I think that Square_peg bringing this subject up has a great value if only just because it attempts to tie the forging process with the shape and the function of the tool.
Some interesting historic questions have come up.How old is the convex axe-blade to begin with?(phantom bevels being a skeletonised form of it)?
(i know the answer to it,it's "very old".Here's a section of an old Piilu,courtesy of Niko Hynninen,
https://imgur.com/a/8UWPB,similar thickening of the edge dates to about 8th-9th c.c. that i know of).
So then why it gained such popularity in the US,all of a sudden?
Americans did have a tough time at the very first,getting their own iron production going.It was a sort of a step-back,by European standards,where they had to rely on rural forges and other small-scale manufacturing.
In hand-forging an axe the material naturally moves easiest the closer it is to the edge,(it can actually be a bit more challenging to forge a flat,non-convex,blade).
Was that it,or the size of trees,or something else again?
As far as the relative value of iron i'm kinda confused...Things like that can be deceptive.A while back more that one scientific thesis had to be reexamined because suddenly it was proven that the iron and steel in ancient Novgorod was dirt cheap,and the knives that everyone assumed were very specially shaped were simply used up and discarded remnants,thrown away vs that careful recycling that emotionally we're all so attracted to...
That's a 1000-year old example.A couple hundred years back,with shipping often ballasted in iron,and it being already fairly widespread,how precious was it?Is there a way to compare a price of a new axe?Somehow in today's terms?...
(In Eric Sloane's "The Diary of Noah Blake",so about 1811,in New Hampshire,his father buys tire strap in 20' chunks,and afterwards sells remnants back to supplier,and it seems like it was fairly inexpensive...it has been a while since i read that...