> It's regressive in that it causes the poor to pay more than they do now. With regard to taxes, progressive and regressive have specific meaning.

Under that meaning, a flat tax is the border between progressive and regressive taxation. It is not an example of a regressive tax.

Increasing the amount that poor people pay makes a tax scheme relatively more regressive, but if it's no more regressive than a flat tax, it's still not "regressive".

> Under that meaning, a flat tax is the border between progressive and regressive taxation.

That depends entirely how you measure. If you measure by total income, then yes. If you measure by income after necessities for living are accounted for, then a very simple progressive tax that takes that into consideration might be considered the baseline, and a flat tax that doesn't take that into consideration could be considered regressive. On the other end of the spectrum, if you make people account for all subsidized government services used and count that as income, then a flat tax is progressive. Without an agreement on the baseline for measurement (which I believe is where a lot of people start disagreeing), you can't even necessarily agree on what is progressive and regressive.

I thought progressive and regressive referred to convex and concave taxation curves, respectively (both monotonically increasing, of course). Then flat tax is exactly the border (i.e. a linear function).

While depending on background, that may be what some people think of as defining progressive and regressive taxes, and may indeed describe at a low level the concept that most people think of and even be the origination of the term, it's not the definition generally put forth currently[1][2] (which is, admittedly, very simplistic). That just goes to my point, which is that people aren't even necessarily in agreement on the terms, and might not even be aware of that.

1: https://apps.irs.gov/app/understandingTaxes/student/glossary...

2: https://apps.irs.gov/app/understandingTaxes/student/whys_thm...

I think these definitions are equivalent. If your marginal tax rate (at least sometimes) increases as you get more income, the curve of tax liability against income will be concave upwards. If it (at least sometimes) decreases, that curve will be concave downwards. Right?

Almost. I think the concept inside many people's heads is simplified to the degree that it doesn't even need to describe a curve. A simple step function, for example (the simplest in this case being if you make less than $X, you pay nothing, otherwise Y%). In that manner, the concept is not approximating the curve, the curve is approximating the concept.

What I meant by the proportion of tax paid is the same at any income level is under a flat tax scenario, leaving out all tax rules today of exemptions/credits.

If the rate is 10%, the proportion of tax is the same, whether you make $10 or $1 million, you are still paying 10% of your income.

Yes, the effective tax rate collected. The reference you provided only supports my claim. It looks like the effective tax rate, which is the actual rate collected by the IRS is around 20% since 1979. I'd have to find it, but if you take the data all the way back to the New Deal, the effective tax rate is around 17%. Most likely due to it being easier to "hide" income back then.

Regardless, the effective tax rate has stayed around 17-22% over almost the last century, despite the large differences in the tax rates at different brackets. Remember, the top rate during the Eisenhower administration was 90%. Yet the effective tax rate average was unchanged.

I think the problem here is that your inference that a flat tax rate would be good because the total amount of income taxes collected each year as a percent of total income hasn't change has had no supporting evidence presented, and it's such a leap to most people that they don't even understand that's the argument you're making.

Exactly why do you believe that since the total percentage of income collected hasn't really changed that means a flat tax is a good idea? The reason for a progressive tax is not to raise the total amount of taxes collected, but to change which people it is collected from proportionally. I'm not sure what the total percentage collected has to do with that.

Someone making $10 is going to miss 10% of that a lot more than someone making $100, and they're going to miss 10% of that a lot more than someone making $10,000. And someone making $1MM isn't likely to miss it at all.