Resources for undergraduates

Courses, podcasts, journals, and books

Math classes: which are most relevant to economics, and why?

  • If you’re thinking about taking more advanced economics courses, whether in preparation for graduate study or simply to tackle more challenging ideas, you’ve probably been advised to take more math classes. But which topics are most relevant, which are least relevant, and when do they come up in practice?

    • Differential calculus (single and multi-variable): Essential if you want to move beyond the basics. Much of introductory and intermediate economics can be (and sometimes is) taught without calculus, often through heavy use of graphical techniques. But differential calculus unlocks the marginal thinking at the heart of how economists view the world, and you can’t progress very far without it.

      • Calculus has an unfortunate recommendation for being intimidating and complicated, but most of the time it actually makes economic math easier, not harder. And the better you understand differential calculus, the better equipped you’ll be to understand and solve economic models.

      • We use plain-old derivatives all the time, and partial derivatives whenever an optimization problem involves two or more choice variables. We use the implicit function theorem to do comparative statics any time we discover that an optimization problem has no closed-form solution.

      • By contrast, vector calculus (in the narrow sense of calculus involving vector fields) is seldom encountered in economics. Confusingly, however, the terms vector calculus and multi-variable calculus are sometimes used interchangeably. Vector calculus gets used in physics and engineering, but I don’t think I’ve ever seen a line integral in an economics paper, nor have I ever needed to brush up on Stokes’ theorem.

    • Integral calculus: Sometimes encountered at the undergraduate level, but increasingly important if you progress towards advanced undergraduate or graduate classes.

      • Key concepts like consumer surplus and producer surplus are usually visualized as areas between demand curves, supply curves, and market prices, which means they can be computed as integrals. Many undergraduate classes sidestep the need to rely on integral calculus by using linear demand and supply curves, which yield triangular surpluses that can be calculated by simple geometry. More advanced classes may use non-linear curves that require integrals.

      • Integrals are central to probability and statistics, at least beyond the introductory level. Since uncertainty is a central factor in most real-world economic applications, graduate students will routinely encounter integrals. Even in models without uncertainty, integrals come into play whenever we want to consider a large number of heterogeneous agents (“a continuum of types”).

      • Integrals are used in advanced undergraduate treatments of consumer theory. You might also see integrals in classes on other advanced topics like game theory and auctions, both of which involve uncertainty about who precisely an agent is interacting with.

    • Linear algebra: Widely used in econometrics, though topics like OLS regression can be taught without it. Linear algebra also comes up in dynamical systems, which encompasses much of advanced macroeconomics. For example, the evolution of employment, unemployment, and labor force non-participation can be expressed in terms of a “transition matrix” listing the probabilities that a person in each of these states will switch to the other states over a given period of time.

      • Beyond its specific applications in economics, linear algebra can be a useful course because (depending on how it’s taught) it can serve as a bridge from calculation-based math coursework to proof-based coursework. The class I took back in college was equally split between plug-and-chug calculations and proofs of theorems, and it eased my introduction to higher math.

    • Ordinary and partial differential equations: Not much encountered in undergraduate courses, at least on the micro side. As an applied microeconomist, I’ve only very rarely come across differential equations in my career. Macroeconomists deal with differential equations more often.

    • Optimization: Highly recommended, if your school offers a dedicated course in optimization techniques and if you’re thinking about pursuing more advanced economics classes. An optimization class would give you a strong grounding in some important mathematical issues that aren’t likely to be covered in a standard calculus course (such as corner solutions and Kuhn-Tucker conditions).

    • Real analysis (sometimes called modern analysis): You aren’t likely to use much real analysis in undergraduate economics courses, unless you take courses in advanced micro- or macroeconomic theory. But if you want to do a PhD in economics, you’ll have to cross this bridge along the way. It’s tough stuff, but it can be an intellectual treat if you’re up for the challenge.

      • For most students who go on to doctoral study, real analysis is the first place they’ll encounter the “definition-theorem-proof” format in which higher-math courses are taught. Because the first-year sequence in an economics PhD program is taught largely in that same format, graduate schools use real analysis as a barometer for whether a given applicant has sufficiently strong technical skills to get through the first year. By all accounts, it gets a lot of weight in admissions decisions.

      • “Epsilon-delta” arguments and other ideas from real analysis are also used widely in economic theory. Many empirical economists won’t use these skills much once they reach dissertation stage, but for better or worse, they’ll be hard-pressed to gain admission to PhD programs (or keep up with first-year classes) without taking real analysis and earning as high a grade as possible.

      • Many math departments offer courses with titles like “Introduction to Higher Mathematics” that are intended to teach the fundamental logic of proof-based math. Taking a class like this first would make real analysis a little less intimidating.

      • Analysis textbooks make for dense reading, and you might find it helpful to consult a second textbook covering the same material. An explanation that’s hard to follow in one textbook might be easier to follow in the other, and vice versa. It might also help to read Wikipedia entries or other discussions online. Differences in notation and slight differences in definitions can make this tricky, but I found that the benefits of reading multiple treatments outweighed the costs.

    • Probability and statistics: Used all over. Alongside traditional econometrics, machine learning is rapidly growing in importance in economic research.

    • Graph theory: Seldom encountered in traditional courses, but also on the rise as economists work to understand social and economic networks. A little knowledge of graph theory can also be helpful for understanding some aspects of research computing, particularly version control.

    • Most applied economists will rarely if ever need to draw on topics like number theory, abstract algebra, complex analysis, or differential geometry. (Economic theorists use a broader array of mathematical tools.) Of course, the more math classes you take, the more accustomed you’ll become to the kind of mathematical reasoning used in advanced economics, so taking classes like this might prove useful to you. But it certainly isn’t expected, and the courses I’ve listed above are much more relevant for getting into and through graduate school.

If you’re considering a PhD in Economics

  • Advice about doing a PhD tends to focus on the mechanics of gaining admission. If you’re intent on doing a PhD, it’s important to rack up the right credentials and to be strategic about applying, and I’ve listed some relevant links below.

  • But advice about applying can sometimes put the cart before the horse. A PhD is a huge investment of time, energy, and self, and you should think long and hard before embarking on one.

    • There are plenty of good reasons to consider pursuing a PhD. If you’re reading this, you likely have a deep intellectual interest in economics, and graduate school can be an intellectual feast. On a practical level, an economics PhD is a versatile credential with many possible career paths: faculty jobs in economic departments and professional schools, positions in government and in international organizations, and private-sector jobs in tech, finance, economic consulting, and other industries. Most PhD economists go on to do well-paid, interesting work.

    • Alongside these positives, you should consider the sacrifices that a PhD entails. PhDs are hard. They’re emotionally and psychologically as well as intellectually challenging, and they take a long time, occasionally five years but typically six or more. Doctoral work is often a stressful and isolating experience, especially once you finish coursework and set off to do original research. Many PhD students struggle with anxiety and depression, not least in economics programs. Ask yourself whether the benefits are worth the costs: both are quite real.

  • If you decide to go for it, here’s some guidance on the application process:

  • Some worthwhile textbooks for the PhD-bound:

Full-time research assistant positions

  • Before committing to the PhD path, seek out research opportunities first: write a senior thesis if that option is available to you, and look for research assistant positions both during and after college.

    • A full-time RA position can help you figure out if a PhD is a good fit. If you decide that it is, your RA work (and the recommendation letter that will hopefully go with it) will increase your chances of admission and prepare you to hit the ground running with research projects once you finish coursework. And if you decide that a PhD isn’t for you, you’ll be spared the personal and professional costs of starting a PhD and realizing you made a mistake.

    • Regardless of what you do next, you can use an RA position as an opportunity to explore your interests, gain experience working with data, and acquire tangible computing skills useful both within and outside of economics.

  • Most economics RAs work for faculty members, government agencies, or think tanks; in addition, economic consulting firms hire entry-level analysts for work somewhat similar to that of research assistants, and such analysts sometimes go on to pursue doctoral programs in economics. Here are some places to look:

  • In general, these employers screen on the same credentials that PhD programs will be looking for: strong grades in economics, math, and statistics coursework; favorable references from professors who’ve gotten to know you; and research experience, either in a thesis seminar or as an undergraduate research assistant. Experience with programming languages and statistical software is a big plus, and attention to detail is de rigueur.

  • Be mindful of deadlines: interviews for these jobs often take place early in the academic year, for positions starting in the following summer or fall.