The public health insurance plan

Here is comment from Ezra Klein, who distinguishes different versions of the public plan idea and also links to further reading.  Matt Yglesias comments in favor of the idea.  Here is a Paul Krugman column.  Arnold Kling is skeptical.  Those are good introductions to the debate.  On the economics, Ezra writes:

Rather, the theory here is simple: If you can't replace them, convert
them. If the public plan works, then private insurance will work better
as well. In this telling, the simple existence of the public plan
forces a more honest insurance market: Private insurers need to offer
premiums closer to their marginal cost, and they have to cut
administrative costs, and they have to work on their reputation for
cruelty and capriciousness. The existence of another option changes the
market. Individuals will have access to private insurers, but they'll
no longer be stuck with them.

I believe Ezra is assuming no direct cash subsidy to the public plan, lower marginal and average costs for the public plan, and some mix of market power and X-inefficiency in the private insurance companies.  The existence of the public plan then "contests the market," which eventually lowers MC in the private plan and leads to lower prices and better service.

My question is what the equilibrium looks like.  Say the public plan has a cost advantage (both MC and AC), as plan proponents suggest.  If public and private plans are to coexist, the public plan must be attracting the higher-cost customers, namely the higher medical risks.  (I am also assuming that the political equilibrium does not allow the public plan to reject these customers outright.)  There is then market segmentation and it is not obvious that there are significant positive competitive pressures on private insurance companies.

Oddly, I believe in some models the public insurer constrains the private companies more tightly when the public insurer does not have an apparent cost advantage.  Even here, the properties of the monopolistically competitive equilibrium would be very tricky.

You might wonder why the public plan does not attract all the low-risk customers and take over the whole market.  I would say that either a) it does, or b) it is tailored toward the high-risk customers.  Since public plan advocates sincerely and correctly claim the policy is not just a back door to single-payer, we are left with b).

Another question: is the "cruelty and capriciousness" of the private plans — cited by Ezra — driven by profit maximization?  Presumably it is and again assume the government plan will not do the same.  Why then would public sector competition force a private firm to throw out a profit-maximizing strategy?  In fact "cruelty and capriciousness" would be a comparative advantage of the private companies and maybe it would be milked more strongly in a more competitive environment. 

Another possibility is that the public company has a bigger cost advantage on AC than MC.  For instance maybe it has a "head start" on the fixed costs, because everyone has heard of it, but its cost advantage for additional service dwindles at some point.  The successive accretion of high-risk customers then threatens to put the public plan under (especially if there are lots of previously uninsured and they are high risks) and the public plan requires a subsidy simply to break even.  I consider this equilibrium to be not totally unlikely.

Obviously I am missing some equilibria, but in many cases the public plan is mainly providing insurance to high-risk customers.  There's nothing wrong with that (and indeed it is a major policy goal), but the resulting equilibrium needn't much improve the performance of private health insurance.  I file this argument under "not yet established."