This post is an attempt to give some intuition for my previous post.
There are two differences between a supertanker and a canoe:
1. Position. You can easily pick up a canoe and make it jump a nautical mile north or south of its present position; while a supertanker really wants to stay where it is.
2. Direction. If a canoe is heading north at 2 knots, you can easily turn it to head south at 2 knots; while a supertanker heading north at 2 knots really wants to keep on heading north at 2 knots.
The Calvo Phillips curve is a hybrid of supertanker and canoe. It is like a supertanker in terms of position, but like a canoe in terms of direction. It's really heavy, so it can't jump; but it can change direction instantly. Imagine a boat that's really heavy, but really short and with a really powerful rudder, so it can turn on a dime. That's the Calvo Phillips Curve. The price level cannot just jump; but the inflation rate can change instantly.
(A Calvo hybrid boat is really a fleet of boats. In each period a small fraction n of the boats, picked at random, get to act like a canoe that can jump to a new position instantly. But the vast majority (1-n) of the boats just sit there anchored, motionless in the water.)
If monetary policy tries to make the boat do something it doesn't want to do, there's going to be a recession (if it tries to move it southerly), or a boom (if it tries to move it northerly).
If the boat is a canoe, that is quite happy to change position or direction, changes in monetary policy cannot cause recessions.
If the boat is like a supertanker, that is not happy to change either position or direction, changes in monetary policy should only change its direction very slowly. Don't try to change the price level too quickly; don't even try to change the inflation rate too quickly.
If the boat is a heavy but short Calvo hybrid, with a very powerful rudder, it's OK to make the inflation rate change quickly; just don't try to make the price level change quickly.
When I look at the data, not just the recent recession, but at previous recessions, I don't think I see a Phillips Curve that is like the Calvo hybrid that can turn on a dime but is too heavy to jump to a new price level. I think I see a fleet of boats. Some of the boats in the fleet are like supertankers, that really don't want to change direction, let alone position. And some of the boats in the fleet are like canoes, that are perfectly flexible and can change both position and direction very easily.
Suppose the whole fleet is heading slowly north, at 2 knots. Then the monetary pilot makes a mistake, and jumps 2 nautical miles to the south. All the canoes follow the pilot easily, and jump to the south. The supertankers desperately try to turn to the south, but keep on heading north at close to 2 knots. The average position of the fleet is halfway between the supertankers and the canoes, so it too jumps 1 nautical mile south. There is a temporary deflation. But the fleet is in recession, because the supertankers are trying to do something they really don't want to do.
Mistakes happen. The important question is: what should the monetary pilot do next?
Inflation targeting says the monetary pilot should make the fleet start steaming north again at 2 knots. But the supertankers are still heading north at 1.9 knots. So he needs to make the canoes follow him north at 2.1 knots. But one hour later the monetary pilot will still be 0.8 nautical miles south of the supertankers, so the supertankers will still be trying desperately to turn south, so the fleet will still be in recession.
Price level path targeting says the monetary pilot should jump 2 nautical miles north, then keep on steaming north at 2 knots. The canoes instantly jump to follow the pilot, and the supertankers keep on steaming north at 2 knots. The recessions ends, because the supertankers stop trying to turn south.
NGDP level path targeting is the same (in this example) as price level path targeting. (The NGDP level path monetary pilot also watches for signs of boom or recession, and not just the path of the fleet).
If the fleet were full of Calvo hybrid boats, returning to inflation targeting following a mistake would work well. All the boats would be steaming south trying to catch up to the monetary pilot. The monetary pilot should just jump north to the centre of the fleet, then steam north again at 2 knots. The Calvo hybrids would instantly turn and steam north at 2 knots too, and the recession would end immediately. Price level path targeting would work less well, because it would cause a boom as the Calvo hybrid boats tried to jump north after steaming south.
If the fleet were half canoes and half Calvo hybrids, it's a little trickier. The best policy would be a mix of inflation targeting and price level path targeting. After a mistake, the monetary pilot should jump to where the Calvo hybrids are, and then steam north again at 2 knots. The canoes will instantly jump north too. But the fleet will still be to the south of its original path, because the Calvo hybrids had been steaming south for the past hour trying to catch the monetary pilot.
I think the fleet has some supertankers in it, as well as canoes, and maybe some Calvo hybrids too.
That's probably about as far as I can push this metaphor. It's complicated by the fact that supertankers and Calvo hybrids will also try to steam towards where they think the monetary pilot will be in future, not just to where the monetary pilot is now.
[Update. let the position of the monetary pilot boat be P*(t). Let the position of boat i be Pi(t). Let the average position of the fleet be P(t). Each boat wants to stay as close as possible to the monetary pilot. An individual Calvo boat knows it can only act like a canoe once every 1/n periods, on average, and stays anchored and motionless in the water otherwise. So it always jumps ahead of where it thinks the pilot boat is heading. That means a fleet of Individual Calvo boats acts like a Calvo hybrid, that can change direction instantly to follow the pilot but cannot jump to where the pilot is. If P(t)>P*(t) there is a recession at time t.]
[Udate 2: there are two known ways to build a supertanker Phillips curve. The first is the Mankiw/Reis method, where the captain of the supertanker rarely pays attention to where the pilot boat is going. The second is the Taylor method. The individual Taylor boat is identical to the individual Calvo boat, with just one imprtant difference: the individual Calvo boat is allowed to weigh anchor and jump at random intervals; the individual Taylor boats take turns to weigh anchor and jump. So the representative individual Calvo boat that jumps comes from the middle of the Calvo fleet; the individual Taylor boat that jumps comes from the tail end of the Taylor fleet. A fleet of individual Taylor boats with a lot of real rigidity so they like to stick together if possible, as well as follow the pilot, will behave like a supertanker.]