I noticed as I was working mine up.

The income tax from the federal government and most state governments is progressive. Basically, as you make more money you pay a larger portion of it in taxes. This is where the idea of a tax “bracket” comes from.

Here’s an example derived from Louisiana’s numbers a few years ago:

• If you have under $12,500 of income you owe 2% of it. • If you have between$12,500 and $25,000 of income you owe$250 (from the first $12,500) plus 4% of the rest. • If you have more than$25,000 of income you owe $750 (from the first$25,000) plus 6% of the rest.

The weirdness comes when you look at how a state calculates part-year residency taxes. Let’s say you made $40,000 last year, but you only made$20,000 as a resident of Louisiana. The rule they use is that you made half of $40,000 in-state, so you pay half the taxes you would owe if you made all$40,000 in state. Here’s a graph:

The red graph is the tax function. The green line is the function for if you made $40,000 overall, but only$$x$ of it in Louisiana. See how you’re always paying more taxes in Louisiana because of all that non-Louisiana income you made? Guess what happens in Connecticut because of all the income you made after you moved out!

To make a more serious point: mathematicians have a word for a function like this. We say that a function $f$ is “convex” if its graph always lies below the straight line between any two points on the graph. That is, if you take two sample points $(x_1,f(x_1))$ and $(x_2,f(x_2))$ (with $x_1), and take the linear function that hits these two points

$\displaystyle g(x)=f(x_1)+\frac{f(x_2)-f(x_1)}{x_2-x_1}(x-x_1)$

then $f(x)\leq g(x)$ for every $x\in\left[x_1,x_2\right]$. In fact, I used this term once before, when talking about upper and lower Darboux integrals.

Here the tax function is defined to be piecewise-linear, with two sharp corners in it where it’s not differentiable. I think tomorrow, though, I may come back to a topic I skipped over in differential calculus: how can derivatives tell you where a function is convex?

4 Comments leave one →
1. Sam Iam permalink
April 15, 2008 23:01

Hi John.

I really enjoy reading your blog. Especially the abstract nonsense material.

Concerning this post. I understand that they go through this hoopla to get more money out of you, but I do think that it is a fair thing to do. Why should you be taxed in the lower bracket just because you moved in the middle of the year?

Imaging what would happen if they did do this. Say you earned $100,000 but moved in the middle of the year. Then your total taxes for both states would be computed as if you were in the$50,000 bracket.

2. April 15, 2008 23:20

So compute federal on your overall income, and state/local at the state and local level.

Incidentally, New York taxes you only on the New York income, so the method I’m talking about here isn’t universal.