This is an interesting question, partly because there's no real answer here yet. At what grade - let's say paved surface - would it be faster on foot, and why?

It's related to the other question, why can we go so much faster on a bike? If you could pin down all of the reasons why, put a formula and numbers to them, then you could figure out the grade. I did a brief search on the internet and this is the other interesting thing about it: almost all of the answers (guesses) are completely wrong! (Mostly it has to do with bouncing up and down. Although leg muscles are partially elastic only a fraction of our running energy propels us forward).

Maybe it's as simple as comparing metabolic power expended. I am familiar with formulas related to cycling and we could write it as a function of grade and speed for a given weight. But I don't have a clue about the same thing with running. It seems to me that running uphill, at least the vertical bounce is more useful - it moves you uphill. Does anyone know of such a formula?