OK, I’ll start a business that delivers mail to your mailbox. Oh wait, that’s against the law. So there is not way of knowing how much cheaper a privately run mail company would be that delivers to your mailbox.
No, no way save for baseless speculation such as that, if operated solely by the profit motive and not with a mandate of universal single-price service[*], a company would immediately drop service to unprofitable rural communities in favour of the lucrative city-to-city market. And that an already somewhat disenfranchised sector of the US, the rural poor, would just disappear and drop off the end of the world. That's
why ending Saturday service prompts endless USPS soul-searching whereas a for-profit company would drop it like it's hot: because it's a service serving up valuable social inclusion.
In fact the example of the postal service is a reasonably good analogue for the benefits of healthcare reform. You get the government involved when the free market would cause unconscionable harm to those at the bottom of the market. USPS = Useful Single-Payer Simulator.
Peter
[*] Um, which I'm assuming they have. Certainly our equivalent Royal Mail monopoly is tied to a universal-service mandate.