# Fooled by Algebra

The late great Armen Alchian taught me that you can be rigorous with words. As a math major, I needed to hear that. Last week, I forgot that. I let algebra distort my thinking.

In a post last week, I expanded on Columbia University economist Wojciech Kopczuk’s statement that a 3% tax on wealth amounts to a 103% tax on capital income from wealth. (This is in a case where the capital earns the riskless return of 3%.) I used algebra to fill in the missing steps.

Reader Daniel was not convinced, writing:

Dr. Kopczuk has a great paper full of compelling reasons on why we might prefer more “vanilla” taxation schemes than a wealth tax, but the “an X% wealth tax is actually a Y% capital income tax” line isn’t one of them.

Daniel also referenced my co-blogger Scott Sumner’s exposition of a similar point. I had read Scott’s piece quickly and had seen that he was referring to a news story in the *Wall Street Journal* in which reporter Richard Rubin had made a strong claim. It didn’t surprise me that an economist could find a flaw in a journalist’s economic reasoning about a complex subject.

Little did I know that Kopczuk made a similar mistake.

And I’ve now figured out how to show it.

The way to see what the marginal tax rate on capital income is is to think on the margin: change the income from capital and see how much extra tax is paid.

So, for example, start with $1,000 at the start of the year that earns what Kopczuk calls the riskless rate of return, 3%. With a wealth tax, $1,030 at year’s end is taxed at 2%, leaving the owner with 98% of $1,030, which is $1,009.40.

Now raise the rate of return to 4%. With a wealth tax, $1,040 is taxed at 2%, leaving the owner with 98% of $1,040, which is $1,019.20.

How much more did the owner of capital net from the investment at 4% rather than at 3%? $1,019.20 minus $1,009.40, which is $9.80. In other words, for an extra income from capital of $10, the owner kept $9.80. The wealth tax amounted to a 2% tax on the income from capital.

Of course, there are other taxes on capital. But I’m focusing here on the marginal tax rate on capital income due only to the wealth tax of 2%. It’s 2%, not the outlandish number I had thought.

Yes, you can attribute the whole tax on wealth to the tax on income. But there’s zero economic justification for doing so.

Bottom line: Think on the margin. With emphasis on “Think” and “margin.” Daniel was right, as was Scott.