## Saturday, November 7, 2015

### If a model result is silly, question the scope before questioning the model

Awhile ago I wrote a bit about scope conditions (or rather the lack of them) in economics. Nick Rowe provides us with a good illustration of this problem with his post on the effects of a delay in ending fiscal policy in New Keynesian models. After reading his post, I commented:
So wait: expectations of a delay in returning taxes to normal produce a change in expectations from an expected temporary change in tax rates to an expected permanent change in tax rates?
Can we add in rational expectations of an expected delay (obviously the government doesn't stop on a dime), so that people don't revise their expectations of a permanent/temporary tax change based on a delay in returning taxes to normal?
I was being a bit tongue in cheek; the result that an infinitesimal delay in the end of fiscal stimulus changing people's expectations of fiscal policy from temporary to permanent (and thus changing the sign and magnitude of the multiplier) is, in a word, silly. It's a bit like a child thinking that because class didn't let out at exactly 3pm, class will instead last (literally) forever.

You could cure this by saying that the theory is valid for government delays dt << 1 so that t + 1 ≈ t +dt  + 1. But then, that's the kind of thing that should have been a scope condition in the theory in the first place.

Nick says:
How long is "one period"? It's as short as you want it to be.
No; that is false. For one thing it can't be shorter than the Planck time. In general, it can't be shorter than the time it takes for light to traverse the entire country in question and return to that point (about 20 milliseconds for the US). It takes a few milliseconds for sensory input to register in our brains.

Less sarcastically, there is a definite period of time over which rational expectations can reasonably take hold. Quarterly NGDP data from the BEA isn't released until a month after the quarter ends. Budgets generally have annual cycles. It takes months (weeks, on rare occasions) to get bills through the US congress. There obviously exist timescales over which macroeconomic and government processes happen.

The New Keynesian theory should have something to say about what "one period" means (it seems to be quarterly from various models I've seen). It should also have some kind of estimate about the relative size of the timescale of government actions (dt) and the reaction of the macroeconomy (dT). Is dt << dT? In that case Nick's analysis is silly, not the result. Nick implicitly assumes dt >> dT (the macroeconomy reacts faster than the government). Maybe that is true (data would help), but it is obviously not a region of validity of the New Keynesian model -- we know this because you get silly results like sudden shifts between fiscal policy being contractionary or expansionary based on a one month delay in changing marginal tax rates.

It's a bit like saying electrodynamics predicts atoms will radiate all their energy and collapse so atoms must not exist. But electrodynamics is not valid for cases where ћ ~ dx dp; you need quantum mechanics. That is to say you need to understand the scope conditions of your theory. If you've found a problem, the problem could well be that you've applied the theory incorrectly.

Funny enough, this is very similar to the mathiness issue between Paul Romer and Robert Lucas. The issue has the same form: Lucas's model is assumed to have infinite scope for its variables and Romer says Lucas's model has different limits if 1/β >> T and 1/β << T where 1/β is the timescale for innovation and T is the observation time of the economy. Those two limits have different model interpretations: innovation is slow (so it never happens) versus innovation is fast (so it has already happened). These are entirely different kinds of worlds.

But no one in economics seems to care about scope [1]. Nick Rowe is just fine with the idea that the New Keynesian model is valid for both extremely fast and extremely slow changes in government fiscal policy. He says the extremely slow version gives us silly results so we shouldn't trust the extremely fast version either.

But you can't extrapolate from one limit to the other. The more logical conclusion is that the extremely slow version is out of scope of the theory.

Footnotes:

[1] Don't just take my word for it; Noah Smith says:
I have not seen economists spend much time thinking about domains of applicability (what physicists usually call "scope conditions"). But it's an important topic to think about.

1. "For one thing it can't be shorter than the Planck time."

Lol... Damn, I wish I'd left that comment for Nick!

2. Boy this is a great post. Thank you Jason for writing this so coherently, with all of the great examples of why perfectly good models aren't perfect predictors in all cases. I am really tired of reading Tribe A writing about how Tribe B's model can't be right because in situation X, model B gives you a bizarre result. *No model is right in all situations*. Models are either useful or not useful to solve a given class of problem. Why is this so hard for people to understand?

Likewise, I'm frustrated by people arguing that Sumner's ideas are not worth discussing until there are sufficiently rigorous model to back them. The best software designs do not involve mathematical models of the software being constructed or the problem it is trying to solve. I suspect monetary policy is more like software design than like physics. At some level, I feel you just have to ask yourself what makes sense. Does it make sense that a five-fold expansion in the monetary base would have no impact on inflation or NGDP? Is it some amazing coincidence that increases in M are exactly offset by decreases in V? Now, does that result depend on whether the central bank is committed to a 2% inflation cap or a 5% marching NGDP level target? Well, *of course it does*. Why do we need a formal model to answer that question?

But maybe this is just because I'm limited in my view of math, regarding it more as a tool to solve problems than a language to express ideas. If a model to explain the "obvious" (to me) insight about the effect of the monetary base on inflation I described above, then I'd really appreciate some help in creating that model. kjd@duda.org.

-Ken

Kenneth Duda
Menlo Park, CA

1. Hi Ken,

Thanks.

I think the software analogy may be very useful here. The key problem I have with the market monetarist model is that it's like a discussion of software without ADDs or ICDs. I'm the software engineer and I'm not sure what to code up. When I do see the ADDs or ICDs, it just seems to be a conventional economic model with an emphasis on NGDP ... sort of like refactoring code to take a different input. The thing is that most conventional economic models have a well-known bug when it comes to the zero bound and I don't see how changing the inputs from inflation to NGDP addresses that.

It's not that there isn't a rigorous mathematical model, it's that some of us are unsure what the model is other than *assuming* targeting NGDP (as opposed to interest rates or inflation) gets you out of a liquidity trap. When we see some details (like here), it doesn't answer our questions.

I don't want Sumner to put his model into math (even supply and demand diagrams) to be formally rigorous (the empirics of economics doesn't really call for formal rigor); I want him to put it into math so I know what he's saying.

Most economists don't have a problem with targeting NGDP, and many seem to think it is better than an inflation target (which is based on particular frictions, but maybe they're real). But in those cases, much like how the central bank can't currently seem to meet its inflation target (where one of the theories is the liquidity trap) how do we know we won't be in the same position with NGDP targeting (or start to undershoot more)?

I don't know what that argument is. Sumner points to countries that aren't in liquidity traps, but that isn't evidence. He points to exchange rates, but they seem to be small movements relative to natural day to day fluctuations.

Overall, I don't know what the argument is. That could well be my own failing.