Would save via administrative costs. It takes a very large administrative dept. for each of these to keep up with the paperwork for each separate insurance company, each with their own brand of paperwork and strange quirks and hoops they put up for caregivers to jump through just to get paid, not to mention the constant re-submissions these office staff personnel have to submit (because insurance co.s routinely "loose " paperwork and stall payment over constant and never ending minutia in order to gain a percentage of money for what they don't pay out when someone fails to accommodate all the stall and loss of paperwork tactics used to avoid paying.
My doctor told me he had 4 people just to keep up with all the differing paperwork and would love to only hire one to fill out what would be a single universal set of papers under a single payer plan.
Single payer would save practices a great deal of money spent on an unnecessarily bloated administration needed to cover all the differing requests and paperwork from several companies rather than from one source, my doctor said he would only need one person that may even have extra time to do other non-medical duties for his practice, such savings on the part of the healthcare providers would cheapen the cost of care. Imagine the effect such a change would have on a large scale caregiver like a hospital, talk about saving on the cost of health care, it would be substantial because of the savings to the caregivers.
Adding that bit of information makes our healthcare savings even larger than under the projections like TYT just talked about that doesn't even address that bit.