Supertanker and canoe Phillips Curves, and inflation targeting

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.]

Comments

[rsj]

So recessions are everywhere and always caused solely by the absolute price level being too high or too low. They can *never* be caused by relative prices being out of whack? Because if the recession was caused by house prices being too high or too low relative to other prices, or by wages being too high or too low, or by debt levels being too high or too low, then you will not have a single answer to this question. And sometimes there is no monetary policy that can help in your fleet of boats of various sizes scenario.

[Nick Rowe]

rsj: you make a good point. It was one I thought of, but set aside to try to keep it simple.

[rsj]

Nick,

But I thought you *were* making the point with the boat and canoe example, because by pointing out that there are many prices with different levels of inertia, you are saying that relative prices are going to be changing, out of necessity, whenever the general price level changes.

Going back to boats and canoes:

Case a)

The boat price should be half the canoe price -- then we want inflation to help us get to the optimal ratio

Case b)

The boat price should be twice the canoe price -- then we want deflation to help us get to the optimal ratio

In which case we can view a general price change as merely the byproduct of different relative price changes.

There should be no target for the aggregate price level path per se, or an inflation target. The target should be whatever is necessary to allow relative prices to adjust with the least adjustment cost. We should be targeting things like output and employment, not inflation or price paths.

[Nick Rowe]

rsj: "We should be targeting things like output and employment, not inflation or price paths."

We tried that in the 1960's and 70's. It didn't work. This is what happened. We now know why it didn't work. The pilot was trying to maintain a position just a bit to the north of the centre of the fleet. And the fleet was trying to be where the pilot was. So pilot and fleet chased each other north faster and faster. And the 82 recession was what happened when the pilot realised it wasn't working, so slowed down heading north, and it was very hard for the supertankers to slow down, so they overshot the pilot.

[rsj]

Nick,

OK, sometimes things work and sometimes they don't. We tried monetary policy in 2008 and it didn't work, so let's never try it again. Japan tried it in 1997 and it didn't work, so that's it for monetary policy, right?

Sometimes you have to turn left to go right (e.g. when you are skidding). But not always.

[Patrick]

Again, a little out of my depth here ... but it occurs to me that you might also build a supertanker Phillips curve by having a flotilla of supertankers all lashed together? It would take time for the captains to figure out how to turn the flotilla while avoiding collisions. The captain could be fully seized of the pilots intentions, but s/he can't follow the pilots advice without first coordinating with the rest of the flotilla.

[Nick Rowe]

Patrick: you are not (obviously) out of your depth. Keynes sketched (very briefly) something like that coordination problem theory of price stickiness in the General Theory. I have vaguely sketched something similar on this blog. But we only have a very vague sketch. Nothing like a blueprint. It's one of those ideas that just isn't easily modelled. Imagine trying to model the 8 oarsmen in a racing 8 trying to change speed yet keep rowing in unison without a cox calling out the beat. All you can really say is "it will be hard".

[Nick Rowe]

Even the Taylor model I have sketched above can't be solved for an analytic solution (the Calvo one can, which is why it gets used). But we can simulate it on a computer. IIRC (it may have changed recently) the Bank of Canada computer model has a sort of Taylorish model in it. But economists keep on assuming an all Calvo fleet when they build models to show Inflation Targeting is best. Because those are the only models we can build. So I have to resort to telling stories about boats, trying to make the point that needs to be made. Pretty pathetic, really. And I could never get a story about boats published in a real journal.

[Kim McPhail]

I think Nick's metaphor is a great one … and it can be pushed even further...

Suppose that, in addition to the supertanker and canoes, there are a lots of houseboats spread out around the ocean.

The supertanker steams along, and wants to make some canoes do what they believe is the wrong thing to do. That creates a kind of “nautical friction.”

That “nautical friction” creates waves.

The people living in their houseboats aren't interested in whether the supertanker wants to go at two knots an hour or three or four, and they don't know much about the way that pilots drive supertankers, and they don't much care. They don't really know or care whether supertankers and canoes can jump from one place to another, or make other canoes jump, and just how many canoes there are and whether they are all rigidly linked together or whether they leapfrog.

They just want their houseboats to be safe.

They go to work each day, and when they come home, they want their houseboat to still be there, and not to have capsized because of the waves that might arise because of the nautical frictions between the canoes and supertanker, many of whose owners go home to their houses on land every night, and are perhaps less affected by the waves on the ocean compared to the houseboats – or at least that's what houseboat owners think.

They might even believe or have heard rumors that supertankers and canoes, in some places at least, like to be unpredictable. They heard some gossip, perhaps, that some supertankers and some canoers believe that only when supertankers surprise canoes and houseboats by unexpected changes in supertanker direction or speed can waves be created that may tip a lot of houseboats. They may also have heard through the grapevine that if a supertanker tries to jump, rather than change direction gradually, that may create a tsunami, not just big waves, and they may be pretty worried about that. They might have heard that what matters is how many waves get created in total, so that changing direction fast may cause bigger waves (at first anyways) which being less spread out and less smooth, create more water turbulence. The houseboat owners may fear that even more houseboats will capsize. And maybe quite a few canoes, too.

Perhaps the supertanker is thinking more about whether a rapid change in direction or speed will cause more houseboats and canoes to sink, than spending most of the time wondering about how many canoes can change direction at any instant and about just how many canoes there are in total (n). Or, at least, maybe that's what the people in the houseboats hope the supertanker's crew are thinking about.

And there seem to be other things rumblings in the ocean too...

The first thing is a rather unusual one compared to thinking about the importance of the speed of many things other than a supertanker for canoes and houseboats on the ocean.

Some people say that the ocean seems endless. So why does it matter, some say, if the supertanker goes at 2 knots an hour, or three, or 4, or 10, or 50, or 100, or 1000? Isn't the most important thing not to create waves? Because the supertanker isn't going 2 knots an hour with the objective of arriving anywhere in particular. It could just go at two knots forever... Some don't see the need for an anchor.

Other people may say that because the world is round, the ocean isn't really endless, even for a supertanker that can replace crews over time. And that if a supertanker speeds along the ocean at higher and higher speeds as it circles the globe, that's not going to make the waves any smaller and may create a very big wake that can capsize houseboats and canoes and even affect or tip even a supertanker.

Some people say that's why it's a good thing for a supertanker to announce what it sees on the horizon, and to try to seem predictable. And canoes, too. Even houseboats.

The supertanker could announce a change in course ahead of time and say – “beware canoes and houseboat's, I'm a supertanker and I'm changing direction. Prepare yourself for the waves.” And do things gradually. These are familiar issues to people who engaged in these kinds of inflation debates decades ago, though they have somewhat fallen out of fashion in this age of squalls and turbulent ocean conditions. Debates are more technical nowadays and therefore often less comprehensible, maybe because ocean turbulence is more chaotic.

Others say, no: surprise canoes and houseboat owners, shock them. Fool the canoes and the houseboats. It's the only way to get anyone to move their boats, they say. But with such tactics you may just end up with a lot of capsized canoes and houseboats and a lot of pirates hovering around hoping to get-rich-quick. Moreover, beliefs about who are the pirates in such situations may depend on whether you live on a houseboat, or own a canoe, or are on a supertanker.

Some people say that if you start to speed up a supertanker, and it gains some momentum, then it's not only hard to change direction, it's not only hard to slow it down once it picks up speed, but it may even be hard to prevent it from continuing to speed up, especially when the canoes and houseboats figure out that the supertanker seems to have been going faster and faster. Most say that can be done, but many say it can only be done by creating even bigger waves down the road (or in the water, to be more precise).

Some say that speeding up a supertanker actually can reduce waves and prevent canoes and houseboats from capsizing. But others say that only works when canoes and houseboats don't know the supertanker is speeding up and that waves are coming. But, even if that works, for how long, others ask? And will the waves the next day or month or year be even bigger?

Some people say that speeding up a supertanker affects canoes and boats differently. Big canoes (e.g., barges) and really big houseboats (e.g. floating casinos) can protect themselves against the waves created by an accelerating supertanker, they say – whether it's accelerating because it's going faster or because it's changing direction. Can little canoes and houseboats do the same? If so, how?

Some say that an accelerating supertanker can actually benefit a lot of canoes, while sinking a lot of houseboats. Other say the opposite. Who is correct?

These days, perhaps a majority of the people discussing such things never owned a houseboat or a canoe or were on a supertanker the last time a supertanker began accelerating pretty fast. Some people, perhaps mostly the ones who have never experienced it, say that would not happen these days. Why not?

Different countries and people of different ages have some, lots, or no experience with what happens in such scenarios. Perhaps some views depend on whether those who hold them are on supertankers, or in canoes or on houseboats? If so, perhaps their beliefs depend on how much they know about canoes and supertankers (if they own houseboats), or about houseboats and supertankers (if they have a canoe), or about houseboats and canoes (if they are on a supertanker.)

Waves on the ocean aren't just caused by a supertanker picking up speed or changing direction. Everybody knows that. In fact, the wake of a supertanker steaming along at a steady 2 knots doesn't cause any waves, most canoes seem to think. Many people on houseboats seem to believe that too, at least if speedometers measure things right and everyone makes the appropriate adjustments to everything. Slowing down to 1.5 knots or speeding up to 2.5 knots might cause some ripples but canoes and houseboats don't really pay much attention to that because they don't have any reason or need to – it doesn't create enough waves to rock their boats, and that's what matters to many of them.

BUT the winds – the real winds – have been unusually calm for quite a few years. There are glimmers that some canoes and supertankers and houseboats believe that may not last for all that much longer, though they can't seem to agree on why. While the real winds were unusually mild over those years, many people built canoes and houseboats that aren't as good at dealing with wave turbulence as they used to be – unlike those that were routinely built when winds were typically more gusty.

Even if the supertanker doesn't change direction, and still trucks (or sails) along at two knots per hour quite steadily, there are some who probably worry that when the real winds pick up to something closer to their normal velocities, that's going to cause big waves on the ocean, and maybe even a big storm. There may be concern that many houseboats and maybe canoes too will capsize – even if the supertanker keeps on chugging along at a speed of two knots. Some might fear that the supertanker's response to the pickup in real winds may even amplify the size of the waves and water turbulence and nautical friction. Something about … leaning against – or with – the wind, perhaps...

Some houseboats have more equity in their boats than others. Some houseboat owners with more experience and history may have realized long ago that the real winds were unusually mild, that such tranquil conditions were unlikely to persist, and they may have taken steps to make sure their houseboats are resilient to a pickup in speed of the real wind to more typical velocities.

Other houseboat owners, less experienced and likely younger – not knowing the typical historical force of the real winds – may be in more difficult situations, with less equity in their houseboats, and with less resilient boats. They may have big mortgages on their houseboats.

Some people believe that if the canoes believed the supertanker when it announced ahead of time that it was thinking of changing speed or direction, then changing course would not create much “nautical friction” with canoes. So there would be fewer waves, and the houseboats would be more insulated.

But many canoes and houseboats know that it's not just a change in direction or speed of a supertanker that can cause waves, and it's not just a pickup in the real wind to more ordinary levels that can cause waves, it's also a myriad of other factors that put houseboats at risk – like unexpected currents, squalls, gales, submerged rocks – and even pirates.

On top of that, there is the fact that turbulence in other oceans can literally spill over into the ocean the supertanker is chugging along in with a current speed of about two knots. A lot of those other factors are hard to predict too. Maybe many canoe owners think more often about such things than they used to.

Some houseboats may therefore take steps to try to make their houseboats more resilient, and so be less buffeted by the waves and less likely to capsize from any of those factors which they did not cause and which are quite beyond their control. Many other houseboats may find that unaffordable at the moment, and feel more vulnerable.

A lot of houseboat owners may be worrying about what the banks will do if there are big waves and a lot of turbulence – or even if the banks just fear there may be bigger waves (even if there aren't), and start to panic – a panic attack by their bank. That bank reaction alone would add to the waves in the ocean, quite apart from anything else the supertanker is doing (but based on what the supertanker might do). An unpredictable, bizarre, superficial, and thoughtless response by the supertanker would greatly compound the turbulence, and might capsize many houseboats. There may also be a belief from houseboats that during a bank's panic attack, even worse, their bank would ignore the law of the sea. Just those things may be enough to capsize many more houseboats and even canoes, too.

I think these are the kinds of things that lots of supertankers and canoes are hopefully thinking about, and houseboats too, in addition to more technical things like what proportion of canoes can move or jump at any given instant in the ocean, and how far, and how big n is (the total number of canoes in the water).

Because there are a lot of Canadian houseboats out there on the ocean.

There's an old joke about that kind of thing. The general commanding a military destroyer yelled through a bullhorn when he saw what he thought was a small boat in his way: “small boat... small boat...yield, yield, we're coming through”. The boat didn't move. The general got out the bullhorn again and said “small boat...small boat...yield, yield, we're coming through...this is the second warning...” The boat didn't move. The general got out the bullhorn again and yelled: “small boat...small boat...yield, yield, this is your last warning. You're just a small boat. And I'm a supertanker. Get out of the way. Yield”

Then a reply came back, flashed to the supertanker in the 'light-equivalent' of Morse Code: “You may be a supertanker...but I'm a lighthouse...”

The supertanker changed course.

That's neither here nor there in the metaphor of the supertanker and the canoes and two knots an hour. But we read that joke, probably in Reader's Digest many decades ago, probably while we were in some doctor's waiting room (just about the only place we'd see such magazines).

It's just to say that perhaps there are many other important things for the supertanker and the canoes and also for all those houseboats other than what proportion of canoes can change direction at any instant (even if they wished to), and what proportion can't, and whether the canoes are in one particular order and closely linked, or whether they play leapfrog when waters get turbulent.

Osprey

[rsj]

Here is some data that supports the theory that even though median CPI prices may have been stabilized, this occurred with great dispersion of prices.

I.e. I'd like to see the trade-off between making the relative price adjustments that are necessary easier and keeping the overall price level moving along some target path.

It seems to me that a CB that sets itself up to follow the latter strategy come hell or high water vis-a-vist relative price changes is going to either fail at forcing the aggregate prices to follow the pre-scribed path, or it will succeed in screwing up relative prices.

[Ryan V]

I don't see why there should be a trade-off between allowing relative price adjustments and keeping the overall price level on target. In fact, keeping the overall price level steady should make the relative price adjustments easier.

The less certainty you have about what other prices will be, the harder it is to set your price relative to them.

[RN]

Those are not criteria on which to qualitatively a supertanker and a canoe. A supertanker going north can be made to go south very quickly. It's simply a matter of input energy.

The same could apply to the economics.

[Doug M]

"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."

This is an ugly metaphor... How about this.. it is a Super tanker. Its postion and motion is slow to change. However, as the water shifts underneath it, its motion realtive to the water can can move rapidly. A canoe, can change direction relative to the land quickly, but is at the mercy of the shifting seas.

[rsj]

In fact, keeping the overall price level steady should make the relative price adjustments easier.

Only if relative prices are constant. I.e. outlook for future conditions is never updated, technology is constant, no new products are introduced, no changes in the foreign sector or demographic changes, no new information about the economy is learned, etc.

But if relative prices need to change, then holding average prices constant may be a real impediment, and it will be a real impediment when some prices have more inertia than others.