Math issues

Felix Salmon's last two posts are brilliant. I strongly recommend reading both of these posts. Seriously!

Heck, I am likely to come back and post on them later today if I have enough time. Paul Krugman and Mark Thoma might be representing the outrage of progressives, but Felix is documenting the damage coldly and dispassionately.

However, this is a statistics blog. So I wanted to visit a comment on one of Felix Salmon's posts:

The top 10% of Americans pays 45% of all taxes, a higher proportion by far than any other country, while the American top 10% earns only 33% of total income.

By comparison, the 10% of the UK earns 32% of total income and pays only 38% of all taxes. And the UK is one of the more progressively taxed countries. The taxes paid by the top 10% of other European countries are on average around 30% of total taxes.

As an epidemiologist (a field that is in love with proportions), I realize how misleading they can be. In the example above, the rich earn roughly the same amount of income. But they pay a higher proportion of taxes. However, is it the proportion of taxes that matters or the absolute tax burden?

Consider a simple example. In two countries there are 10 people. Of these people, nine make 10 units of income per year and one makes 50 unites of income. This is pretty close to the income distribution of the US and UK cited above.

In country A, the nine people pay 2 units of tax per year and the rich person pays 16 units of tax. Total tax revenues are 34 units (or 24% of GDP). The rich definitely pay more as a proportion of taxes but pay 47% of the taxes overall.

In country B, they pay something more like the OECD average (35%). So the nine people pay 3.5 units of tax (31.5) and the rich pay 17.5 units of tax. That is 49 units of tax collected (exactly 35%). The rich pay 36% of the tax burden.

Which country is a low tax country for rich people to live in? One where they pay an effective tax rate of 32% versus 35%? Or one where they pay a higher proportion of total taxes collected?

The trick here is that the US total tax burden is so light that the top 10% can both pay a higher proportion of taxes than in other countries and, at the same time, still pay less absolute tax. That can make proportional analysis potentially misleading.