Extremely rough: A note on James Bullard (2010), "Seven Faces of 'The Peril'" http://research.stlouisfed.org/econ/bullard/pdf/SevenFacesFinalJul28.pdf...

Here is the graph:

Bullard:

In this paper I discuss the possibility that the U.S. economy may become enmeshed in a Japanese-style, deflationary outcome within the next several years. To frame the discussion, I rely on an analysis that emphasizes two possible long-run outcomes (steady states) for the economy, one which is consistent with monetary policy as it has typically been implemented in the U.S. in recent years, and one which is consistent with the low nominal interest rate, deáationary regime observed in Japan during the same period.... When the line describing the Taylor-type policy rule crosses the Fisher relation [i.e., Wicksellian Balance], we say there is a steady state at which the policymaker no longer wishes to raise or lower the policy rate, and, simultaneously, the private sector expects the current rate of ináation to prevail in the future.... In the right-hand side of the Figure, short-term nominal interest rates are adjusted up and down in order to keep inflation low and stable. [Point A]... [A]s we move to the left... the two lines cross again, creating a second steady state [at Point B].... The policy rate cannot be lowered below zero, and there is no reason to increase the policy rate since well, inflation is already "too low." This logic seems to have kept Japan locked into the low nominal interest rate steady state. Benhabib, et al., sometimes call this the "unintended" steady state...

As I understand this model, the "Policy Rule" line shows what the Federal Reserve wishes it had set its policy nominal interest rate i--the Federal Runds rate--given the rate of inflation π. When i lies above the policy rule line, monetary policy is "too tight" and the Federal Reserve will reduce i. When i lies below the policy rule line, monetary policy is "too loose" and the Federal Reserve will raise i. As the vertical arrows in the version below show, i is falling everywhere above the policy rule line and rising everywhere below it.

The "Wicksellian Balance" line is what Bullard (and Benhabib et al.) call the "Fisher Relationship." I prefer to think of it in Hicksian IS or Wicksellian terms: for aggregate demand Y to be equal to potential output Y*, the market real interest rate r must be equal to the natural real interest rate re. When the market real rate r exceeds the natural real rate re, investment spending is too low for aggregate demand to match potential output and there is downward pressure on the rate of change of prices. If the market rate r is less than the natural real rate re, investment spending is too high for aggregate demand to match potential output and there is upward pressure on the rate of change of prices. Above the Wicksellian Balance line, there is downward pressure and so inflation is falling. Below the line, there is upward pressure and so inflation is rising.

Complicating the dynamics further is the zero-bound requirement: i cannot fall below zero.

Complicating the dynamics still further is downward price stickiness: π cannot become large and negative (although we see hyperinflation in the real world, we have never seen hyperdeflation with governments adding zeroes to the denomination of their currency). Just as i cannot fall below zero, π cannot fall below the vertical line starting at point C.

The natural dynamics of this model separates the graph into two regions. The first region consists of a subset of those points above the Wicksellian Balance line and to the right of the Policy Rule line for which the economy's dynamics lead it to eventually hit the X-axis to the left of point B, and then converge to point C. The economy starts out with monetary policy "too tight" and with aggregate demand below potential output, hence both inflation π and the policy interest rate i fall until the economy hits the x-axis. Then i stops falling--it can't fall any further--and inflation π continues to fall until the economy reaches its sticky-price deflationary speed limit at Point C. There the economy sits.

The second region consists of all those points that generate paths that at some point fall below the Wicksellian Balance line--that at some point produce situations in which aggregate demand is higher than potential output. Those paths then spiral into the stable equilibrium at point A. There the economy sits.

Note, under these dynamics, that point B is not of especial interest as an equilibrium of any kind, but rather because it marks the bottom of the curve that is the boundary between the two regions: the region of states that ultimately converges to the good equilibrium A and the region of states that converge to the absorbing deflationary-Japan equilibrium C. If the economy is sitting at point B, we do not expect it to stay there. A small downward shock to aggregate demand or a small upward shock to the policy interest rate would throw the economy into the zone that converges to C. A small upward shock to aggregate demand would throw the economy into the converge-to-point A zone.

Bullard suggests two possible policies to deal with the danger of convergence to Japan-style chronic deflation a la point C.

His first policy suggestion is to engage in a policy of quantitative easing if deflation threatend: have the Federal Reserve expand its balance sheet even after pushing its policy interest rate i to the floor and take duration, systemic, and possibly default risk onto its own books. In this model, such a policy of quantitative easing can be interpreted as changing the position of the "Wicksellian Balance" line when inflation is very low. With more risk of various kinds taken onto the government's books in response to threatening deflation, firms are more willing to invest at higher real interest rates. Thus the "natural" real interest rate falls, and falls discontinuously if the quantitative easing program is switched on discontinuously when, say, inflation falls below zero. The effect is to change the model to:

When inflation hits zero and the quantitative easing program begins, the policy interest rate i consistent with Wicksellian balance rises--points lying below the now-discontinuous blue line produce upward pressure on inflation. The effect on the dynamics is to eliminate the zone that converged to point C, and thus to eliminate the danger of a Japan-style chronic deflation.

A second policy suggestion is for the Federal Reserve to contract rather than expand its assets when deflation begins: to engage not in quantitative easing but rather in contractionary open-market operations to raise the policy interest rate i away from zero to some higher level, like so:

Bullard's thought is that this would eliminate the danger of an adverse outcome by eliminating the point B where the red and the blue curves cross. If point A is the good stable equilibrium and point B is thought of as a bad alternative equilibrium, adopting a monetary policy that keeps the curves from crossing at point B and so eliminates point B entirely might help the sitaution.

A little investigation, however, shows that under these dynamics the correct conclusion is otherwise. The problem is not the existence of point B. Rather, the problem is the existence of a zone within which the dynamics of the system drive it to the absorbing point C. This second policy suggestion increases the size of that unfortunate region.

The dynamics of our new system are given by:

Tracing paths through these dynamics, we see that the boundary between those states that converge to C and those that converge back to normal affairs at A has shifted from the green curve to the orange curve. Basically, almost every path that ends in any deflation at all now converges to point C--only those where the pace of deflation is trivial and is accompanied by aggregate demand well above potential output escape the black hole at point C:

Even though the point B at which the curves crossed is indeed eliminated, raising interest rates if the economy actually goes into deflation does not diminish but rather increases the peril of a bad outcome. This should not be surprising: if expanding the money stock via quantitative easing helps, how can contracting the money stock--which is what raising the policy interest rate is--fail to hurt?

To recap: I think Bullard has been led astray in part of his analysis. I think the flaw is in his conceptualization of point B--where the curves cross--as an equilibrium at which the economy might sit. The actual equilibria in his model are, I think, point A--where active monetary policy keeps inflation near its target and aggregate demand near potential output--and point C--where the policy rate i is at its zero bound and where the deflation rate π is at its lower bound as determined by the economy's resistance to nominal wage declines.

Thus Bullard thinks in terms of eliminating point B where the curves cross, when he ought to be thinking of how to eliminate the zone of initial conditions that converge to point C. He--correctly--concludes that policies of quantitative easing can help eliminate 'The Peril." But he also, I think incorrectly, concludes that policies that reduce the set of states in which very low interest rates are pursued (or that shorten the duration of very low interest rates) can help. However, if my analysis is correct, such contractionary policies would oly hurt. True, they eliminate the point B where the curves cross. But we are interested in where the curves cross only when those points are where dynamic trajectories end. And that is not the case here: here eliminating point B via raising policy interest rates in deflation states increases rather than decreases the zone that converges to the bad equilibrium C.

Jess Benhabib et al. (2001), "The Perils of Taylor Rules," Journal of Economic Theory http://www.columbia.edu/~mu2166/perils.pdf

James Bullard (2010), "Seven Faces of 'The Peril'" http://research.stlouisfed.org/econ/bullard/pdf/SevenFacesFinalJul28.pdf