Math for Economists

Consolidated mathematics advice for undergraduate economics majors
looking at PhD admissions

Cliffs
------

You basically need a minimum of a math minor. A good plan of action is:
Freshman year:  single-variable calculus
Sophomore year: multivariable calculus, linear algebra
Junior year:    probability, statistics, real analysis
Senior year:    topics as needed or desired

If you walk into college with calculus credit, you may shift things around
accordingly. I strongly encourage you to finish the calculus and linear algebra
sequence by the end of sophomore year, and preferably earlier. That gives you
two full years to take upper-level courses.



What Textbooks Recommend
-------------------------

What sort of mathematical preparation is necessary for graduate school
in economics? Let's look at a few first-year graduate textbooks to see
what they require or recommend.

Mas-Colell, Whinston, and Green, "Microeconomic Theory," is the standard
first-year graduate microeconomics textbook. It recommends:
	multivariable calculus, some linear algebra, some probability

Kreps, "Microeconomic Foundations," is an advanced first-year or second-year
graduate microeconomics textbook. It recommends:
	multivariable calculus, real analysis, some abstract algebra

Hayashi, "Econometrics," is a first-year graduate econometrics textbook.
It recommends:
	multivariable calculus, linear algebra, probability

Greene, "Econometric Analysis," is a graduate econometrics reference
textbook. It recommends:
	multivariable calculus, mathematical statistics

Amemiya, "Advanced Econometrics," is a second-year graduate econometric
theory textbook. It recommends:
	multivariable calculus; linear algebra; probability; mathematical statistics

Stokey, Lucas, and Prescott, "Recursive Methods for Economic Dynamics"  is
a graduate macroeconomics textbook. It recommends:
	multivariable calculus, linear algebra, probability, real analysis



Lessons
-------

It is clear that graduate work in economics requires substantial mathematical
prerequisites. Or, at least, it's clear that the leading graduate textbooks
assume considerable mathematical background. A sensible course of study would
begin with the following skeleton:

Calculus I-II-III
Linear Algebra
Probability
Mathematical Statistics
Real Analysis

The Calculus and Linear Algebra sequence typically comprises a two-year
continuous lower-division sequence. I recommend completing all four courses
by the end of your second year of undergraduate study. These courses will
provide you with the mechanical skills necessary for writing down, solving,
and analyzing economic models.

The Probability and Mathematical Statistics courses would ideally be a
continuous yearlong sequence and would have Calculus III as a prerequisite.
This sequence will prepare you for graduate work in econometrics and empirical
economics.

Real Analysis is uniquely valued by admissions committees because it tends
to be difficult everywhere and it is typically the first math course where
you are expected to follow all the proofs and write proofs yourself. Graduate
work in economic theory follows the theorem-proof style and familiarity with
that style is considered a positive signal.

Beyond these seven courses, it is perhaps useful to take further coursework
in functional analysis, measure theory, topology, and optimization. Economics
uses so little abstract algebra that a full course in the subject is likely a
poor use of your time; similarly for number theory. Those who wish to do work
on time-series econometrics will find exposure to complex variables, signal
processing, and Fourier series useful; but again, a full course on such
topics may be too much time investment for benefit gained. Focus your further
coursework on linear algebra, analysis, topology, probability, and statistics.



Why take math?
--------------

Multivariable calculus. Economics is about choice under constraints. Hence
virtually any economics problem boils down to a constrained maximization or
minimization problem, which means you're going to need to take first-order
conditions and find optima, which means calculus and optimization. On a
practical note, multivariable calculus will be your bread-and-butter during
the graduate core.

Linear algebra. Empiricists need this because linear models are ubiquitous in
econometrics and form the foundation for nonlinear models. By the end of grad
econometrics, you'll be doing things to a nonsingular matrix X that you never
dreamed were possible. It's also useful for computational reasons; many models
can be represented as systems of matrices and can be solved/estimated/simulated
via the tools of (numerical) linear algebra. Macroeconomists need this because
linear approximations are everywhere in macro.

Probability. Empirical economics (econometrics) is all derived from probability
theory. Read Haavelmo. Some aspects of microeconomic theory (expected utility)
assume familiarity with basic probability. Macro people, you don't get off
easy either: any model with forward-looking elements will involve expectations,
which means you need to know probability theory.

Mathematical Statistics. Econometrics dovetails with mathematical statistics and
it's useful to see how the statisticians do things before learning all the weird
stuff we have to do because our data isn't nice. At a bare minimum, knowledge of
basic inferential statistics will make your econometrics coursework easier.

Real analysis. Modern economic models are mathematical and modern economic
theory follows the theorem-proof style. If you are comfortable with real
analysis -- meaning proofs that involve limiting arguments, sequences and
series, epsilons and deltas, and the like -- then you will be able to expend
brainpower on figuring out the economics of an argument, rather than expending
brainpower on understanding the formal or mathematical aspects of the proof.
Mathematics is a language. Master the language so that you can spend your
brainpower trying to understand the substance.

Substantively, you'll need to understand the bare basics of fixed point theory
to understand general equilibrium theory in the micro core.

You also need real analysis because, at some point, you're going to work with
forward-looking recursive dynamic models. That means you need chapter 9 in
Rudin's book, which means you need chapters 1-7 of Rudin's book, which
means you need real analysis.

Topology. Analysis is just applied topology. By understanding topology, you'll
have a deeper appreciation for the core concepts of continuity, connectedness,
and compactness. In turn, this allows you to understand some aspects of
microeconomic theory that appeared after 1950. (Micro went through a topological
phase for a while before everyone calmed down and started working on sensible
things like matching theory.) Also, topology is plain fun, unlike analysis.
But it's probably not essential. You only really need topology two or three
times in first-year micro, and you can pick up the tools along the way if
need be.

Optimization. A full course in nonlinear optimization may be useful if you
feel the need to brush up on your Lagrangeans, Hamiltonians, and Bellmans.
These courses often also have a computational or paper requirement, which
can be quite nice. Courses in linear optimization are less useful for
economics.

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