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I am doing a PhD in economics and I have the chance of taking one subject a semester in the maths department (I would like to do more, but "unfortunately" I have to work on my thesis). I want to have a plan so that I start with subjects I might know a bit but making sure I have all the basis, while at the same time ensuring I get the chance to sit in more advanced courses. My plan, for now, is the following:

(I am skipping calculus I and II, because even though I don't know all of it, I know enough I think to be able to do the next course without trouble)

  1. Linear Algebra
  2. Real Analysis
  3. Discrete Maths and Operations Research or
    Engineering Maths or
    Group Theory and Linear Algebra
  4. Numerical & Symbolic Mathematics or
    Vector Calculus

Can you anyone give me a hand on making the decision or giving suggestions? In economics we use software such as matlab, lots of differentiation, Euler equations, difference equations, dynamic programming, statistical analysis (but for that I plan on taking Probability and Stochastic Modelling).

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off topic note. I was surprised you are doing a phd in econ and don't know all of calc 1 and 2. My knowledge in econ is limited, but I always thought mastery of calculus is a requirement even for undergraduate econ. – Chao Xu Jul 21 '10 at 1:48
I don't know eigenvalues, which are taught in calculus 2, I think. I guess I know most of the techniques in calculus 1 and 2, but I never studied them formally, I mostly picked up as I went. The bottom line is: I am sure I would learn something if I sat through the course, but I know enough (or a lot) of it to make the time I would spend there not worth it. But yes, my maths knowledge is weak and I would very, very, very likely not be able to get on a PhD in the US. That is why I want to take maths courses :) – Vivi Jul 21 '10 at 1:57
While the question is very specific, it is also very discussiony. This is a definite community Wiki I think – Casebash Aug 10 '10 at 23:26
Transferred some off-topic comments to this question's meta-question. Also, converted the question to community wiki. – Larry Wang Aug 10 '10 at 23:30
I think a course on either statistics or probability theory would be very practical. – Matt Calhoun Nov 22 '10 at 22:38

12 Answers 12

up vote 9 down vote accepted

If you do take vector calculus, it's probably best taken before real analysis (where you'll learn many of the results from vector calc in a more general setting). If it were me, I'd take discrete math and operations research over the other two in (3), because they'll be useful for economics and because those subjects are the most different from linear algebra and real analysis. The group theory wouldn't, as far as I know, be tremendously applicable to economics except as it relates to analysis, and you'll get a sense of an algebraic way of thinking through linear. Numerical and symbolic math, if it includes numerical analysis, would be very relevant.

A main criterion I am using here is that you'll want to learn more math as you go on in econ, and so the most important thing, for your ability to learn various things on your own later and for developing an appreciation for different kinds of math, is to take courses that cover many different kinds of mathematical thinking (including that which is more directly relevant to econ and other applied areas).

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Oh, wow, thanks for that! I talked to 4 people here in the department - two economists and two mathematicians working in finance, and that is how I got the short list written in the question. None of them told me that Vector Calculus should come before Real Analysis! I also really like the way you went about it: "the most important thing, for your ability to learn various things on your own later and for developing an appreciation for different kinds of math". Thanks so much for your answer! – Vivi Jul 21 '10 at 1:46
Vivi - that said, if you feel comfortable learning some things on your own, I'd skip vector calc and just do real analysis. You should be able to pick up vector calc with minimal extra work having done real analysis. The ideas are all in the latter, you must might need to do some computational problems for practice. – Jamie Banks Jul 21 '10 at 2:28
I think that will be the way to go, since I can only do 4, and I want to take Real Analysis, Linear Algebra, and some numerical methods, and it seems like Operations Research has been recommended by a lot of people (including you). Thanks again for your answer :) – Vivi Jul 21 '10 at 10:05

You have a choice of a course called "Group Theory and Linear Algebra"? You want linear algebra but not group theory if you have limited time. If you have an interest in understanding stochastic models in a serious way then you want to understand stochastic differential equations, which requires serious doses of measure-theoretic probability (not lower-level probability using no measure theory).

Have you looked at Greg Mankiw's blog? He has a section of links called Advice for Students and one is about math courses. The link is

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Some advice from the eminent macroeconomist Thomas Sargent

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That looks like good advice. One of the standard tools of economists is stochastic calculus. To get there you'll have to start with topics like real analysis, linear algebra and eventually a probability course. – Ryan Budney Sep 23 '10 at 3:29
Stochastic Calculus isn't really a standard tool in economics (it is of course in finance). It is used by many, but it is not obligatory. – Michael Greinecker Apr 8 '12 at 1:15

You may consider focusing on Operations Research, and anything numerical. I think you can pick up the tools of vector calculus as you go (which for your specialty will mostly be partial-differentiation, I think). Maybe you can take two semesters of OR and a semester of something like Numerical Solutions to PDE, and one of Numerical Linear Algebra or Numerical Analysis.

And, I suppose all of this depends on what you plan to do with Econ.

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Sorry, I am a bit confused. You mean 1. take one of Numerical Linear Algebra or Numerical Analysis, 2 and 3. two semesters of OR. What about the fourth one? Is numerical analysis the same as real analysis? Is numerical linear algebra the same as linear algebra? – Vivi Jul 21 '10 at 1:41
Oh boy,I'm sorry. I think that a class like OR would be beneficial because it is very "applied" (in a good way!), but it may be too big for one semester (you want to study it until you get through stochastic optimization and/or nonlinear optimization). Numerical Linear Algebra is typically heavily computational, which is a very good thing, and they teach ways to solve large, real-world problems. Taking another computational class such as Numerical Solutions to PDE develops models real-world problems and what it takes to "solve" them. (By now, Katie has chimed in and I like her advice, too.) – Tom Stephens Jul 21 '10 at 1:58
I strongly disagree. These things matter only for certain parts of macroeconomics. Real analysis is needed in all of economics and linear algebra is mandatory for all basic econometric work. – Michael Greinecker Apr 8 '12 at 1:12

During my graduate studies, I had a friend who was extremely interested in differential topology and focused on that in his work toward an MS in math.

In his PhD work, he switched to Economics, but was still essentially doing Differential Topology.

Economics can get just as abstract as math!

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I would be very interested to hear how financial mathematics and differential topology are related. I honestly don't see it; is there a paper or some reference material you can point to? – ItsNotObvious Jul 12 '11 at 20:18
I've lost touch with this friend and don't have a reference to his thesis, but googling "econometrics differential topology" turns up some potentially interesting connections. – donroby Jul 12 '11 at 21:34
@ItsNotObvious I'm probably a little late to the party here, but I'd look up anything to do with general equilibrium. If you'd like some specific references I can probably oblige, but almost all GE at a deep level is just differential topology (and thus all your finance models that are GE based are too I'd imagine, though that side is less my cup of tea). As an example: consider the classic result that "a generic GE has an odd number of equilbria". This is simply a rephrasing of the idea that the excess demand function generically transversally intersects the zero submanifold. – May 22 '15 at 15:35

Looks like you're set on linear algebra and real analysis, both of which are great choices. You can never know enough linear algebra, and if you haven't taken an analysis class yet, it'll definitely get you thinking more "mathematically" than in many of your previous classes, which like Katie said, is definitely one of the main goals.

You might also want to take a look into probability theory. You mentioned that you talked to a couple of finance professors, so I'm guessing that finance might be a possibility in your potential field of research. In that case, probability will be a pretty key component. It's also useful to know it in a lot of other areas of economics as well.

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From a PhD student in economics, the answer is: linear algebra and real analysis. Because those courses are designed to improve your mathematical maturity and provide you with an ability of "picking up the math you need" on your own.

To the poster above, differential topology is used heavily in a subfield called "General Equilibrium Theory". To check what it looks like;

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You should certainly know ordinary differential equations, partial differential equations, probability theory and statistics.

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Where do you need PDEs in economics? – Michael Greinecker Apr 8 '12 at 1:20
@MichaelGreinecker That depends largely on the bent of the person pursing the econ degree. It would not be unreasonable for an econ PHD to end up doing quantitative finance - in which case, PDE's are bread-and-butter. – ItsNotObvious Apr 8 '12 at 2:20
That is true, but finance really is a different field. – Michael Greinecker Apr 8 '12 at 9:16

I think game theory is becoming increasingly important for economists. In addition to classical models such as matrix games (zero-sum and examples like Prisoner's Dilemma and Chicken) and the work of John Nash on equilibria and bargaining, there is the Gale-Shapley model for two-markets, mechanism design (a recent Nobel Memorial Prize was awarded for this work), auctions, fair division methods, group decision making, etc..

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  1. Go back and review and upgrade your Calc 1 and 2. These course are foundational. You are building on a house of cards if you lack comfort here. Get a Schaum's Outline or some programmed workbook. You can't express supply and demand functions without them.

  2. Linear Algebra (lots of applications in econometrics)

  3. Operations Research (very business oriented field of math). The other courses are more foundational, but given you are a weak calculus student, I think emphasizing more relevant (and less difficult) topics would do you better. [It is VERY COMMON to find Ph.D. Econ students who loved micro-econ but who hate the mathematical detail of how graduate econ is done. To be honest, I think they have a point. I have seen many academic economists who get themselves lost in time series correlations, especially in macro, but who can not really break down an industry using Porter 4 forces, who can not organize their thoughts into main effects and included subordinate effects, who can not write an NPV model in Excel for a project or company, who get confused with accounting versus cash impacts, who make sunk cost fallacies, etc. etc.]

  4. Vector Calc ("Calc 3"): basic course, you should know this prior to any work on PDEs. Need to get used to thinking about more than x and y. This is multivariable (welcome to z and w and the like!) Note that there is an important conceptual point here, INDEPENDENT of learning the details of manipulating the equations. This is the point that an outcome variable is affected by many inputs and that they may interact (not be independent). This is quite commonly the situation in economics and other social sciences (things are "messy"), is why we do multiple regression, etc. So even if you aren't strong on the div/grad/curl, if you just start thinking more multivariable, it will do you well in economics and, heck, in business.

  5. Ordinary Diff Eq. ("Calc 4"), basic course. More important than PDEs (if you had to do one or the other). And common prereq. for the material in PDEs. [It is technically possible to do PDEs first, but is uncommon and not optimal for absorbing the material, which is "harder" (less intuitive) than ODEs. Also, many texts or professors will assume you have ODE already.]

If you are finance-oriented, could consider to do a PDE course rather than OR. However given you say you are weak on basic calc, I doubt this enjoyable for you. If you do go down that path, take an applied course NOT a theoretical one. Engine math will probably be a PDE course, but you have to check the content. Really, I think you enjoy the OR more.

Avoid real analysis (it's theoretical calc, a waste for you).

I assume probability and statistics WELL covered in econ department?

[P.s. I edited my answers above to lower the rankings for calc 3/4. I just know the nature of econ students.]

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Of interest to anyone who finds this page, a new stack exchange Economics site is nascent and needs your commitment if you can lend it:

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As a professional economist, I have found linear algebra to be the most necessary. "Can't leave home without it."

Next, is a choice between engineering mathematics and (advanced) vector calculus. They'll cover some of the same ground, but engineering mathematics is more "practical" and vector calculus more theoretical.

The final two courses you'll want to take are real analysis and complex analysis, to round out your math education.

I wouldn't advise group theory or discrete mathematics unless your planned progam/career path is HEAVY in statistical analysis.

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