Monday, December 13, 2010

The Intensive Margin: Math vs Econ

I was recently told, by an undergraduate a top school who had been planning to major in economics, that the required courses had turned out to contain a great deal more mathematics than economics. That report was confirmed by a senior faculty member at the same school with whom I raised the question, who agreed that the situation was an unfortunate one.

Presumably, the content of such courses reflects what professors believe that their students must learn in order to go to graduate school and end up as academic economists publishing articles in leading journals. That fits my not very expert impression of the current state of academic economics, that it is heavy on what Gordon Tullock used to refer to as "ornamental mathematics," advanced tools used to demonstrate the author's mathematical sophistication but contributing little to the substance of the analysis.

I have not been much involved with the world of journal submissions for a long time—I prefer to write books and blog posts—so am in a poor position to make blanket judgments. But some years back, reading an interesting article by Akerlof and Yellin on why changes that should have reduced the number of children born to unmarried mothers had been accompanied instead by a sharp increase, I was struck by the fact that they had used game theory to make an argument that could have been presented equally well, perhaps more clearly, with supply and demand curves. Their analysis was simply an application of the theory of joint products—sexual pleasure and babies in a world without reliable contraception or readily available abortion. Add in those technologies, making the products no longer joint, and the outcome changes, making some women who want babies unable to find husbands to help support them.

Assume, for the moment, that I am right, that both economics in the journals and economics in the classroom emphasize mathematics well past the point where it no longer contributes much to the economics. Why?

The answer, I suspect, takes us back to Ricardo's distinction between the intensive and extensive margins of cultivation. Expanding production on the intensive margin means getting more grain out of land already cultivated, expanding it on the extensive margin means getting more grain by bringing new land into cultivation.

In economics, the intensive margin means writing new articles on subjects that smart people have been writing articles about for most of the past century—new enough, at least, to get published. One way of doing it, assuming you don't have some new and interesting economic idea, is to apply a new tool, some recently developed mathematical approach,. It has not been done before, that tool not having existed before, so with luck you can get published.

The extensive margin is the application of the existing tools of economics, and mathematics where needed, to new subjects. Examples include public choice theory, law and economics, and, somewhat more recently, behavioral economics. The same thing can be done on a smaller scale if you happen to think of something new that is relevant to more conventional topics. I have considerable disagreements with Robert Frank, some exposed in exchanges between us on this blog a while back. But when, in Choosing the Right Pond, he showed how the fact that relative as well as absolute outcomes matter to people could be incorporated into conventional price theory, he really was working new ground and, in the process, teaching the rest of us something interesting.

My conclusion is that, if you want to do interesting economics, your best bet is probably to work on the extensive margin—better yet, if sufficiently clever and lucky, to extend it.


Darf Ferrara said...

This is an interesting post, but I believe it could have been more clearly expressed using galois cohomology.

Neolibertarian said...

Assume, for the moment, that I am right, that both economics in the journals and economics in the classroom emphasize mathematics well past the point where it no longer contributes much to the economics. Why?

In your example, you juxtapose calculus (supply and demand curves) and game theory. Both forms of mathematics are generally comparable in difficulty, so it's not clear what you mean by "past the point". It seems like an especially odd example since game theory is most notably used in economics.

Maybe it would clarify things if you gave some details on:

the required courses had turned out to contain a great deal more mathematics than economics.

What were all these math courses (beyond high school calculus) that were required?

Anonymous said...

Neolibertarian is missing the point of this post--what math courses are being taken or what kind of math being used is not as important as the heavy emphasis the current study of economics places on math rather than on economics itself. I saw this repeatedly -- great mathematical 'performances' demonstrating the tools, with little or no understanding of the logic of economics. Really a sad sight.

Neolibertarian said...

what math courses are being taken or what kind of math being used is not as important as the heavy emphasis the current study of economics places on math rather than on economics itself

Math is just a way of representing concepts. As such, there is a distinction to be made between teaching math for the sake of math, and using math to discuss economic concepts. Requiring a class on Lie Algebra or Computational Geometry would be a bit silly, but discussing the extensive form of a game hardly sounds like a "heavy emphasis".

There is somewhat of a language barrier that needs to be crossed in order to understand mathematics, so if you don't understand the language it may be difficult to appreciate what's being said.

For example, the Black-Scholes equation is just a mathematical description of very simple and elegant concepts and how they relate to pricing. I suppose the equations look like gobbledy gook if you aren't comfortable with the language of mathematics.

It seems like there may be some confusion between what is being said (stuff about economics) and how it's being said (with mathematics).

Robin Hanson said...

If one thinks of academic economics as dominated by signaling concerns, the question becomes: which margin offers the clearest way to signal your abilities? For most smart students, complex math offers a more reliable way to distinguish themselves than being clever about finding new insight via old tools.

Mike Huben said...

There we have a perfect example: Robin Hanson distinguishes himself as clever by using new tools such as signaling theory rather than the classical tool of cynicism.

I think that DF is missing some important points.

First, it is valuable to show that different representation can be equivalent: in mathematics, it's extremely highly regarded.

Second, because there isn't perfect isomorphism between the different representations, important assumptions may be revealed, other findings in the new representation may apply in novel ways for this subject, and some problems might be much simpler in some representations than others. For example, while you may find supply and demand curves easy to comprehend, it might be easier to model in complicated cases with game theory because the computation might be simpler. You might also be able to test robustness of outcomes when parameters are tweaked more easily with game theory.

And of course from the cynical Hansonian point of view, you also may be signaling phatically, attempting intellectual intimidation, etc.

Anonymous said...

But the math in econ isn't that advanced, is it? Or am I missing something? Isn't symbol pushing the name of the game? (perhaps to signal, intimidate or whatever).

lelnet said...

"If one thinks of academic economics as dominated by signaling concerns"

If one thinks that, then one has already answered the implied question, and declared that academic economics has crossed the event horizon and become yet another department staffed principally by folks sniffing one another's excretions, as opposed to folks who have anything useful or interesting to say to those who aren't part of their circle.

I, for one, think that this would be rather vastly overstating the case, even if the underlying phenomenon is real.

If you think "math is Good", go study math. If you think "math is Bad", go study humanities. Math is, at best, _useful_. (And I'd contend that, to an economist, the proper definition of "useful" ought to focus either on answering questions no one has answered before, or correcting answers from the past that have turned out to be wrong, rather than coming up with cleverer ways to get to the same answers that people in the field already have.)

RKN said...

Economics is not my field, but I think students of economics might be best served learning what the outstanding problems are within the field of economics (surely there are such problems), and then setting themselves to making an incremental contribution to the solution to one or more of those problems. Unless you're truly exceptional, this is the best most people working in any field of study can hope for. Before branching out into new areas where an "economic approach" may not be appropriate, or where it strains credulity to think it is. If my models don't explain much within my field of study, people will, understandably, be more skeptical about their relevance outside the field.

Wayne Conrad said...

Mr. Friedman, Apropos of nothing, but perhaps you'll enjoy it: Home Sapiens vs. Homo Economicus

Anonymous said...

well at this point I would like to add the following link - economists suffer from "phsyics envy" for all the reasons the above comments make and more:

at the end of the day, hans christian andersen got it completely right with his story "the emporer's new clothes". think about it!

came to you via a blog: financial follies

Tibor said...

I doubt anybody will read this, since the post is quite old, but it reminded me of a good mathematical anectode/problem:

You teach undergraduates in maths and you encounter the concept infinite sums. Then you present them this problem:

There is a man and his dog and they are walking back home from the town which is N kilometers from the man's house. The man walks at a constant speed of X, the dog runs at an also constant speed Y, where Y>X. The man's dog is a very happy and active dog. He keeps running the distance from the man to the house and back while the man is walking (and stops when the man gets there obviously). Now, what is the distance covered by the dog?

The approach that begs to be used is exactly what you just explained at the lecture - infinite sums. It will yield the right answer, but it is also a very clumsy (a mathematician would probably say inelegant) way to do it...since all you need to do is to realize that the dog runs as long as the man walks and since his speed is constant it all comes down to multiplication.


It seems to me these patterns in economics are exactly that...and it is probably at least partly caused by the way mathematics is usually taught at not math-centred schools - as a way of making simple things complicated when real mathematics is much more about making complicated things simple. No mathematician is going to be amazed by a long technical proof that uses a variety of tools but can be replaced by a much shorter proof which uses a few clever ideas. New proofs of old theorems are published if they belong to the second group, but not the the first one. The result is that things that are worked with extensively become clearer and their proofs simpler, shorter and more elegant.

A great essay about that by G. Hardy:

I guess the ecomonists could learn their lesson from mathematicians - by trying to use simpler arguments rather than complex ones whenever they lead to the same results (and are correct, obviously).

David Friedman said...

To Tibor:

The version of that puzzle I know is set in a story about John Von Neumann. In that version, two cyclists are converging from a stated distance at a stated speed, while a fly flies back and forth between them, and the question is how far the fly flies.

The story has a researcher putting the puzzle to various people in various fields to see how they will answer it. Von Neumann gives the correct number immediately. The researcher is disappointed, and explains that he expected mathematicians, such as Von Neumann, to set the problem up as an infinite series and sum it.

Von Neumann: "You mean there's another way?"