Recently there has been a lot of attention given to the two equilibria that one gets from the interaction of a Fisher rule and a monetary policy rule in the presence of the zero bound. The typical depiction uses a diagram with the nominal interest rate on the vertical axis and inflation on the horizontal axis. I wanted to present the material to my students and thought it would go better if I translated the problem into AD-AS terms. In doing so, the issue of the comparative stability of the two equilibria - which has been debated- pops up in a somewhat new light - or so I think, but I'm probably wrong. Anyway, here's the example:
The monetary policy rule: R = Rn + a(I - It), where R = the real interest rate, Rn is the natural rate, I is inflation and It is target inflation. The zero bound on the nominal interest rate, i, entails the following:
i = R + I = Rn + a(I - It) + I greater than 0 implies that I is greater than (a/(1+a)) It - Rn/(1+a)
I also have an IS relation : Y= 500 - 100*R, where Y is real output, and
Y*= 800, where Y* is potential output
Now let a= 1/4, Rn = -3, and It = 4
AD ( in I, Y space) - or IS-MP, if you prefer, has the following characteristics:
At any inflation rate above 3.2 (which is where the zero bound starts to bind) AD is given by Y = 900-25*I
For inflation rates below 3.2, R = -I, so AD is Y= 500 + 100I, so it is negatively sloped above I= 3.2 and positively sloped below
AD thus looks like the nose-cone of a rocket pointing to the right, and intersects the vertical LRAS (Y=800) twice, at inflation rates of 4 and 3 on the negatively- sloped and positively-sloped segments, respectively.
When inflation is 4, the Fed sets the nominal rate at 1, giving a real rate of -3, consistent with Y at potential. When inflation is 3, the Fed sets the nominal rate at 0 (we've hit the lower bound ) and the real rate is again -3, the natural rate. (The inflation target cannot be met in this second equilibrium)
Now for stability. If we are at an inflation rate greater than 3.2, on the downward-sloping portion of AD, with Y less than potential, all is well. Put in your (output) Phillips curves with expected inflation moving down when inflation is less than was expected and we make our way back to potential.
If we are at an inflation rate below 3.2 and to the left of potential, on the other hand, we have instability as long as the SRAS curves are flatter than the upward -sloping AD: we move further away from potential in a disinflationary spiral. If, on the other hand, the Phillips curves are steeper than the AD curve, we may have stability, or cob-webbing or a stable orbit - depending on the details of the adaptive expectations process. If expected I in t is actual I in t-1, for example, then we have stability in this case