The ergodicity problem in economics (wonkish)

6 Dec, 2019 at 15:59 | Posted in Economics | 9 Comments

A surprising reframing of economic theory follows directly from asking the core ergodicity question: is the time average of an observable equal to its expectation value?

At a crucial place in the foundations of economics, it is assumed that the answer is always yes — a pernicious error. To make economic decisions, I often want to know how fast my personal fortune grows under different scenarios. This requires determining what happens over time in some model of wealth. But by wrongly assuming ergodicity, wealth is often replaced with its expectation value before growth is computed. Because wealth is not ergodic, nonsensical predictions arise. After all, the expectation value effectively averages over an ensemble of copies of myself that cannot be accessed.

This key error is patched up with psychological arguments about human behaviour. The consequences are numerous, but over the centuries their root cause has become invisible in the growing formalism. Observed behaviour deviates starkly from model predictions. Paired with a firm belief in its models, this has led to a narrative of human irrationality in large parts of economics. Scientifically, this deserves some reflection: the models were exonerated by declaring the object of study irrational.

Ole Peters / Nature Physics

Paul Samuelson once famously 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 — as Samuelson and most other mainstream economists — that ergodicity is essential to economics?

Ole Peters’ article shows why ergodicity is such an important concept for understanding the deep fundamental flaws of mainstream economics:

Sometimes ergodicity is mistaken for stationarity. But although all ergodic processes are stationary, they are not equivalent.

Let’s say we have a stationary process. That does not guarantee that it is also ergodic. The long-run time average of a single output function of the stationary process may not converge to the expectation of the corresponding variables — and so the long-run time average may not equal the probabilistic (expectational) average.

Say we have two coins, where coin A has a probability of 1/2 of coming up heads, and coin B has a probability of 1/4 of coming up heads. We pick either of these coins with a probability of 1/2 and then toss the chosen coin over and over again. Now let H1, H2, … be either one or zero as the coin comes up heads or tales. This process is obviously stationary, but the time averages — [H1 + … + Hn]/n — converges to 1/2 if coin A is chosen, and 1/4 if coin B is chosen. Both these time averages have a probability of 1/2 and so their expectational average is 1/2 x 1/2 + 1/2 x 1/4 = 3/8, which obviously is not equal to 1/2 or 1/4. The time averages depend on which coin you happen to choose, while the probabilistic (expectational) average is calculated for the whole “system” consisting of both coin A and coin B.

Instead of arbitrarily assuming that people have a certain type of utility function — as in mainstream theory — time average considerations show that we can obtain a less arbitrary and more accurate picture of real people’s decisions and actions by basically assuming that time is irreversible. When our assets are gone, they are gone. The fact that in a parallel universe it could conceivably have been refilled, are of little comfort to those who live in the one and only possible world that we call the real world.

Time average considerations show that because we cannot go back in time, we should not take excessive risks. High leverage increases the risk of bankruptcy. This should also be a warning for the financial world, where the constant quest for greater and greater leverage — and risks — creates extensive and recurrent systemic crises.