4. Mathematics
Having a good understanding of mathematics is crucial to success in economics. Most undergraduate students, particularly those coming from North America, are often shocked by how mathematical graduate programs in economics are. The math goes beyond basic algebra and calculus, as it tends to be more proofs, such as "Let (x_n) be a Cauchy sequence. Show that if (X_n) has a convergent subsequence then the sequence is itself convergent". I've found that the most successful students in the first year of a Ph.D. program tend to be ones with mathematics backgrounds, not economics ones. That being said, there's no reason why someone with an economics background can not succeed.Mathematical Economics Material You Must Know as a Bare Minimum
You'll certainly want to read a good undergraduate "Mathematics for Economists" type book. The best one that I've seen happens to be called Mathematics for Economists written by Carl P. Simon and Lawrence Blume. It has a quite diverse set of topics, all of which are useful tools for economic analysis.If you're rusty on basic calculus, make sure you pick up a 1st year undergraduate calculus book. There are hundreds and hundreds of different ones available, so I'd suggest looking for one in a second hand shop. You may also want to review a good higher level calculus book such as Multivariable Calculus by James Stewart.
You should have at least a basic knowledge of differential equations, but you do not have to be an expert in them by any means. Reviewing the first few chapters of a book such as Elementary Differential Equations and Boundary Value Problems by William E. Boyce and Richard C. DiPrima would be quite useful. You do not need to have any knowledge of partial differential equations before entering graduate school, as they are generally only used in very specialized models.
If you're uncomfortable with proofs, you may want to pick up The Art and Craft of Problem Solving by Paul Zeitz. The material in the book has almost nothing to do with economics, but it will help you greatly when working on proofs. As an added bonus a lot of the problems in the book are surprisingly fun.
The more knowledge you have of pure mathematics subjects such as Real Analysis and Topology, the better. I would recommend working on as much of Introduction to Analysis by Maxwell Rosenlicht as you possibly can. The book costs less than $10 US but it is worth it's weight in gold. There are other analysis books that are slightly better, but you cannot beat the price. You may also want to look at the Schaum's Outlines - Topology and Schaum's Outlines - Real Analysis. They're also quite inexpensive and have hundreds of useful problems. Complex analysis, while quite an interesting subject, will be of little use to a graduate student in economics, so you need not worry about it.
Advanced Mathematical Economics that would be Helpful to Know
The more real analysis you know, the better you will do. You may want to see one of the more canonical texts such as The Elements of Real Analysis by Robert G. Bartle. You may also want to look at the book I recommend in the next paragraph.What Advanced Mathematical Economics Book You'll Use When You Get There
At the University of Rochester we used a book called A First Course in Optimization Theory by Rangarajan K. Sundaram, though I don't know how widely this is used. If you have a good understanding of real analysis, you will have no trouble with this book, and you'll do quite well in the obligatory Mathematical Economics course they have in most Ph.D. programs.Be Sure to Continue to Page 4
