A Cato guy wrote this article on the growing doctor shortage. I sense that many people will read it, get the gist of it, and say "well the solution here is easy, the government should just pay more to providers."
A shallow, stage-one analysis makes the suggestion seem okay, but a deeper look reveals that whether the government is under-paying or over-paying providers, in both cases we have essentially the same thing going on: the decisions of panels of government bureaucrats setting prices are taking the place of the normal, market-based, supply-and-demand determination of prices (of course, given the existence of occupational licensing and a host of other government interventions into the health care industry, we aren't even close to a free market system, Medicare or not).
As it exists right now, the government pays doctors too little, which means that to be profitable (or to break even) providers must charge more on their private patients--those with private insurance or people who pay out-of-pocket. Or opt out of the government system as much as possible.
If the government decided to start paying doctors more, it would attract more doctors to the government programs and do something towards easing the (artificially-created) health care labor shortage. But the story doesn't end there, at stage one. This means doctors will change what they charge private patients. As government reimbursement per service went up, doctors would be able to charge private patients less as a result and the prices paid by both welfare recipients and private patients would be closer to the same dollar amount per service. But this would cost tax-payers more and create friction there.
The government could also overpay providers and have two choices about what to charge private patients, either extract the same amount of money from them as they get from the government (which would make people unhappy, imagine if an envelope company said "sorry, we get dollars a box from the government now, so now we're charging you ten dollars a box instead of the usual three"), or charge them less than they charge the government. That is a politically unsustainable situation (to say nothing of the economics). Voters would see they are being ripped off by providers and not stand for it. The government would pay less to providers.
The third option is for the government to not set prices at all, but to pay whatever providers charge. The government would be saying to providers "just think of us as another customer, and whatever you charge them, charge us." This might actually work if we had a properly functioning health care industry that could be more receptive to consumer demand and hence prices would be more liquid, but what with the health care industry being one of the most tightly regulated and inefficient industries in the country, I don't see it happening. If the government made it known that it would pay whatever providers want to charge, then providers, knowing they can exploit the tax-payers, will do so, and overcharge.
Another option, the best option (for as long as we have some kind of socialized medicine), is for the government to give every recipient a dollar amount per year, possibly with some sort of bonus for using fewer dollars, and providers simply have to work with that. Say, for example, $4,000. Then providers know they can't extract more than that per recipient, they will have to be more economical and less likely to mandate high-cost, low-return tests, treatments, etc.