Greg Mankiw (rhymes with "thank you") says that there's no one way to measure the progressivity of the tax code, giving as an example:
There are two people. A rich guy earns $200,000. A poor guy earns $20,000. At first, the rich guy pays $50,000 in taxes, and the poor guy pays $1,000. Then a new President takes office and cuts the rich guy's taxes to $48,000 and the poor guy's taxes to $800.
- You could say the rich guy gets the better deal: The rich guy gets an extra $2000 in take-home pay, while the poor guy gets only $200. After the tax cut, the difference in take-home pay between the two guys is larger.
- You could say the deal is evenly balanced: Everyone gets to keep an extra 1 percent of his income.
- You could say the poor guy gets the better deal: The poor guy gets a 20 percent tax cut, while the rich guy gets only a 4 percent tax cut. After the tax cut, the rich guy pays a larger share of the total tax burden.
It is impossible to say on purely economic grounds which of these perspectives is better. All of these statements are mathematically correct, even if they leave the reader with very different impressions. If you are a politician or a journalist trying to argue that this tax cut is good for the rich, good for the poor, or somewhere in between, you can do it!
I disagree. There is one clear problem with the first two metrics: They both allow us to eliminate the poor guy's tax burden altogether while making the system less "progressive." For example we could cut the poor guy's taxes by $1,000 (5% of his pre-tax income) and the rich guy's taxes by $12,000 (6%). I don't see how any metric of progressivity which suffers from this flaw can be considered valid.