I'm in between your two measurement at 16", with cedar beds. I would love 24" to really minimize bending down (which is a primary reason I have raised beds in the first place) but 16" isn't too bad. I find that adequate for tomatoes and squash, plus it's tall enough to keep wild rabbits out of my greens.
If you're going to go deep then you can fill in the bottom with branches, cheaper soil, etc. to help lessen the costs - really no reason to fill an entire bed with expensive soil. I fill about halfway with cheap bagged stuff, then the rest with a good quality locally mixed amended soil, topped with my own compost.
If you're going to go deep then you can fill in the bottom with branches, cheaper soil, etc. to help lessen the costs - really no reason to fill an entire bed with expensive soil. I fill about halfway with cheap bagged stuff, then the rest with a good quality locally mixed amended soil, topped with my own compost.