Monday, May 5, 2014

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.

1. Jason, I notice that you link to one of my favorite Sumner articles: "HPE explained." I refer to that one a lot, and used case 7 as a basis for exploring a difference between Sumner and Sadowski on one side and Nick Rowe on the other:

http://banking-discussion.blogspot.com/2014/03/toms-epsilon-example.html

This has come up several more times, and I'm more inclined than ever to agree with Nick on that particular debate.

I'm really enjoying your series here on expectations. Can you elaborate a bit on exactly what you mean by "worst ad hoc theoretical model?" Can you think of a similar example in another discipline? Thanks.

1. Thanks. And I'll have to read that.

I should have said the worst kind of ad hoc. Ad hoc can be good (as e.g. Paul Krugman defends here).

As mathematics is somewhat unbounded you can quite literally come up with anything. The kind of ad hoc that's "good" is when you restrict your palette ... in Krugman's case, to supply and demand curves. In macro, the data is seriously limited. There just isn't enough data to allow theorists to invent new mechanisms on the fly. Actually, this paragraph sums up my original intent with this blog:

There appear to be no limitations on what "expectations" can do in economics. Rational expectations says that expectations can do anything you can get your model to do -- and with DSGE models, that is pretty much anything. According to Sumner, it doesn't seem matter if you show the price of gold falls to p' via the "hot potato effect" or just say it falls to p' simply by announcing the mining discovery. The former explanation comes from supply and demand, the latter is just assuming the answer.

Assuming the answer is what I mean by the worst kind of expectations. I'm going to do a cheeky post about this with some pictures I found :)

2. Good point about the equivalence of "rational expectations" and pure ad hockery. :)

BTW, as far as the fall in the price of gold goes, there are at least two historical episodes of the discovery by Europeans of large amounts of gold. The first is the discovery by the Spanish conquistadores of the gold of the Incas. It did not cause as much of a drop in the price of gold as might have been expected, because most of it was absorbed by the Indian and Chinese markets. The second is the discoveries of gold in the ground in the 1890s, which were instrumental, during a time when most of the Euro-American world was on the gold standard, in bringing an end to the Long Depression. I suppose that there was inflation during the 1890s, but maybe less than might be expected, because the depression made for slack in their economies.

3. Hi Bill,

According to the EMH, the first one should have already been priced in :)

Those events would be an interesting place to look for the impacts of monetary policy ...