This is in response to John Cochrane's good post.
First off, you need to understand that Keynesians don't actually believe their own New Keynesian models. When they reason informally, they always assume inertia in actual or expected inflation. But the standard New Keynesian model, with the Calvo Phillips Curve, doesn't have inflation inertia in it. So I am going to build a New Keynesian model that does have inflation inertia in it.
If there's inflation inertia, so actual and expected inflation can't jump, then an upward jump in nominal interest rates means an upward jump in real interest rates, and that means a recession. What happens next?
The second thing you need to understand is that Keynesians don't actually believe their New Keynesian models. When they reason informally, they worry about the economy going off on an equilibrium path that does not eventually converge to the natural rate. They don't believe the New Keynesian model when it just assumes eventual convergence to the natural rate. Your diagram was neat, but it only showed the equilibrium paths that converge eventually to the natural rate. Keynesians believe that a permanent rise in the nominal interest rate could cause the economy to have ever-accelerating deflation and an ever-worsening recession. My model shows that as possible.
The third thing you (and the Keynesians too) need to understand is that the nominal interest rate is a signal. A really stupid way for central banks to signal, but a signal nevertheless. And the effects of any signal depend on how it is interpreted. How does it change people's beliefs about the central bank's beliefs and preferences? My model illustrates that.
So, here is my model:
At the beginning of every period, 99% of firms announce their prices for the beginning of the next period. There is a one-period lag in changing prices. This is a quick and dirty way to get inflation inertia into the model, but it's no worse than the Calvo fairy as a "microfoundation". The inflation rate is 99% pre-determined one period in advance. The remaining 1% of firms can change prices any time they like.
Next, Nature chooses the natural rate of interest. She gives the central bank, and each individual firm, imperfect information about her choice, but she tells them all different things, so nobody knows what she told the central bank.
Next, the central bank announces the nominal interest rate.
Next, people choose how much to buy. Standard Euler equation stuff, with a stochastic natural rate of interest.
Central bankers differ by type. All the sensible types of central banker have a (different) inflation target. They don't like inflation deviating from their target, and they don't like output deviating from the natural rate. Standard assumption, except firms don't observe the inflation target and must infer it from the nominal interest rate. But there is also an evil type of central banker who wants ever-increasing deflation and an ever-worsening recession, and so always wants lower inflation than firms think he is targeting.
The Phillips Curve for this economy will be (roughly):
p(t) = E(t-1)[p*(t)] + 0.01y(t)
where p(t) is actual inflation, p*(t) is the central banker's inflation target, E(t-1)[p*(t)] is firms' prior expectation of the central banker's inflation target, and y(t) is the output gap. The output gap has a very small effect on current inflation, because only 1% of firms can adjust their prices any time they like. All the action in this model comes from the remaining 99% of firms updating their priors about the central banker's inflation target.
Suppose we start in an equilibrium where the central banker has been on the job long enough for firms to have inferred his type with certainty. And suppose he is a sensible type, and is targeting 2% inflation. Every year, the 99% of firms set their prices for the following year 4% higher than the price level in the preceding year. Actual inflation will always equal 2%, plus or minus a small amount of noise from the 1% of firms who have perfectly flexible prices. The central banker sets the nominal interest rate equal to his best estimate of the natural rate, plus target inflation. But his estimate will be imperfect, because Nature did not give him perfect information. If he sets the nominal rate too high, there is a recession, the 1% of firms cut their prices below those of the 99% of firms who can't, and actual inflation comes in slightly below target. If he sets the nominal rate too low, there is a boom, the 1% of firms raise their prices above those of the 99% of firms who can't, and actual inflation comes in slightly above target.
That's what we see in normal times, when the inflation target is known. And we will observe zero correlation between nominal interest rates and inflation. (That is an immediate implication of rational expectations on the part of the central banker, since his inflation forecast errors must be orthogonal to his information set).
Now suppose that the old central banker retires, and a new one is appointed. Firms don't know his type. They will need to infer it from how he sets nominal interest rates.
Initially, just to keep it simple, suppose that firms do know the new central banker is one of the sensible types. They just don't know his preferred inflation target.
Suppose the new central banker has a higher inflation target than the old one. But firms don't know this. So he has a higher inflation target than they expect. He observes the inflation rate implied by the 99% of firms who have already announced their prices for next period. He sets a nominal interest rate such that the real interest rate will be below what he expects the natural rate to be, so the 1% of firms will raise their prices. There is a boom, and inflation rises above the previous target, part way towards the new target.
Firms reason that he might have a higher inflation target than the old central banker, or he might just have made a mistake about the natural rate of interest. They slowly revise upwards their Bayesian priors about his inflation target, if he makes repeated "mistakes" in the same direction.
When the inflation target changes, we should observe an immediate negative correlation between nominal interest rates and the inflation rate. Then a transition period with a negative correlation between real interest rates and the rate of change of the inflation rate. Then finally a positive correlation between nominal interest rates and inflation, once firms have figured out the new inflation target.
Now let's introduce the possibility of an evil type of central banker.
Suppose firms see that the new central banker has set nominal interest rates too high, given the 99% predetermined inflation rate, so that inflation falls, and there is a recession. There are three possibilities: maybe the new central banker just over-estimated the natural rate; maybe the new central banker is sensible, but has a lower inflation target than the old one; or maybe he is evil. They will adjust their priors each period according to the evidence.
If they see the central banker set real interest rates too low in subsequent periods, so that inflation rises back to target, they figure it was just a mistake, and the inflation target has not changed.
If they see the central banker cut nominal interest rates as inflation falls, so that real interest rates return to normal as inflation stabilises at the new target, they figure the inflation target is lower.
If they see the central banker keep real interest rates permanently too high, they figure he must be evil. Inflation falls without limit. If he wanted ever-increasing deflation and ever-worsening recession he would set the nominal interest rate too high and keep it there.
Any competent grad student could do the math and solve this model formally.
Of course, if we changed the model, so that new central banks could simply announce their type, or else signal their type in a more sensible way, the relation between nominal interest rates and inflation could be very different.