However one of the more likely methods of construction (from this distant point anyway) would be a launch of lots of material, tie it all to some asteroid or other large counterweight that has been put in space a good 90,000+km away, and then unspool it all to the surface, which would require a good and cheap heavy launch capacity
(I suppose you could say if we are going to do such an undertaking as build a space elevator, with all the billions it may entail, then launch costs to get it going will probably be the least of our worries, but as far as I see it, every little bit helps!)