I think the subset of people who would have a box installed and never use prime again and the people who would give away a key to their house for 1 delivery is

I think the subset of people who would have a box installed and never use prime again and the people who would give away a key to their house for 1 delivery is pretty small. At $50/box, I can outfit 20 million homes for 1 billion which is a 1x cost. If they are really loosing 7b on package theft every year, the one off loss of box cost should start offsetting that right away. A quick google shows that there are 90 million prime members in 2017, so they could do a box per prime member and pay it off in 9 mo. The second year they just save 7 billion. Am I missing something?