To be a professional economist requires a large mathematical toolbox. The math everyone is speaking about would require every econ student to come close to completing a math major as well. Until we want to make linear algebra, differential equations, real analysis, and topology required for the major, there is no way to teach with the rigor y’all are asking for. And I am not sure I see the benefit of doing so anyway.

To address your questions without whipping out my math:

Remember, a Nash equilibrium is just a place where no one wants to move from if no one else is going to move. For example, you are on a date at a shitty movie. The movie is so bad you would love to be grabbing a beer instead. Your very attractive date is thinking the same thing. Because its a movie, you can’t talk and so you don’t leave because you would prefer to watch a shitty movie with a hot date than drink alone. Your date thinks the same thing (lucky you).

The problem is the Nash concept guarantees an equilibrium but often gives us too many and some of these may be really illogical. I don’t need a fix point theorem to understand this. I also don’t need a fixed point theorem to find solutions.

Fixed point theora are used for one reason and one reason only: to prove the existence of an equilibrium. It is an important detail, but just a detail, and one that can wait until grad school.

You write as if you expect Williams undergrads are going to be publishing in AER right out of college. That is just silly. Williams will send plenty of good people to grad school who will advance the field. And this should not be the goal of a Williams education in the first place. The point should be that they understand enough economics to see the flaws in many arguments and enough to formulate their own arguments.

It is my belief that they should leave undergrad understanding that economics is not applied mathematics. Math will never lead you to building your own model, and that should be the goal of a Williams undergraduate education. How can we get you at the end to write down and solve a simple 2 period model where you make a choice between 2 options. I don’t care if you can re-invent the proof that simultaneity biases OLS results. I care that you can think up a simple example of simultaneity (stadium attendance and winning percentage: do fuller stadia cause more winning or does winning cause fuller stadia) and come up with a way to solve it. Yes, I am ok with someone just using ivreg in stata, as long as they understand why they are using IV instead of OLS and they meet the conditions for doing so. That is doing economics, not twiddling around with system of differential equations.

]]>That out of the way, I guess that I am the serious economist that contradicts the preceding proof. While I might disagree with anon’s prose, his points are more valid than the dismissive treatment that they receive. Having employed both economists on Wall Street and taught graduate students, undergraduates, and high school students in the discipline, I can certainly assert that “Leavitt(sic) style” economics is not sufficient for serious inquiry, or continued employment. And as a side note, I’ll add that Steven Levitt has exceptional math skills which are on display in his published articles; none of these stopped him from committing serious econometric oversights that others, also with exceptional math skills, are now rectifying.

Of course you don’t need a fixed point theorem to understand a Nash equilibrium. Hell, I saw the movie. To prove that a strategy is actually a Nash equilibrium it is a help though, and last I checked, applying equilibrium concepts to the study of (for instance) corporate or labor decision making was still a major thrust of economic theory. Simply put, if I wanted to merely absorb knowledge, your characterization might suffice (and it would be really cool to watch a movie in class!); if I’d actually care about advancing the field in a manner I’d hope most Williams students would aspire to, your prescription is woefully inadequate.

And no, Nash didn’t use Brouwer in his original proof…he used a variation of Kakutani’s extension (which is a bit harder to grasp actually) in his NAS article, and then reverted to a “simpler” Brouwer explication in his “Non-Cooperative Games” article. And while it is true that all finite games have Nash mixed-strategy equilibria, the truly relevant issue is usually how to find reasonable candidates among a plethora of candidates in infinite games (i.e. those with a continuum of players or time). For this, both theoretical, and more importantly, numerical implementation of fixed point theorems are essential.

I won’t begin to dignify the comments on Black-Scholes. No, there aren’t economists who do it, since we all understood it a long time ago. And yes, Ito calculus is important, essential even, to access any recent results on interest rates or credit insurance…and to ask it to explain the cross-section of individual stock returns is a straw man of Ozian proportions. B-S actually doesn’t rely on any presumption of rationality in stock returns at all, and can be directly derived via no arbitrage and assumptions regarding the diffusive motion of an individual underlying asset.

And thanks for suggesting that we relocate the econometrics class to the interplanetary biology department. The sad fact is that probability is an especially slippery subject, and that intuition, even that of accomplished statisticians is not so reliable. Think about the theory of long leads for instance, or the return to origin tests of randomness. I hardly think knowing when to type xtreg into Stata is becoming of a Williams education. That is, as they say, what manuals are for.

And if you’re looking for econometric role models, Levitt might be not be the best choice; first, because he clearly understands the theory behind the technique, and still let intuition lead him astray.

Sad really, that I like defending the anonymous.

]]>It’s not an intro class, but it only requires one econ and one poli sci class as prerequisites.

]]>Maybe you are a professional economist, but I doubt it. I don’t know a single economist who would agree with you. Your opinion of economics is so far removed from what we think of economics I don’t know where to begin. The answer to all your questions is: Yes, you can understand all of those with out deep knowledge of the math.

Very few economists can reproduce Brouwer’s fixed point theorem, but we do know the requirements for it and invoke them when necessary. These results are never in the paper proper, but relegated to the Appendix (browse AER or Econometrica).

Sorry to bore those who aren’t economists, but no, you do not need to know Brouwer’s fixed point theorem to understand Nash equilibrium. First, Nash’s original proof did not make use of it. Second, you need Brouwer to prove the incredible result that every game has at least one Nash equilibrium, but you don’t need Brouwer to find the Nash equilibrium.

Here is the concept of a Nash equilibrium: If you don’t think anyone else is going to change what they are doing, then you don’t want to change what you are doing. If everyone thinks this way, then you are at a Nash equilibrium.

See, I didn’t need Brouwer at all. If you understand that, the only other thing you need to do is translate my words into math so the analysis is neater.

Given the choice between people being able to find the Nash equilibria and people understanding the proof that they exist, I’ll take the latter. Indeed, given the choice between finding Nash equilibrium and understanding what the concept means, I’d still take the latter.

As for Black-Scholes, I simply don’t know economists who do it. This is a useful relation from the finance literature, but economists will admit that our models of stock prices based on rational expectations are woeful when it comes to predicitons of individual stock prices. A great number of economists work in discrete time rather than continuous, in which case you don’t need Calculus of Variation at all.

To really understand econometrics you have to be from another planet. There are plenty of people who are excellent at linear algebra who don’t grasp econometrics. You don’t just sit there with your matrices and go to town. There is an intuition behind it all that is directing you all the time, and the algebra, as we say, just keeps you honest.

I suggest you actually read a paper by Steven Leavitt. He has all the mathematical tools an economist could want.

The point is, if you understand the intuition behind a decision, you become less dependent upon the math to get you there. I read papers, look at long equations with many constraints; wonder at how the person is ever going to solve this; and discover their result is a simple application of marginal cost equaling marginal benefit. They probably knew what the answer would look like at the end before they started all the calculus.

Anon, you are far scarier to economics than anything Leavitt has or will ever do.

]]>“Nothing could be farther from the truth. The most beautiful math is often the simplest requiring little more than an inquisitive nature.

Math, in large part, is an organizational tool for economists. Yes, we need math to establish nasty things like “existence” and “uniqueness,” but for the majority of econ work, this is the unsavory march that must be made before the economics can actually begin.”

Mr. Dunn, can one really understand the concept of Nash equilibrium without having a basic grasp of Topology and the Brower Fixed Point theorem? Can one really comprehend the Black-Scholes equation without having an understanding of the calculus of variations? Can one really do the econometrics without being able to explain what the eigenvalues of a matrix are?

I am appaled by the people who think that the

“Steven Levitt”-like analysis is what the economics is all about. Anyone can hypothesize, put a buntch of numbers in a regression, and see what it spits out. To be able to advance the Economics, one needs to be able to advance, first and foremost, the Economic Theory, and that is impossible without at least a graduate-level mathematics in your tool box.

Nobel prizes in Economics are not given to the people who do regressions. Nobel prizes are given to the people who can prove the ‘ [***]nasty things[***] like “existence” and “uniqueness.”‘ And it should be that way, because that is what Economics is about.

]]>That aside, I suspect that Richard and I agree on what an ECON 101 course should look lie. It should give people an overview of economics in a serious fashion, but with a focus on classical readings and contemporary debates.

The math can be avoided, but students should be able to understand “adverse selection”, “moral hazard’, “diminishing returns” and all the other economic terms that make regular appearences in public policy debates.

After this class, majors would go on to take 251 and 252. There is no need for 110 and 120.

At some point, I would like to design — perhaps in conjunction with our next CGCL! — such a class.

]]>The fact that “decreasing marginal utility” is still a concept widely discussed in intro classes is disturbing. Instead of learning silly play models, we would be better off with a first course in economics that was a combination of econ history and the philosophy of economics with simple models used as illustrations.

Too many students walk away with an interpretation of economics that is perversely incorrect and far different from how professional economists think.

The goal shouldn’t be a course for non-majors, but a better intro class that placed econ in a fuller context…you know, in the liberal arts style. There is no need to teach econ just like everyone else teaches econ. ]]>

I do disagree with getting rid of the oral exam though. I think that this was a very important part of the economic major. Being able to articulate an argument is an important skill, and I think that it should be kept in the major.

]]>Math is kind of a prude, and does not reveal her even her low-level secrets easily to the dilletante, in contrast to certain other fields, which are more like the last folks “standing” at Brooks Late Night.

Nothing could be farther from the truth. The most beautiful math is often the simplest requiring little more than an inquisitive nature.

Math, in large part, is an organizational tool for economists. Yes, we need math to establish nasty things like “existence” and “uniqueness,” but for the majority of econ work, this is the unsavory march that must be made before the economics can actually begin.

]]>It is shocking how much can be “derived” in econ without resorting to complicated math, even calculus. Math is not an end-in-itself and to varying degrees it is a crutch and a language. We resort to calculus because it is difficult to understand immediately all the conflicting forces on a particular decision. At the end of the day, however, the longest string of greek symbols must be met with–“this is intuitive because each symbol is representing a force and here is how the interact.” I am not sure of the benefits of teaching people how to produced these equations as undergraduates versus understanding the intuition behind them.

I would prefer to see students learn the ability to translate between mathematical symbols and economic theory than plug through derivatives and linear algebra. Those are skills best left to grad school. Instead, focus on the fundamental relations the recur time and again across economic fields. By far, this is a harder skill to learn and occurs much later in the progression through grad school than the mindless use of calculus. Students should learn how to take an equation apart and explain it in basic economic terms rather than spend hours churning through arithmetic

As for econometrics, I believe an undergraduate econometrics course should focus on applications rather than theory. The theory of econometrics is adequately confusing that many a grad student leaves with only enough metrics to accomplish the simple techniques they require. I just don’t see the benefit of doing linear algebra and matrix derivatives. At the undergrad level, you should be able to explain why something is a bad estimator without resorting the mathematical proof. People should be able to read a table of results without having to know why precisely the estimator was the best estimator available. It is just too difficult to do so with little real benefit at that level.

I think the “mathing” of econ at the undergraduate level as a weeding tool is unnecessary and speaks of a little math arrogance.

]]>Furtheromre, it’s also a lot harder for the average person — even the average Williams student — to develop math skills on their own time than it is to learn about most other areas. Math is kind of a prude, and does not reveal her even her low-level secrets easily to the dilletante, in contrast to certain other fields, which are more like the last folks “standing” at Brooks Late Night.

Then again, I just really like Math, so just about any “more math” requirement is fine with me!

]]>If you were to run a very mathematically advanced econometrics/regression class, you would clearly require familiarity with methods in multivariable calculus (Math 105) that would be absent in Math 103. This knowledge would be largely useless, however, if the students were not also familiar with concepts in Linear Algebra, which would be key in understanding advanced concepts in multivariate regressions.

I think that Math 105 should be a prereq for Econ 255 because it would raise the bar for the required level of mathematical/quantitative skills for students taking the class. This would in turn allow the professor to conduct a more mathematically intensive course than would otherwise be possible.

]]>This carries the added benefit of giving Econ majors a true taste of modern economics and a better background in empirical data analysis. That said, it’s slightly disappointing to see that, as the anonymous poster says above, the department will only require Math 103 for the empirical methods course. Stat 346, by comparison, requires that the student take Stat 201 and Linear Algebra (Math 211). Both Linear Algebra and Stat 201 require having taken Math 105, which fosters a level of mathematical rigor that I’ve found invaluable to my understanding of empirical data analysis/econometrics.

I truly hope that this does not become a policy of finding the “lowest common denominator” among students’ abilities. With those concerns I mention above, I fear that this may eventually happen. Hopefully, however, these new policies will weed out those who have very little real interest in the material, and raise the bar for all Econ majors.

]]>As for Jeff’s view on the benefits of making the major harder, I couldn’t agree more. When I was a major, there was no 251M (it came around during my senior year, I think), but it was clear that at least some majors (including myself) were frustrated at the lack of seriousness and rigor in the required classes.

The department was especially weak on math–Econ 251 didn’t even require any calculus. And until DeVeaux arrived (again, I think it was my senior year), there were no statistics classes offered on campus except for Econ 255, so econ majors couldn’t even look to the math department for further study.

As for Frank’s question about the appropriateness of setting a level of rigor for the “primary purpose” of discouraging certain students, I think it’s off-point. Williams is (supposed to be) a rigorous institution. If the “certain students” that will be discouraged are less intellectually serious students, then by all means it should discourage them.

I should say that as a liberal arts institution, Williams should not be in the business of primarily preparing its students for professional careers, including careers as professional academics. I would certainly not approve of the econ department (or any department) shutting its doors to anyone who wasn’t aiming for a PhD. However, that is not to say that Williams students should be able to get by without taking their chosen course of study seriously, and that’s just what used to happen with econ majors.

]]>On another note, Econ 101 was offered when I was an econ major, and it was a major waste in my view. It was basically a joke and totally duplicative of material later covered in 251 and 252. I think 251 provides a great introduction to economic thinking for various disciplines and that you don’t really need 252 if you have a casual interest in understaning the economic approach to thinking — 252 is more about policy and whatnot, while 251 gives you an elementary tool kit that is applicable not only to econ but a whole variety of disciplines (see, e.g., freakonomics). Just like having a poly sci 100 that attempts to cover all political science topics for an interested non-major could never really work; better to avoid a class being so overbroad that it can cover material in only the most cursory fashion.

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