How well should Real Analysis be learned for econ phd preparation?

[Jaminli]

In ascending levels below:

1. Able to follow Rudin (or other similar level texts) and to write simple proofs.

2. Able to replicate the proofs of all well-known theorems without looking at the textbook.

3. Able to write complicated proofs following proof formats presented in the textbook.

4. Beyond (or others)...

 

I appreciate any thoughts and/or experience sharing.

[zurich_econ]

I think 2 and 3 are pretty useless - apart from the fact that learning proofs may help you coming up with our own for different propositions. But the knowledge on its own seems pretty useless to me. There are some useful small things, however, that might come in handy if you know the strategies, like the 3-epsilon trick, use induction if you see n, continuity always makes things easier etc.

 

4 may be useful however, I think some time spent on trying to write not so simple proofs may be quite valuable.

[APJ]

I think, even if one is well prepared with major results in real analysis (both single variable and multivariable), then applications of those results is what you can easily achieve during the course. I mean, most of the MWG uses main results from real analysis (Including fixed point theory and optimizations) to prove the theorems. So, I think ability to apply the theorems/results will develop gradually during the program. But Rudin may not be a good text to do real analysis in my opinion for PhD for two reasons: 1) It simply does not contain enough material which you may be interested 2) His approch is more sequential rather than fundamental, in the sense that he uses previous results little too much to prove the theorems. Berge on the other hand try to use basic definitions to prove most of the thorems. But again, Berge's book is topology, and sequential proof approach may not be as bad as I am thinking. But use Rudin with the caveat that few things have to be done from other sources. (some fixed point theorems, some convex analysis etc). I do not think that writing proofs of the theorems (with full understanding of the proof) for main results/theorems is such a bad idea. Excercises may be useful applications of the results proved in main text.

 

though most may disgree, the best book I have seen is Real Analysis for economic applications by Efe. It is simply amazing, albeit rigorous. But it just gove you good sound mathematical background to start with.

[dreck]

Given that your question is about preparation rather than signaling, I would really love to hear from some current grad students. Most prospective students, AFIK, overestimate the amount you must learn beforehand.

 

My PhD Micro was less rigorous than one would expect at, say, a top 20 school, but I made an A without a course in Real Analysis (which I am currently completing). Strong mathematical intuition is more valuable--again in my limited experience--than memorizing the results of any given subject.

[informer]

I'm a second year grad student at a top five place. I don't think you need any analysis at all to do well in the first year. Analysis may be helpful after that, depending on what you want to focus on.

 

Good mathematical intuition and prior exposure to the type of problem-solving peculiar to mathematical economics are much more important determinants of first year success in my experience. Unfortunately it's a bit harder to tell beforehand whether you have those things than it is to check the real analysis box.

[treblekicker]

if you aren't doing theory, then i suppose one's goal in learning proofs is to pass the first year and qualifying exams. the degree to which proof-based mathematics are employed varies depending on who is teaching the course. micro, macro, and metrics could all present material in a fashion in which a passing understanding of analysis was not necessary.

 

a few hints to tell if this is the case in at your school:

 

for micro, look at the syllabus. if the section on monotone comparative statics mentions super-modularity, then be prepared for a proof intensive class (i know MIT and Stanford cover the subject this way, as is the case here [uNC]; however, both Duke and Harvard do not).

 

for macro, if the class uses Stokey, Lucas, and Prescott, be prepared for the worst, seriously.

 

metrics will depend on whether the professor is a fan of measure-theoretic probability -- i know Rochester teaches this way.

[rob59404]

This seems to go back to the notion that real analysis isn't important per se, but mathematical sophistication and a comfort for proofs is. And it just so happens that those with a high degree of mathematical sophistication have aced analysis prior to graduate school.

[Guest _nanashi]

I'm at a M.A./PhD program at a lower ranked school and have used Varian with some supplementary readings from Mas Collel Winston Green. Most of the top tier M.A. programs, and 2nd tier (schools outside of the top 30) would be using either Varian, Reney or Kreps with a few spoon feeding mas collel. Most top 20 programs would be using mas collel and may supplement with Varian. I have friends on the spectrum and have compared their experiences as well, and most of my class mates at top Universities (LSE, UBC, Chicago) didn't have Real Analysis. I just finished my first course in the subject. If you open any of the 4 big intro Micro Texts (Kreps a course in Micro theory, Mas Collel, Varian , Jehle Reney) they will state that all you need is calculus I,II and a course in matrix algebra to work through their texts and so far I do agree.

 

There are a few things Real Analysis does for a student.

 

1. Introduction to Real Analysis is theoretical calculus. Theoretical Economics papers often contain proofs of very general ideas, R.A is the first math coruse where your expected to prove results using properties, regardless of how it is taught to you. Linear Algebra, Calculus at most schools tends to focus on calculuting and solving problems. The core math stream for most economics graduate students is Calculus I-III, Matrix (linear) Algebra, Probability and Ordinaray Differential Equations, Econometrics. One could feasibly have taken all of these courses and never have to write a single proof. Therefore it is a perfectly feasible idea that someone could have had this level of mathematics and still nto be able to prove something.

 

2. R.A is a course on reading and comprehending theorems. It will simply make you less intimidated to read and work with graduate level text, which will require you to read and comprehensive theorems. Most of the proofs you write are pretty simplistic, and you can do them without having R.A.

 

Beyond a signal, the direct uses of Analysis in first year Graduate courses is little to nonexistent. It simply though makes your life a bit easier to have seen it. I know people at top 5, top 20 and top 30 graduate programs who've never had it and have been successful. I really do genuinely think if you have a natural aptitude for math, calculus I-III and Matrix Algebra is enough to get through PhD program. The main thing is very few people have the kind of natural aptitude.

[kimbanator]

Calculus I-III, Matrix (linear) Algebra, Probability and Ordinaray Differential Equations, Econometrics.

 

How necessary is ODE to take for signaling and preparation? I heard it has a small marginal value.

[Guest _nanashi]

Absolutely none. Some older models in macro use differential equations and your likely to run in those problems if your following a book like say Romer thoroughly, which almost no good ranking program does. Its barely used in new macro. What is important is learning how to solve difference equations, which is often taught in a Differential Equations courses. Difference Equations are essential for applied macroeconometrics. That being said you can certainly pickup the use of difference equations quite easily from a math econ text book. Walter Ender's Time Series (An easier graduate level Timeseries text spends the first chapter covering difference equations as they apply to economics.

[APJ]

About singnaling, If having taken real analysis is a good signal for Phd Admissions, then there must be at least some use of it. I mean signal has to be credible. Then by that logic, taking any proof based math course should send equally positive signal. (Say logic or even stochastic processes, where proofs are equally logical and based on properties / definitions). But i feel real analysis (including theoretical calculus) is of great help. Otherwise how one would prove existence of utility functions for certain class of preferences? or prove that optimial solutions exists for a particular prblem etc? It must be useful to know the theorems beforehand which are used to prove these and other theorems in Micro theory. Why would Efe take out time to write a book on real analysis in economic applications if R.A was of not much use in economic theory (even macro, dynamic programming etc). I don't know when one should study real analysis (before PhD or during Phd or both) but I am sure that you must have good knowledge of R.A. if you want to strivr in theory

[Guest _nanashi]

Then by that logic, taking any proof based math course should send equally positive signal. (Say logic

Most of the people I know in top 5 either got a B in Analysis, or stopped at a math course called proof. Very few undergrads from my school take R.A. (the difficulty is harder than most other schools and majority of math majors are discouraged from taking it), and the program I did undergraduate consistently places people strong GPAs into top 30 econ PhD.

 

All that being said I really think R.A is more of a weed out course than anything. Simply there is oversupply of people wanting to do Econ PhD's relative to the number spots, and its anotherway to reduce the application pool. It also helps control for variance among american candidates. Foreign candidates especially Canadians, Europeans and Australians are likely to have about 2 to 3x's the amount of economics courses their American counter parts have, their courses will be more quantitative. At the same time this design means they won't have the room to take the math Americans do.

 

That being said I do think the reason R.A is considered a bigger signal than say stochastic processes or any other proof writing course is that it is a core course in any pure math stream. Almost all credible math graduate programs require students to have had a year of Undergraduate Real Analysis. Its a lot like Intermediate Microeconomics for economist. Its the only course that offers a small taste (some schools nothing more than a nibble, at other schools a large bite) of what graduate level theory looks like.

[heraclitus_junior]

You can't do serious math without basic real analysis. It's a gateway course. If you've done stochastic processes without background in analysis, then the course probably wasn't very serious.

[APJ]

You can't do serious math without basic real analysis. It's a gateway course. If you've done stochastic processes without background in analysis, then the course probably wasn't very serious.

 

True. That just reiterate that R.A is important for economic theory, whether you do before PhD or during PhD as may be the case

[wind up bird]

I've heard that Caltech's core sequence is a little more heavy on analysis than most programs, so not surprisingly I have found my background in analysis helpful many times this year. That said, I really had no idea what was going on when I was an undergrad in my analysis courses, even when it comes to the basic stuff. I just didn't have much of a feel for it at the time. After analysis was over, I forgot about it for a year, and somehow by the time I got to math camp and it was presented to me a second time, it made sense. So I don't think it's necessary to be completely comfortable with real analysis even for fairly technical core sequences, as long as you're somewhat familiar with the notation and concepts.

[treblekicker]

you also need to keep in mind that even if you get some coursework that is proof intensive, likely you will neither be tested in an final nor a qualifier on the trickier proofs. so while analysis will help in learning the material, mastering an undergraduate analysis course is far from necessary (nor sufficient) to succeed in the first year. a vast majority of the problems one is expected to solve in the first year reduce to constrained optimization and disgustingly messy algebra. really, the hardest problems are the ones with the worst algebra.

 

analysis will help in understanding lectures. if you haven't had analysis or a solid proof-based course, then you'll likely be lost when the professor shows that a cantor utility representation is continuous in prices and wealth, or in showing that the feasible set is upper hemicontinuous. nearly everyone who has applied aspirations ignores this stuff while it is being taught, and completely forgets it after the semester is over.

 

however, if you want to do theory, only having mastered undergraduate analysis is far from sufficient.

[Guest _nanashi]

It also depends on the kind of theory. I would argue that Microtheory requires more mathematics than Macroeconomic theory.

[Gecko]

Really? ^_^

[treblekicker]

yeah, that's not at all the case.

 

just page through SLP or take a gander at sargent's math recommendations...

[wuwo]

I am slightly confused. Is T. Sargent also recommending to take graduate real and complex analysis and so on... as they were listed on his website, under the graduate Courant courses.

[Guest _nanashi]

Really? ^_^

The crux of the macro literature I've been exposed to so far, didn't require anything more than Dynamic Programming, Calculus to understand, v.s. some of the micro literature I've read has used analysis. I've also found macro professors to far more than likely to assume CRRA utility, with micro literature to leave utility unspecified which is why I'd make that comment.

 

I've asked this question several times to my professors directly, ones who were well known as theorist, and ones who were not, none of them actually have claimed that analysis at any level has a particular sue to economics though they all have studied it. The only value any of them have ever said R.A. has is that it helps with a type of though process, and states that you can form a mathematical argument. Most of the theorist I've met said the only thing you need is to use Lagrange Multipliers.

 

(ironically enough I am enrolled in graduate analysis next fall.)

Edited May 12, 2010 by _nanashi

[abcblan]

^^I am a first-year and this has not been my experience. In macro we do a lot of constrained optimization with Lagrangian techniques. But we also draw heavily and real analysis, measure theory, and stochastic processes. In micro, on the other hand, I have done almost no constrained optimization since the first quarter. Since then it has been a lot of general equilibrium and game theory... and the professors expect proficiency with analysis and even a bit of point-set topology. The focus has been much more an analysis than on solving explicit examples.

[kimbanator]

^^I am a first-year and this has not been my experience. In macro we do a lot of constrained optimization with Lagrangian techniques. But we also draw heavily and real analysis, measure theory, and stochastic processes. In micro, on the other hand, I have done almost no constrained optimization since the first quarter. Since then it has been a lot of general equilibrium and game theory... and the professors expect proficiency with analysis and even a bit of point-set topology. The focus has been much more an analysis than on solving explicit examples.

 

Region/ranking of school?

[abcblan]

I am at a top 10ish department. I should add the caveat that while the professors expect proficiency and use advanced mathematics freely in their lectures, some people manage to pass with only a tertiary understanding of these topics (for instance, using differential topology to develop results on smooth economies). Although, this tends to come with a lack of depth of understanding. My point is that the mathematics required for our micro sequence is way above Lagrangian techniques.

[informer]

I agree with other posters here about at least one thing-- you will definitely see a lot of advanced math during your first year classes-- real analysis topics, point-set topology, measure-theoretic probability, etc. There are two questions, though-- a) whether you need to learn this part of the material to succeed as a first year, and b) whether prior exposure to these topics is essential for learning it.

 

The answers to both a) and b) appeared to be no, at least to me. You didn't need to master that fraction of the material to succeed, and, if you decided you wanted to anyway, you could learn it from scratch. Of course, as I said before, what you need beyond first year depends a lot on what your planned specialty is. And there may not be time to catch up by taking lots of math classes after you've arrived at grad school.