The greater fool theory 1: A mostly verbal mathematical model
Here is a brief description of the idea that price bubbles are caused by people buying something, not necessarily because they think it’s worth anything, but because they think they can find an even greater fool to buy it at a higher price. This continues until no more such fools can be found, and this bust drives prices back down to what they were before the boom began.
I didn’t see any references to mathematical models of the theory at Wikipedia or through Googling around a bit, so I made one up today at Starbucks since I didn’t have anything to read to pass the time. Because I’m not an economist, I don’t know how original it is, or how it compares with alternative models of the greater fool theory (if they exist). So, this is intended just as an exercise in modeling, explaining the model, and hopefully shedding some light on how the world works. I’ve kept most of the exposition straightforward and largely verbal, so that you don’t need to know much math at all to understand what the model says and what its implications are.
In part 1, I lay out the logic of the model and explain enough of it to show that it is capable of producing a single round of boom-and-bust for price hype. Part 2 will provide more mathematical detail about how the dynamics unfold, a phase plane analysis, and graphs of how the variables of interest would change over time, to better wrap your brain around what the model predicts.
This is a dynamic model, or one that tracks how things change over time — after all, we want to see how price, the number of fools, etc., evolves. It is made of several differential equations, and all these equations say is what causes something of interest to go up or go down over time. (You may recall that the sign of a derivative tells you whether a function is increasing or decreasing, and the magnitude says by how much.) I’ll only explain what is absolutely necessary for the reader to see what’s going on, with the less necessary math being confined to footnotes.
First, we set up the basic picture before we write down equations. My version of the greater fool theory goes like this. There is a population of people, and during a price bubble they can fall into three mutually exclusive groups: suckers (S), who are susceptible to joining in on the bubble; investors (I), who currently own the speculative stuff (such as a home bought for speculation); and those who are retired from the bubble (R), who used to be investors but have gotten rid of their investment. And of course there is the price of the thing — I model only the extra price that it enjoys due to hype (P), above its fundamental value, since this is the only component of price that changes radically during the bubble.
I set the population to be fixed in size during the bubble, since growth or decline is negligible over the handful of years that the bubble lasts. I also set the amount of speculative stuff to be fixed, which is less general — supply should shoot up to meet the rising demand during a bubble. So, this model is restricted to cases where you can’t produce lots more of the stuff, relative to how much already exists, on the time-scale of the bubble’s boom stage (say, 5 years or less). Or perhaps no more of it will be produced at all, such as video game consoles from decades ago that the original manufacturers will never bring back into production, but which nostalgic fans have taken to buying and selling speculatively (like NEC’s TurboDuo). Last, the amount of stuff that each investor has is the same across all investors and stays constant — say, if each investor always owned just one speculative home.
At the start of the bubble, there is a certain number of early investors. In order to sell their stuff, they need to meet a sucker to sell it to. When they meet — and I assume the two groups are moving around independently of each other — there is a probability that the sale will be made. If they make a deal, the sucker is now an investor, and the former investor is now retired. In this model, retireds do not again become suckers — they consider themselves lucky to have found a greater fool and stay out of the bubble for good afterward. That’s the extent of how people change between groups.
As for price hype, again I’m not an economist, so the exact formula may differ from what’s standard. I take it to respond positively to demand — namely, the number of suckers — and that there is a multiplier that serves as a reality check. This reality check should be weak at the start when most non-investors are suckers, and should be strong near the end when most non-investors are retired. In other words, the price hype at the beginning is a near total distortion — nearly 0% accurate — whereas the price hype near the end is nearly 100% accurate. This will make more sense once we write down formulas.
Now we get to the differential equations for how these things change. We write down one equation for each variable whose values we’re tracking over time. I use apostrophes to denote the derivative with respect to time (i.e., rate of change):
S’ = -aSI
Since suckers can only lose members (by turning into investors), there is only one term, and it shows how suckers decline (negative sign). Remember, retireds do not go back into the pool of potential buyers. And investors either make a sale and go into the retired group, or they sit on their stuff in hope of selling, so they never contribute to the growth of suckers. Thus, there is no growth term. The parameter a shows the probability that, when a sucker and an investor meet, the investor will transfer his stuff to the sucker. (“Parameter” is another word for “constant,” in contrast to a variable that changes.) The reason we use the product of S and I is that this is essentially the rate at which the two groups encounter each other when they move around independently of each other. [1]
I’ = aSI – aSI = 0
Investors both grow and decline, so one term is positive and the other negative. They grow by having a sucker join their ranks, which as we saw above happens at rate aSI. However, each time that happens, the investor loses his stuff and becomes retired. That happens at the same rate, and the negative sign just shows that this causes I to decline. When we simplify, we get I’ = 0 — that is, the number of investors does not change over time. That makes sense because each bundle of stuff always has an owner, regardless of how it may change hands, somewhat like the game of hot potato. When something doesn’t change, it is constant, so whenever we see I from now on, we’ll know that this is just another parameter, not a variable that changes. In particular, it refers to the initial number of early investors who get the bubble going.
R’ = aSI
Retireds never join the suckers again. And recall the mindset of a retired person — they knew the stuff was junk and are glad to have gotten through the selling process, so they cannot be sold the stuff again to become investors once more. Thus, there is no way for them to lose numbers. They grow by former investors making a sale and becoming retired, which once again happens at rate aSI.
Here’s the neat thing: notice that S’ + R’ = -aSI + aSI = 0. The sum of the two derivatives equals zero, and since taking a derivative shows the distributive property, this also means that (S + R)’ = 0. That is, the sum of suckers and retireds does not change over time. This makes sense since, if the number of investors stays constant, the leftovers — suckers and retireds — is constant, regardless of how each separate group grows or shrinks. We can take this further to note that S’ + I’ + R’ = 0, which means (S + I + R)’ = 0. That is, the combined size of all three groups does not change over time — which is just what we claimed by keeping total population size constant. (Otherwise, each group would have birth and death terms, aside from the terms that show how their members switch between groups.)
We’ll call this constant total population size N. So, S + I + R = N. Now, I is just a constant, so we’ll move it to the other side: S + R = N – I. We have two variables, S and R, but we just wrote an equation connecting them, so we can re-write one in terms of the other. I’ll choose R, but it doesn’t matter. So, R = N – I – S, and anywhere we see R, we can replace it with N – I – S. In other words, we’ve removed R from our focus — we can always get it from knowing what the variable S is, as well as the two parameters N and I. That means the equation for R’ only gives us redundant information, and we can ignore it. We can also ignore the I’ equation, since it just tells us that I is constant, and we’re only interested in things that change. So we’re left with just the S’ equation.
Now we move on to the price hype formula and how it changes over time. First, the formula for price as a function of demand and the reality check, since hype is never totally irrational and at least tries to take stock of reality:
P = bS(R / Rmax) = bS(R / (N – I))
Demand is driven by the number of suckers — the ones who eventually want to get in on the bubble — and the parameter b says how strongly demand responds to the number of suckers. The multiplier (R / Rmax) provides a reality check. If you landed from Mars and only knew the number of suckers, you would also want to know how many retireds there were — if there were few retireds, that would tell you the bubble had only just begun, so that hype is likely to be high and to go even higher short-term. Thus, this filter should not let much of the demand information through. Indeed, when R is very low compared to Rmax, the multiplier is near 0.
However, if you saw that there were many retireds, that would say the bubble was near its bust moment, and that the information from demand is very accurate by this point. Indeed, when R is near Rmax, the multiplier is near 1 and the filter lets just about all of the demand information through. What is Rmax? It is the value when no one is a sucker and everyone is retired, aside from the constant number of investors. Looking above at the equation S + R = N – I, we see that when there are no suckers, R = N – I.
Now we need to find the differential equation for how P changes over time. Using the product rule for derivatives [2], we get:
P’ = (abI / (N – I)) * S(2S + I – N)
Since a, b, I, and N – I are always positive, and since S is positive except for the very end of the bubble when it is 0, in the meantime, whether price hype shoots up or crashes down depends on whether the term 2S + I – N is positive or negative. It is positive and price hype grows when S exceeds (N – I) / 2, which is half the size of non-investors. It is negative and price hype declines when S is below (N – I) / 2. It is 0 and price hype momentarily stalls out when S is exactly (N – I) / 2.
Because the bubble starts with all non-investors being suckers, S is initially N – I, which is greater than (N – I) / 2. So at first the price hype shoots up. However, remember that S only declines — as more and more of the suckers are drawn into the bubble (some of whom may also make sales and become retireds), S will inevitably fall below (N – I) / 2 and price hype will start to contract.
When S inevitably reaches 0 — when all non-investors are out of the bubble for good — then P = 0 (recall that P = bS(R / Rmax)). Moreover, at that time P’ = 0 too. Thus, at the end, price hype has completely evaporated and it will stay that way. This is a single round of boom-and-bust for price hype.
In this post, I’ve shown how some pretty simple “greater fool” dynamics can lead to a boom-and-bust pattern for price hype. You can quibble with all of the assumptions I’ve made, but the model shows that the greater fools theory is a viable explanation for price bubbles. I’ve relaxed some of the assumptions to see if it makes a difference, like making the decline of S be a saturating rather than linear function of S, and so far they don’t seem to affect things qualitatively. A more realistic model would have P appear in the equation for S’ — that is, to have price hype affect the probability of making a sale. Or rather, the trend of prices (P’ ) should affect sale probability — if suckers see that price hype is increasing, they should want to get in on the bubble, and to stay put if price hype is dropping. Also, allowing retireds to re-enter the pool of suckers would be more general and would almost certainly lead to sustained cycles of boom-and-bust, rather than a single round. But that’s for another slow afternoon.
In part 2, I’ll go into more mathematical detail about how we see what states this system is at rest in, and whether they are stable to disruptions or not. I’ll look more at the formula for the maximum level of price hype, and interpret that in real-world terms in order to see what things will give us larger-amplitude bubbles. I’ll provide a picture of the phase plane, which shows what the equilibrium points are, and how the variables will change in value on their way from their starting values to the final ones. I’ll also have a couple of graphs showing how the number of suckers and retireds, and the amount of price hype, change over time.
[1] Draw one person at random, and the chance that they’re a sucker reflects S. Draw another one at random, and the chance that they’re an investor reflects I, since the draws are independent. The chance of doing both is just the product of the two separate probabilities.
[2] P’ = (bS(R / Rmax))’ = (b / (Rmax)) (S’ * R + S * R’)
A little algebra, which you can confirm by hand or using Maple, gives the equation in the main body for P’.
Labels: Behavioral Economics, Economics, mathematics, Modeling
Bubbles are supposed to be cyclical. Presumable, before the bubble there is a steady state situation, what causes the bubble mechanism to start? On the light side, your model does not take into account that a new sucker is born every minute.
Someone has actually come up with an economic theory on why we have price bubbles:
http://en.wikipedia.org/wiki/Hyman_Minsky
Minsky believed that stability bred complacency in financial systems and that this encouraged borrowers and lenders to become ever more reckless.
Minsky’s Actual Paper (PDF form)
http://www.levy.org/pubs/wp74.pdf
Another author also wrote something about economic bubbles:
http://www.amazon.com/Dollar-Crisis-Causes-Consequences-Cures/dp/0470821027
Duncan that the dollar acted like steroids for economies running large trade surpluses with the US and believe this led to hyperlending and the Asian currency crisis of 1997
Last but not least there is too big to fail:
http://www.amazon.com/Too-Big-Fail-Hazards-Bailouts/dp/081570304X/ref=sr_1_1?ie=UTF8&s=books&qid=1250767678&sr=1-1
The author of this book believes that financial institutions are inherently reckless because of government guarantees that underwrite reckless behaviour.
Yes I did get a degree in economics but I didn’t read any of the aforementioned texts for school and hardly covered any of the issues that the authors raised in my classes.
You may wish to look at some of the results coming out of experimental economics.
How steady state passes into the bubble phase?
Bubbles are supposed to be cyclical.
Right, this model just shows that we can at least get boom-and-bust. I’ll bet that if I allow the retireds to re-join the suckers, it will produce cycles. It’s like when an epidemic hits, and the people who are recovered and immune eventually lose their immunity and become susceptible to infection again.
Presumable, before the bubble there is a steady state situation, what causes the bubble mechanism to start?
In this model it’s an exogenous event. Somehow, a group of people get it in their heads to start speculating in some market. In the early 1990s, there was speculation in comic books. Starting in the late 1990s, they were speculating in houses. Why is it one thing at one time, and another thing at another? The model doesn’t try to explain that, only how the dynamics unfold once the initial speculators have arrived.
agnostic,
Retards DO rejoin suckers all the time, because they retired with money from the market, that is, they feel good about their chances to do it again.
Be kind enough to “allow” them to rejoin.
I just checked what happens when we let retireds re-enter the suckers, i.e.:
S’ = -aSI + lR
Where l is the rate at which retireds decide to come out of retirement. This assumes that they decide to do so independent of what other retireds are doing, and again ignores price or price trend, which is unrealistic.
It does not produce cycles. Rather, there is a single stable fixed point that has a positive value for both S and P — meaning that the suckers are never completely used up (now they can be replenished by retireds) and there will always be some degree of price hype. Moreover, if l is less than aI, the price hype will go up and then down, although not to zero; but if l is greater than aI, price hype only goes up to its stable value and never declines. So for some parameter values, this model doesn’t even produce a semi-boom-and-bust.
So, even allowing for retireds to get back in on the bubble won’t produce cycles in this model. We probably need a more realistic picture where P’ affects the rate at which suckers become investors, as well as the rate at which retireds decide to re-join the suckers.
There’s a paper by Doyne Farmer, among others, carrying out this kind of modeling experiment in more detail. IIRC, his three groups are fundamental investors, momentum investors and noise investors. The main result was that his market didn’t settle down to an equilibrium. And his momentum investors aren’t necessarily dummies. There are models in which information about fundamental values percolates down slowly to retail investors and stodgy fund managers. Under those assumptions, the movement of share prices does convey useful information.
The characteristic features of bubbles include the involvement of new players and the use of debt. The sign of new players is the magazine cover indicator in which news magazines at the peak all have covers devoted to real estate, internet startups, etc. So perhaps an epidemic model would provide better insights.
I haven’t read Minsky, but the secondary accounts emphasize his point that debt acquired for productive investments is ok, but debt acquired to buy assets with the intent of flipping them is a Bad Thing.
The standard source for the descriptive history is Kindelberger. He has a bestiary of perhaps 200 of these cycles from the GD down to real estate speculation in a single European city. And yes, the tulip bulbs are there.
the magazine cover indicator has been improved on by google search word count and other data mining indices. Not that it did any good in recent housing bubble case.
Let me see if I can summarize your model.
First, you define a system wherein “investors” sell an asset to a pool of “suckers” until the pool of suckers is depleted. Then, you define the (extra) price of said asset to be 0 when the pool of suckers is full, 0 again when it is empty, and positive everywhere in between. Finally, you declare that this system – which is tautologically defined to produce a bubble – produces a bubble. And people wonder why Austrian economists don’t like mathematical models?
Not to be rude, I don’t think you need to do all this work. The logic behind the greater fool theory is already simple and clear: If Steve thinks he can find an idiot to pay $100 for a $75 barrel of oil, he’ll probably buy some oil. If enough Steves are doing this, oil prices will rise. QED. To me, grafting equations onto this logic just distracts from and conceals the logic. Maybe there’s some use to this I’m not seeing, but I’m really not seeing it.
I do mean to be rude, and you need to learn more math before you criticize models. It is not built in that there will be a bubble (a boom-and-bust). It happened that way. That’s the point of modeling — sometimes it will turn out that the price will only increase or only decrease from start to finish. It’s harder than you think to build models where there is a boom-and-bust, let alone where there are sustained boom-and-bust cycles.
Indeed, earlier in these comments I noted that when we allow the retireds to re-join the suckers, if their rate of leaving (l) is greater than a*I (the leaving rate of investors times the number of investors), we do NOT get a bubble. Price hype only increases to a point where it stays put.
This is why we model — to see what assumptions about the world can produce what we observe. You can argue among those that can produce real patterns, but we can immediately discard those that cannot produce them.
As an aside, whenever someone uses the word “tautology,” the conditional probability that they don’t know what they’re talking about is high.
I think the implicit question that everyone needs to ask is this: do markets price things properly? That may sound like an easy question but it’s underlying assumption that underlies all economic pricing.
paleontologist)
For example during the bubble days a few years ago the average median house price in California was $450,000. I think everyone implicitly believed that a house was worth that much. If not no one would have bought a house, because no rational investor will buy something if believes the prices will come down substantially.
I haven’t read Minsky, but the secondary accounts emphasize his point that debt acquired for productive investments is ok, but debt acquired to buy assets with the intent of flipping them is a Bad Thing.
This is where ecology and economics in theory converge. One of the unstated assumptions of both is that systems tend to converge towards some sort of equillibrium state. However as a believer of the Medea Hypothesis:
http://en.wikipedia.org/wiki/Peter_Ward_(
I believe life is inherently self destructive we only have to look at the “oxygen holocaust” that occurred because the first life on earth produced oxygen,a substance that was poisonous to those organisms but which allowed higher life to flourish. Or own produced of C02, incidently the ideas for curbing global warming by spraying sulfer particles in the air strike me as really dumb and probably even worse than global warming.
Economic systems are very similar in the fact that stable systems lead to ever increasing levels of instability. In this case it’s investors who invest with a false sense of security, which is my read on Minsky. All economic systems eventually evolve into systems of “Ponzi Finance”. What a great name it describes our credit system to a tee.
Of course if you really want to get crazy with this analysis there is another great book on economics written by a religion professor called. Confidence Games about how the markets are really based on faith:
http://www.amazon.com/Confidence-Games-Redemption-Religion-Postmodernism/dp/0226791688/ref=sr_1_1?ie=UTF8&s=books&qid=1250880709&sr=8-1
The author doesn’t come out and say it but he infers that all markets function because of faith. Once that faith is lost things go to hell in a handbasket. From a GNXP point of view this is more evolutionary psychology than biology.
As far as mathematical modeling is concerned Nicholas Taleb wrote some great stuff on it:
http://www.amazon.com/s/ref=nb_ss?url=search-alias%3Dstripbooks&field-keywords=nicholas+taleb
In his lay books he believes that most models make an assumption that most distributions are gaussian when in fact they are not. He believes in “fat tails” or black swans. Hence the title of his one book. The fact that quants and economists seem to get things consistently wrong supports this hypothesis.
Personally I think the problem with economics is that we fail to ask some obvious questions. Is this price rational? Do we act on the basis of faith(much like religion)? Is the system inherently unstable?
If not no one would have bought a house, because no rational investor will buy something if believes the prices will come down substantially.
Many speculators buy into bubbles, even believing that the price will come down eventually – because they think it still has a ways to go up, and they think they can get out before the tipping point. Or soon enough after to still make a profit. This kind of buyer contributes to the asymmetrical shape of the rise and collapse in speculative bubbles (they jam the exits once the bubble pops).
I think globalization encourages bubbles because it increases the plausibility of the idea that there are enough greater fools out there eager to be bilked. In 2006, it was obvious that very few people currently living in California could afford to buy a house there, but the feeling was that there was a near infinite supply of people from somewhere else (China? Mexico? The ex-Soviet Union?) who would pay to live in California. And since factual discussions of groups of people are discouraged, it was easy to dream.
agnostic,
Perhaps the language of of math is not as intuitive to some people as it is to others. Personally, I find it much more satisfying to take the logic from your models and see it expressed it in English.
For instance, will the price of an asset experiencing “greater fool demand” eventually collapse, popping the bubble, or will it not collapse, preserving the bubble? The answer to this question is clear in English. If your investors run out of suckers to sell to, then the price of your asset will collapse. If they don’t run out, then it won’t.
On the other hand, you can set up two models, one with an exhaustible supply of suckers, and one with an inexhaustible supply. Then, you can define the “extra price” of the asset to be the supply of suckers times some other stuff. This means that the “extra price” will reach zero when the supply of suckers reaches zero, i.e. it will collapse when the supply of sucerks collapses. Now you can check your models and find that the price in the exhaustible supply model eventually collapses, whereas the price in the inexhaustible supply model permanently remains afloat. Thus, you reach the same conclusion. To me, you have just taken a more unintuitive and roundabout path to this conclusion.
In my opinion, math is most useful when you need to make precise calculations. For example, if I thought that you could use your models to make precise, real-world predictions, then I would think them useful. I.e. if you could predict how high real-world bubbles would go, when they would collapse, etc. But you can’t make those kinds of predictions, because in economics there is no method to determine your input variables with any kind of precision.
Most of the data available in economics is qualitative and order-of-magnitude, and this sort of data is best analyzed and integrated using your God-given ability to reason with language. Think about this fact: the best economic predictor in the world is Warren Buffett. Buffett ridicules mathematical models. His method is purely one of common sense and sound judgment. This suggests to me that economic models are extraneous.
One other issue I have with models is that they allow previously accessible fields of study to be transformed into mysterious arcanum, only meant to be understood by trained experts. This effectively shields the field from scrutiny by outsiders, making it much easier to be dishonest (either with yourself or with others). I’m not saying you were being dishonest here, since you clearly weren’t. But think about the ways you could massage the model in your favor, if you wanted to. Just drop in covariance amplifier term here, an animal spirits coefficient there, and suddenly the behavior of your model changes completely, and math illiterate folk like me can just scratch our heads.