Are you saying whether the size of a population in a region warrants the buying of a mammogram machine, a hospital would not buy its own and cede potential customers to a hospital that does have one?
That kind of answers its own question. If the size of a population warrants a buy, it buys, and competes for the business.
The larger question here is if the overpurchase of medical equipment is a significant factor in the elevated cost of health care services in the US. Possibly, although I think it pretty obvious that recurring high costs (salaries, etc) are more distortive than discrete costs (equipment purchase) because the former compounds itself over time.