## Paul Samuelson and the ergodic hypothesis

7 Aug, 2012 at 18:40 | Posted in Economics, Statistics & Econometrics, Theory of Science & Methodology | 2 CommentsPaul Samuelson claimed that the “ergodic hypothesis” is essential for advancing economics from the realm of history to the realm of science.

But is it really tenable to assume that ergodicity is essential to economics?

The answer can only be – as I have argued

and

here – NO WAY! ** **

Samuelson said that we should accept the ergodic hypothesis because if a system is not ergodic you cannot treat it scientifically. First of all, that’s incorrect, although I think I understand how he ended up with this impression: ergodicity means that a system is very insensitive to initial conditions or perturbations and details of the dynamics, and that makes it easy to make universal statements about such systems …

Another problem with Samuelson’s statement is the logic: we should accept this hypothesis because then we can make universal statements. But before we make any hypothesis—even one that makes our lives easier—we should check whether we know it to be wrong. In this case, there’s nothing to hypothesize. Financial and economic systems are non-ergodic. And if that means we can’t say anything meaningful, then perhaps we shouldn’t try to make meaningful claims. Well, perhaps we can speak for entertainment, but we cannot claim that it’s meaningful.

In what sense would saying something that’s patently false be “meaningful,” or “scientific” rather than “historical”? You can see where I’m going with this. Important models that economists use are not ergodic, so what’s this debate about?…

In finance or economics the situation is different. Take the most basic model of a stock market, Louis Bachelier’s random walk. Is that model ergodic? No. A little later, in the 1950s, maybe starting with M. F. M. Osborne, the popular model in finance became geometric Brownian motion—basically a random walk in log-space …

Since geometric Brownian motion is a mathematical model, you can answer the question of whether that’s ergodic by scribbling a few lines of equations. Of course it is not. It’s a model of growth, after all, so it can’t be ergodic, but you can actually make this completely formal and do the math, and not even the expectation value of the growth rate is equal to the time average of the growth rate. At the end of the day, what’s more important in finance than growth rates?

So Samuelson’s comment makes little sense. A hypothesis is about something we don’t know, but in the case of finance models this is something we do know. There’s no reason to hypothesize—the system is not ergodic. It’s like hypothesizing that 3 times 4 is 0 because it makes the mathematics simpler. But I can calculate that the product is 12. Of course, a formalism that’s based on the 3-times-4 hypothesis will run into trouble sooner or later. In economics, that happens with the ergodic hypothesis when we think about risk, or financial stability. Or inequality, as we’re just working out at the moment.

The reason this is so important is quite simple, and stems from a basic question: what does risk mean if the notion of time is not irreversible? The only reason risk exists is that we cannot go back and make decisions over again. Economics got very confused about the point of dealing with risk, and had to resort to introducing psychology and human behavior and all sorts of things. I don’t mean to say that we don’t need behavioral economics. What I mean is that there are lots of questions in economics that we can only answer behaviorally at the moment, but at the same time we have a perfectly formal natural physical analytic answer that’s very intuitive and sensible and that comes straight out of recognizing the non-ergodicity of the situation.

To be blunter, I’m pointing out that economics is internally inconsistent. I accept all the models that economists have developed. I could critique them, but I’m not worried about that. I didn’t make them up, the economists did. But when the economists treat the models as if they were ergodic, that’s when someone has to say “stop, that’s enough.”

## 2 Comments

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Lars, I enjoyed this thoughtful (as always) post and the related links.

Notwithstanding this I am certain that some lurker hiding behind the moniker of anonymity will attempt to refute this as he continues to subscribe to his fantasy world view that there is nothing wrong with neoclassical doctrine.

Two points –which you have no doubt discussed in previous posts in greater detail:

(i) risk management in the world of financial markets clings to the ergodic hypothesis; the notion of ‘black swans’ popularized by Taleb but written with greater precision, clarity and brevity by the incomparable Joan Robinson in 1962, has lulled risk practitioners into thinking that “six sigma” events are common in finance (especially in the post GFC world) occuring as the fat tails of asset returns. There still is not the acknowledgement of non-ergodicity in these markets.

(ii) I don’t see the behavioral economics movement as a saviour as I personally favour the Berg and Gigerenzer critique (ref: http://mpra.ub.uni-muenchen.de/26586/ )

Making ‘risk management’ manageable in finance –if this is possible– requires (in my biased opinion as a former engineer) a subscription to robust systems thinking: make it simpler and build in very large buffers. This is done in the real world and when corners are cuts –as in the case of the BP Deep Horizon ‘accident’– disaster ensues inevitably ensues.

In the world of high finance, every bit of financial innovation is a fancy way of bringing leverage into the system (thus increasing the chance of profitability) and large buffers mean greater capital consumption which decreases profitability. Hence the erogodic hypothesis will remain, as will the notion of unforseeable ‘black swans’. When ‘uncertainty’ is brought into the conversation, members of the guild invariably interchange Knightian and Keynesian version but I do believe that there are subtle distinctions. CROs tend to think in terms of probability distributions and this the legacy of Samuelson and his ergodic hypothesis.

Comment by Arijit Banik— 7 Aug, 2012 #

I got confused halfway through this post.

It seems that an economic *model* can be ergodic. But whether or not the real economy is ergodic is an open question. So there is nothing wrong with economists treating their models as ergodic if that’s a mathematical property of the model.

Comment by marris— 7 Aug, 2012 #