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.
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Aug '10
Re: Mathematical Models in Economics or, Tom Sargent, Part II
I worry that the main 'feature' of a model - mathematical precision - is also its biggest potential liability because it is a false precision if the model itself is incomplete or wrong. Garbage in, garbage out.
Friedrich Hayek's criticism of the over-reliance on mathematics in economics comes to mind. An economy is a complex adaptive system full of very smart actors who do not necessarily behave in predictable ways, and whose responses will change as they assimilate information. The actual structure of the economy and the billions of inter-related factors is not knowable to people looking from the top down. The information required to understand it is locked up in the heads of economic actors.
There are ways to model these kinds of systems, but such models are more along the lines of what you described as models that are not used to predict specific outcomes, but to help understand the behavior of the system in order to learn more about it. I do not believe this includes the simple Keynesian models currently being used to predict things like the number of jobs saved/created when X dollars of stimulus are injected into the economy.
Nov '10
Re: Mathematical Models in Economics or, Tom Sargent, Part II
Just by way of background - I spent 23 years doing math modeling - 6 in electroquasistatics, the rest in torpedo defense and underwater acoustics. Although the math in those is technically more difficult than economics, the problems are infinitely simpler. That's because there are few, if any, feedback mechanisms, and the underlying statistical structures are generally static and well known. Because the problems are "well-behaved", one could, by combining classical science, statistical methods, and frequent recalibration, achieve useful results in reality.
Now my experience in mathematical economics is limited - I used to pick up grad level textbooks and read through them for interest and the occasional mathematical challenge, and I enjoyed adapting various Bayesian algorithms to trading systems - but I did get familiar enough to realize that at least at the grad level, there's huge holes in the mathematical approach, particularly with respect to assumptions regarding the underlying distributions. Nassim Taleb is a wonderful, non-technical read on this (for those who are interested). Those fat tailed distributions (the unknown unknowns) always get you in the end.
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Nov '10
Re: Mathematical Models in Economics or, Tom Sargent, Part II
Because of this, I would say that the real orthodoxy to be addressed is the assumption that mathematics is, in fact, a suitable tool at all. My own sense is that it is not (at least above the simplest, illustrative level). Even rule-of-thumb folk philosophy seems to work better. Don't put your eggs in one basket would have considerably helped the Nobel Prize winners at LTCM, for example. Don't borrow above your means would have helped many immensely. And of course, TANSTAAFL ought to be tattooed backward on the forehead of every politician, so they can read it as they gaze in the mirror.
But math is sexy these days, and it tends to attract the same way Latin Mass did. (Who knows what they are saying? But it sure sounds good...).
.Now maybe someday economics will become a real science, and the problems mathematically tractable - say along the lines psychology is just beginning to become. But in the meantime I know I would be a lot happier if the math was kept to the undergrad level, and the real arguments allowed to take place along philosophical, historical, and moral lines.
Edited on Oct 11, 2011 at 8:51pmAug '10
Re: Mathematical Models in Economics or, Tom Sargent, Part II
You're right that non-precise mathematical models are useful for conveying qualitative information... but that already moves us away from the kind of precise models which engineers distrust, which Hayek critiqued, and which central bankers borrow from modern macroeconomics.
It may be more accurate to say economic relationships can be expressed using mathematical notation, not models, to convey useful information. Examples include the IS/LM equations, the Laffer curve, and the geekiest shirt ever.
One of my earliest "aha!" moments was in my high school economics class**. The teacher asked me to draw "the supply and demand curves" on the board. But when I picked up the chalk, he announced it was a trick question: there's no such thing as the curves, they differ with each good produced. The important thing is how the two relate to each other; the rest is just simple graphing exercises with arbitrary numbers.
If you're saying use models to represent theories, I think we're in agreement. If you're saying use mathematical models for prediction... isn't that copying Arrow's weather unit?
**Do they still teach economics in high school? If not, the OWS protests start to make sense.
Re: Mathematical Models in Economics or, Tom Sargent, Part II
Wow. That is the most fascinating, informative, and lucid discussion of mathematical modeling I've ever read.
Thanks, David. But be careful. Tuition at Stanford business school these days is a lot. You're going to undercut your own institution if you start giving education away for free.
Re: Mathematical Models in Economics or, Tom Sargent, Part II
To Blue Ant: What do you mean by "prediction?" I think it means something less precise than you do. To give an example: When I teach microecon to MBAs or execs in our Exec Ed programs, my favorite case concerns GE and Westinghouse struggling to restrain themselves from engaging in cutthroat competition. What is stopping them? Halfway through the case discussion, we stop and play/analyze a pretty simple game (repeated prisioner's dilemma with noise). This, to me, is a (mathematical!) model of the situation facing GE and Westinghouse. It is very, very far from the situation they face, but it captures beautifully some critical aspects of their situation, and it allows the students to predict: IF GE can find a way to strip out the noise, GE and W can probably do a lot better (and consumers a lot worse---important insights for the Antitrust Division of the DoJ). (And, btw, GE found a stunning way to do just that.) If that's a model (and it sure is mathematical) and a prediction (NOT mathematical), we're using math to make predictions in a very fruitful way. Which, I contend, is precisely what Tom Sargent has in mind.
Aug '10
Re: Mathematical Models in Economics or, Tom Sargent, Part II
The models you're describing are very different than the climate models or the macroeconomic models that HalifaxCB and myself are talking about, though.
I don't think you'll find anyone here arguing against the general concept of mathematical modeling or the utility of using mathematical models to test certain scenarios where boundaries and behaviors are well known, or where you are searching to understand the general behavior of a system in a testable way.
So I don't have a problem with mathematical models, I've used them many times. What I have a problem with is the use of them where they are not applicable or where there are too many unknown variables, feedbacks, or potentially chaotic responses to inputs, and then to use the apparent precision of the models to pretend that you have a deep understanding of what's going on. This is especially problematic when those models are applied to systems for which there are no controls to test against or no way to determine what was wrong if reality deviates from the model.
Aug '10
Re: Mathematical Models in Economics or, Tom Sargent, Part II
Another problem I have with models of complex systems is that they necessarily require the system to be simplified in order to make it tractable to modeling. In macroeconomics, we have the use of aggregate measures of supply, demand, labor, and money. Climate modeling also involves numerous simplifying assumptions.
Friedrich Hayek argued against the use of aggregates in macroeconomics, because he believed the microeconomic factors were critically important, and by aggregating them you hide the information you need to actually understand the economy. Furthermore, by injecting fiscal stimulus or other changes into the economy by a central authority, you would damage the underlying fine structure of the economy but this damage would never show up in a model that reduces economic data to aggregate values.
More than one wag has likened the use of aggregates in economic modeling to a drunk looking for his car keys under a streetlight half a mile from the path he took from the bar to his car, under the theory that it's the only place lit well enough to find them.
If we are going to model a system as complex as a national economy, agent-based modeling seems more appropriate to me.
Jan '11
Re: Mathematical Models in Economics or, Tom Sargent, Part II
"...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."
As in chaos theory? Seems appropriate considering the social, political, and economic state of the world.