In this post, I review a number of the resources I’ve used to learn the mathematics required by PhD courses in economics. Some of these texts were required reading for UPenn’s math camp, while I have used others for self-study in the past. This reference list is incomplete and will be updated as I encounter new texts.
UPenn’s summer math camp used the first five chapters of Real Mathematical Analysis, by Charles Chapman Pugh, to teach real analysis. I believe Penn uses this book because the material that shows up frequently in economics is compiled concisely in the book’s first five chapters and because the problems in Pugh are *very* difficult. However, I was not particularly fond of this book, as it provides few examples to clarify the concepts it introduces. Also, there is little to no commentary explaining where its proofs come from and why they work. For the topological concepts that were not explained well in Pugh, the first three chapters of Topology by Munkres were a useful reference. Munkres is a great text. There are plenty of diagrams and examples to aid in understanding the material. The only problem from an economists’ point of view is that it does not cover real analysis. For that I recommend Sohrab’s Basic Real Analysis. Unlike the more famous Principles of Mathematical Analysis by Rudin (which offers mostly theorems and proofs but little discussion), Sohrab provides examples that build intuition and commentary to clarify the proofs. Sohrab explains real analysis more clearly and more thoroughly than other books I have seen.
In short, if you already know real analysis and topology and just want to refresh your memory and do some really hard problems, check out Pugh. If you want to understand topology, the first three chapters of Munkres are enough for most economists. If you are trying to teach yourself real analysis, read Sohrab.
There is no shortage of good books on linear algebra. Simon and Blume’s Mathematics for Economists has several chapters on linear algebra which provide a basic, but not quite sufficient, overview of everything you will use in your first year as a PhD student. Note that if you are using Simon and Blume, you also need to study the “Advanced Linear Algebra” topics at the back of the book (never fear, they aren’t particularly advanced). In particular, we used the Rank-Nullity Theorem a lot in math camp at Penn, and that isn’t covered until the end of Simon and Blume.
If you have more time, Elementary Linear Algebra, by Edwards and Penney, is an easy-to-understand book that covers more material than Simon and Blume. In particular, you get more practice with basis vectors, projection matrices, transformations, and eigenvalues. I would recommend this book to anyone who has not studied linear algebra before. It is approachable enough that a diligent student could learn linear algebra from it without the aid of a teacher. Note that this book is out of print, but it can be found at many university libraries and through used book sellers on Amazon.
Univariate and Multivariate Calculus
Simon and Blume is more than sufficient for refreshing your memory about the computational side of calculus (taking total and partial derivatives, using the implicit and inverse function theorems, etc). Penn’s math camp, however, delved into a more technical examination of multivariate analysis. See the fifth chapter of Pugh (above) for this material.
I *highly* recommend A First Course in Optimization Theory by Rangarajam K. Sundaram. We followed chapters 2-9 in Penn’s math camp. Sundaram does an excellent job of explaining the Lagrange and Kuhn-Tucker methods, why they often work, and when they fail. He also explains concavity and quasiconcavity well, and he shows how determining that an objective function is concave or quasiconcave can allow you to relax some of the assumptions of the Kuhn-Tucker and Lagrange Theorems. The author works through many examples (often with unusual functional forms) to show why the Lagrange and Kuhn-Tucker methods usually yield solutions and why they sometimes don’t.
Simon and Blume also has several chapters devoted to optimization. The book covers a few things that Sundaram does not : bordered hessians, envelope theorems, and the intuition behind the sign of the Lagrange multiplier in equality- and inequality-constrained problems. The advantage of Simon and Blume is that the text is easy to read and the problems are not too hard. Before I began studying economics at LSE, it was possible for me to teach myself the Lagrange and Kuhn-Tucker methods by reading Simon and Blume and working through the problems. For anyone who has lots of time, I would recommend studying chapters 16-19 of Simon and Blume and then reading chapters 2-9 of Sundaram to fill in any gaps in your understanding of optimization.
Also useful is this pdf explaining when to use Kuhn-Tucker and the intuition behind the complementary slackness conditions:
(highlight and copy the link into your browser)
Probability and Statistics
Penn’s math camp used the first five chapters of Statistical Inference by Casella and Berger to teach probability and statistics. The book is good because the authors work through several examples to explain each concept, and the examples use a variety of distributions (not just the normal distribution!) to help build the reader’s statistical fluency. There are also a ton of exercises to work through on your own, and the solutions manual is available online. However, the book makes for dense reading, and it is probably not the best resource for someone who last studied statistics several years ago.
If this is you, I recommend you read the review chapter of Christopher Dougherty’s Introduction to Econometrics. Among other things, this review chapter covers probability distributions, hypothesis testing, unbiasedness and consistency, and central limit theorems. The slides, which are available online, are a very useful accompaniment to the text and provide a good visual aid for things like how a central limit theorem works. The slides are available here:
Good books on differential equations are a dime-a-dozen. There are three chapters on them in Sydsæter and Hammond’s Further Mathematics for Economics that are probably sufficient to begin studying at the PhD level. Dynamic Programming and Optimal Control are hard, and only about half of the students entering the PhD at Penn seem to have encountered them before. Upper-year students at Penn swear by Stokey, Lucas and Prescott’s Recursive Methods in Economic Dynamics, but I have yet to read it. The paper copied below by Robert Dorfman provides a good explanation of optimal control, but most students will probably need to work through a few problems in addition to reading the paper in order to understand how it works.
Lastly, for anyone who has already studied everything listed above and who just wants a resource filled with formulas and theorems useful to economists, there is the Economists Mathematical Manual by Sydsæter, Strøm, and Berck.
Blume, Lawrence, and Carl P. Simon. "Mathematics for economists." New York, London (1994).
Casella, George, and Roger L. Berger. Statistical inference. Duxbury/Thomson Learning, 2001.
Dorfman, Robert. "An economic interpretation of optimal control theory." The American Economic Review 59, no. 5 (1969): 817-831.
Dougherty, Christopher. Introduction to econometrics. Oxford University Press, 2016.
Edwards, C. H., and D. E. Penney. "Elementary Linear Algebra. 1988." Prince-Hall, Englewood Cliffs.
Munkres, James R. Topology. Prentice Hall, 2000.
Pugh, Charles Chapman. Real mathematical analysis. Vol. 2011. New York/Heidelberg/Berlin: Springer, 2002.
Rudin, Walter. Principles of mathematical analysis. Vol. 3. New York: McGraw-hill, 2006.
Lucas, R. E., and N. L. Stokey. "Recursive methods in dynamic economics." (1989).
Sohrab, Houshang H. Basic real analysis. Vol. 231. Birkhäuser, 2003.
Sundaram, Rangarajan K. A first course in optimization theory. Cambridge university press, 1996.
Sydsæter, Knut, Arne Strøm, and Peter Berck. Economists' mathematical manual. Vol. 3. New York, NY: Springer, 2005.
Sydsæter, Knut, Peter Hammond, and Atle Seierstad. Further mathematics for economic analysis. Pearson education, 2008.