The effect of expectations in economics (third addendum)

Here's a point I left off that didn't fit well in the post from yesterday. In the paper [pdf] Mark Sadowski links to, the mechanism for how expectations enter into the model is illustrated (as an extreme case) in their Figure 1. This is ad hoc in the worst way. The model does not actually add any information aside from the original observation that "calling" a recession has an impact -- a classic example of (the original meaning of) begging the question. If I have an empirical effect and I assume a mathematical model that gives me that effect, what have I learned?

This is an example of what I called the useless power of expectations. I should have been more explicit by calling it the useless power of theoretical models of expectations. I can achieve whatever I'd like by assuming some piece of a model that accomplishes that effect.

For another example, Scott Sumner put forward a verbal model of expectations. Let gold sell at a price p. Sumner first asked what happens if a mining company discovers a huge amount of gold and brings it to market. Well, as supply and demand goes, the price of gold falls. He even provides a just-so story mechanism (the "hot potato effect") whereby individuals unload gold because now there is more gold than individuals want to hold in equilibrium. Eventually the price falls over these transactions until everyone wants to hold as much gold as they have at the new price p'.

Sumner then asks what happens if the company just announces the gold discovery. He says gold prices plunge. I'd agree, but I ask: how far? The worst ad hoc theoretical model of expectations would be to say they fall to p'. But if it's not p', what is it? And why does this new p'' differ from p' derived from the "hot potato effect"? It is possible people will use the heuristic of the relative size of discovery versus the amount of gold previously in the market (e.g. a discovery of 10% the size of the gold market should cause the price to fall about 10%, ceteris paribus). Maybe that heuristic will anchor the expected value at p''. Even then, the distribution of prices achieved [1] in the market grinding down to p' via the "hot potato effect" in the first case will in general differ from the distribution of the expected price p''. Since those probability distributions differ, there is information loss in Hayek's information aggregating function of the price mechanism [2] (i.e. the KL divergence of the two distributions).

This is why rational expectations is one of the only ways forward-looking expectations [3] have been incorporated in economics in a tractable way -- it assumes away the inconsistency between not only the prices p'  and p'', ascribing it to random error, but also the distributions. But that means rational expectations is precisely the assumption that the price falls to p' -- the worst ad hoc theoretical model.

[1] In different possible worlds, the price p' is different -- that's the distribution.

[2] In the information transfer model, we'd say that the information loss due to expectations means that p'' is initially lower than p', eventually drifting back towards p' (ceteris paribus).

[3] Backward-looking expectations (like adaptive expectations or the simple martingale of this post) don't present the same issues as forward-looking expectations. Generally, when these are dramatically wrong it's because of an unforeseen shock.