The AMA doesn't restrict the number of doctors really either.. that's an old trope that isn't really true. The number of doctors is restricted by the number of residency slots available, which is restricted by congressional funding since they're all paid for via Medicare. If congress were serious about increasing the number of doctors, they could pass a bill tomorrow authorizing more money for new residency programs.

We've dramatically increased the number of medical students but haven't kept up in residency spots.

NEJM lays it out:

http://www.nejm.org/doi/full/10.1056/NEJMp1306445?viewType=P...