Mathematical Models in Economics or, Tom Sargent, Part II
(On Monday, I started a conversation concerning brand-new-Nobel-Laureate Tom Sargent's admonition that, to model situations that are dynamic, uncertain, and ambiguous, the proper course of action is to roll up one's sleeves and learn and use some mathematics. In starting the conversation, I said that I agreed with Sargent, but I reserved my reasons until others had commented. My reasons---a rant, actually, as this is at the heart of what I do professionally---are much too long for a comment box, so Diane Ellis kindly suggested I start a Part II conversation. If the topic interests you, it probably makes sense to read the original conversation first, especially, the very interesting comments. And before embarking seriously, I observe that, like CJRun, I too am an Asimov fan. And I don't think we need to fear an outbreak of psychohistory unless and until R. Daneel Olivaw appears to assist.)
A lot of the comments take a fairly narrow view of what is a “model”; namely, a set of mathematical relationships to which one feeds a bunch of parameters and out pops a numerical “answer” on the order of: set interest rates at 1.24% (as if that were even possible); or require banks to hold precisely 12.7% of their capital in the form of equity.
But there are many other kinds of mathematical models, whose output is of entirely different character. Among these are, so to speak, mathematical parables, which teach the modeler and reader something more qualitative about the situation. The numbers that come out of the model are not taken seriously; it is the insights that inform. In his discussion about bank regulation, Sargent gives a good example: The two models he mentions each focus on one aspect of deposit insurance and neglect another. No one would take seriously a number that emerges from either model; anyone in the bank regulation business should take seriously both models, learning from them (both) the qualitative things they can teach, while simultaneously comprehending how each is flawed.
Sargent says to roll up sleeves and learn and use mathematics. The first and crucial step is to learn. If you build a mathematical model at the edge of your understanding of the math involved, the model is apt to hoodwink you. Think of the people who bought so-called “portfolio insurance,” based on the Black-Scholes option pricing formula. That formula comes out of model that trades very, very heavily on some very subtle and deep properties of Brownian motion (that the martingale multiplicity of the Brownian filtration is one). And if you have no idea what I just said, rest assured that most of the folks who blithely bought portfolio insurance didn’t, either, and they got badly burned because of it.
The positive side of mathematical models is their precision. They make precise assumptions. They come to exact conclusions. If you understand the assumptions and the link from assumptions to conclusions, you can learn a lot from them. Their precision---their mathematical character---facilitates this. And in a dynamic, uncertain, and ambiguous world, the need for a more precise idea what our models are saying and why they are saying those things is only greater. Which is why (I believe) Sargent has high regard for mathematical models. It is certainly why I do.
A good argument can be made that when the situation becomes too uncertain, dynamic, and ambiguous, any attempt at modeling will fail. Kenneth Arrow, an even more distinguished economist than Sargent (and we are talking the absolute apex on the pyramid here) tells a story relevant to this point, about his experiences in World War II. Arrow was assigned to a weather prediction unit, tasked with forecasting the weather weeks in advance. He and his colleagues knew---because they understood statistics---that their models were useless. And they tried to convince their superiors that their unit should, for that reason, be disbanded, so they could do something useful. “The General knows your forecasts are worthless,” was the response they got back, “but he needs them for planning purposes.” Faced with complexity and ambiguity, people are impelled to try to model the situation. Better to use good models, meaning those that employ mathematics that the modeler (and her audience) understands.
Peter R asks a great question. One must always worry about a stultifying orthodoxy, which lands on a “consensus opinion” and chokes off detractors. I’m less worried about this in macroeconomics than in, say, long-term climate studies, because cornering and controlling the resources needed to do research (government research grants) is a lot tougher in economics than in “big science.” And, indeed, we see lots of very open controversy in economics; any current orthodoxy has a hard to impossible time shutting down skeptics. Indeed, Sargent’s Prize is for work he and Bob Lucas and others did that confronted and largely defeated the previous Keynesian orthodoxy.
But if one is worried about stultifying orthodoxy---and it is a very legitimate worry in all sorts of fields---mathematics, because it sets up precisely the orthodox position to be attacked, gives skeptics a firm target at which to aim. This isn’t to say that there shouldn’t be more empirical tests of models. But orthodoxy can be overthrown if you have the facts on your side---the story of this year’s Nobel Prize in chemistry and quasi-crystals is right on point---and the employment of mathematics in models helps, not hinders.