This is a reply to a comment on the previous post, but I thought I would just post another entry. There are two equations here determining a steady state nominal interest rate R and an inflation rate, p. They are, first, a Taylor rule:
R = a(p-p*) + r + p,
where a > 0 is a parameter, p* is the target inflation rate, and r is the long-run real interest rate, assumed constant. The other equation is the long-run Fisher relation,
R = r + p.
Now, all this tells us is that there is one steady state, where p = p*. However, we really want to account for the zero lower bound on the nominal interest rate, so write the Taylor rule like this:
R = max[0,a(p-p*) + r + p].
Now, just as in Bullard's picture (though he has a nonlinear Taylor rule instead of this piecewise linear one), there are two steady states. In one, we have p = p* and R > 0, and in the other we have p = -r and R = 0. The second steady state is the Friedman rule steady state. Benhabib et. al. supply the dynamics, and the examples with optimizing models. Krugman is making up some dynamics, but maybe he can supply us with the details.
Of course, baseline monetary theory, from Friedman's optimum quantity of money essay on tells us the Friedman rule steady state is optimal, and the Taylor-rule central bank should set p* = -r, and you'll get one steady state. An important point is that something has to be going on in the background to support the Friedman rule steady state. For a serious look at this problem see this. One condition you require is that the money stock go to zero in the limit as time runs off to infinity. From a practical policy point of view, I think we can agree that the Fed is committed to not having that happen.
An important policy question is what p* should be. New Keynesians say it should be closer to zero, if not zero, to eliminate sticky price frictions. There may be other good reasons (taxing currency transactions) for having a positive inflation rate, maybe even 2%, as the Fed seems to think is a good idea.