I’ve been listening to NPR’s Planet Money podcasts recently, and a couple of days ago I listened to this one about economic bubbles. Bubbles are what some people call it when the price of something goes up and up and then crashes, and often you could kind of see that a crash was coming sooner or later. Sometimes people say that people were silly to spend all the money buying the thing when it was expensive. The podcasters spoke to Nobel Prizewinner Robert Schiller, who thinks the people driving up the price were being silly, at least sometimes. They also spoke to Eugene Fama, who shared the prize, and thinks that people aren’t being silly; you just can’t tell when something’s a bubble and when it isn’t. It was all pretty good-natured. Schiller and Fama shared their prize with another guy called Lars Peter Hansen, if you’re interested.
Now, regular readers will probably have noticed that one of my hobbies is duplicating the intellectual efforts of other people, and today is no exception. I had a bit of a think about bubbles. Silliness is hard to model, so I tried to think of a kind of situation where the price of something might go up and up and then crash, even though the investors are going into it pretty much with their eyes open.
Here’s what I came up with. You have some kind of commodity which is generating a pretty good income at the moment, but you don’t know how long it’s going to carry on doing it. Maybe it’s a tulip farm and at the moment people are paying top dollar for tulips, but you don’t know how long the craze is going to last. Maybe it’s a share in Justin Bieber’s record company. How do you value something like that?
Well, you need to quantify how long you think it’s going to last. Maybe you think it’ll last something between one and ten years, and your credences are distributed equally over one year, two years and so on up to ten years. That lets you put an expected value on it. Now, if you’re a Bayesian and it’s still popular after a year, you’ll think it has between one and nine years left, and your credences will be equally distributed over one year left, two years left, and so on up to nine years. This lets you put a new expected value on it, and it’ll be lower than it was a year ago. So the expected value of Bieber's tulip farm goes gradually down, until sometime during this decade people lose interest in his tulips and the value of the farm crashes. That’s not a bubble. In a bubble the price is meant to go up until the crash.
Suppose you model your uncertainty about how long the craze will last differently. Instead of thinking how long it’ll last, you think about what the craze’s half-life is. Maybe you think it’s got a half-life of between one and five years, and you distribute your credences equally over half-lives of one year, two years, and so on up to five. (The craze’s half-life is the length of time in which it has a 50% chance of ending. In general, if the half-life is h years, the chance of the craze surviving the next h*n years is 0.5n.)
Now what happens if the craze is still going after a year? Well, that was more likely to happen if the half-life was long, so you end up redistributing your credences to make a long half-life more likely and a short half-life less likely. This means that after a year the value will go up. And it’ll keep going up until the craze ends. That’s got the rise-rise-crash character of a bubble, but nobody has had to do anything silly. This is true even though it’s predictable that the price will rise until the crash, and even if the investors are all sure the crash is coming sooner or later. I guess that if prices going up and up and then crashing was always this kind of phenomenon, that would mean Fama was right and Schiller was wrong. But I don’t really know; I just listened to one Planet Money podcast. (Well, actually I’ve listened to about a hundred Planet Money podcasts, but only one was about bubbles.)
Is this a reasonable way of dealing with uncertainty about how long a craze will last? Sometimes, it probably is. People have been into Barbie dolls for longer than they’ve been into Loom bands, and this inspires confidence that Barbie will still be around after Loom bands are gone. The longer Loom bands stick around, the longer they might seem to have left. When something’s been around as long as Barbie and Coke, it’s hard to imagine it ever going out of fashion.
So here’s another question: do real commodities exhibiting the price-rise-then-crash phenomenon fit this model? Well, no. Not exactly. The dotcom bubble was based on a load of companies which often weren’t bringing in much income at all at the time (right?), with investors betting on future income. But I think that can still fit into the model. Even if you only projected that the income would come, say, five years into the craze, the expected value still goes up as the probable half-life goes up, and the probable half-life carries on going up until the craze ends (or until a bunch of similar crazes end). So maybe the dotcom bubble was like that too. And the tulip bubble. Anyway, I recommend the podcasts.