[schoen]

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?

[kbenson]

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.