# The Behavioural Impossibility Theorem

It’s been a few years since I’ve posted here but here goes…

Many non-economists (and economists alike) seem to think that behavioural economics is the new and important big thing in economics. I’m writing this blog post for myself as a quick reference I can send to people on why I think behavioural economics is fundamentally limited, but it might be interesting to others. The fundamental limitation I will describe is what I call `The Behavioural Impossibility Theorem’. David K. Levine has a great and accessible discussion here too.

The Impossibility Theorem of Behavioural Economics is the simple insight that a behavioural theory of beliefs cannot simultaneously be believed and true. Or, more strictly, a behavioural economic theory cannot be acted upon and be true. The only theories that can be both acted upon and true at the same time are those that satisfy rational expectations (RE).

Why is this the case? Consider an example of forecasting a stock-market crash. Suppose there were a theory that could accurately predict that the stock market was mis-priced and that there would be a crash in a week. This theory can only be true if it is not widely believed or acted upon. If it were believed, then everyone would short the stock market and asset prices would fall today rather than in a week. Thus the theory now becomes false.

Such a theory would be predicated on a systematic flaw in agents’ reasoning. When agents correct the reasoning by adopting the theory they then disprove the theory they now believe. By contrast, in a rational expectations theory, agents necessarily reason as best they can. They do not have perfect foresight or anything like that, but rather, they agree on on an underlying probability distribution for events which is optimal. In this world, the only possible shocks to the stock market that can occur are those which are essentially unpredictable (or predictable only at great cost of things like information acquisition or skill).

Mathematically this idea can be described as follows (ignore this italic text if you don’t want the math). Consider a mathematical space, $X$, where theories describing the world are objects of the space and agents’ beliefs are functions on the space, $f$. Since the economy is composed of peoples’ actions following their beliefs, the belief functions map to real world outcomes, $Y$, i.e

$f:X \to Y$.

In particular, $f$ is a bijection from beliefs, $X$, to outcomes, $Y$. Formally you might have to appropriately condense the belief space into equivalence classes blah blah but that’s by-the-by. Since the bijection exists, beliefs and outcomes can be considered isomorphic spaces, so $X=Y$. In this space, behavioural theories are never fixed points i.e $f(a) \neq a \in X$. Only rational expectations theories are fixed points, i.e $f(a^*)=a^*$. Fixed points are the theories which can be both true and believed.

This is why I, and many others, think that RE theories are the most beautiful and likely to endure. They embrace the intrinsic randomness of human civilization. Much commentary focuses on Economists’ inability to forecast the financial crisis of 2008, but that’s the whole point! If Economists could forecast such a thing then it would signal a much deeper disfunction in our economy and our theories. Rational expectations is the truest embrace of randomness. It is our abyss in which to float, as Professor Mark Fabian would perhaps phrase it.

Note that Rational Expectations is also not efficiency. Perhaps the stock market is mis-priced, and that everyone knows it is but that there are short-sale constraints.* Then the stock market is mispriced, everyone knows that, but the mispricing persists for a while due to frictions. This is why economists like frictional explanations for things (including informational frictions).

David K. Levine nicely analogises economists’ RE theory to the Heisenberg uncertainty principle in physics. The Heisenberg uncertainty principle arises because the physicists’ very act of measuring a quantum system disturbs the quantum system in an unpredictable way. The same thing happens to economists. The act of generating a theory which is acted upon changes people’s behaviour and potentially disturbs the theory. Luckily, economists have Rational Expectations where physicists don’t!

*Actually, short-sale constraints cannot keep the price elevated in a world of pure RE, so the example is technically wrong (you need belief dispersion). I think the example is instructive nonetheless so I’ll keep it.