A guide to survive Math Camp
Summary: This is the time of the year when you should feel sorry for 1st year Econ PhD students as they start their long and painful journeys through Math Camp, a summer course required for all incoming graduate students. It is a challenging time of the graduate program both because of the amount of material covered in a short span of time and because many incoming students don’t have the necessary background to perform well in the course. In this review, I provide references to review prerequisites and learn (some of) the mathematical methods needed for the rest of the PhD sequence.
Some prerequisites about proofs
Incoming students who didn’t major in math or didn’t take real analysis classes often have a hard time in Math Camp because proofs play a significant role. Daniel Velleman’s “How to prove it” is useful to help students make the transition from solving problems to proving theorems. The author shows how complex proofs are built up from smaller steps, using detailed “scratch work” sections to expose the machinery of proofs. “How to read and do proofs: an introduction to mathematical thought processes” by Daniel Solow is also a useful reference for this.
Some proofs are hard to follow simply because there are not intuitive. Leo Simon shows why in a series of lectures that help understand the intuition of the most important proofs in mathematical economics. Simon often starts by working out a more intuitive version of a proof (which basically follows our graphical understanding of what’s going on), then explains why this version of the proof in incorrect (in general because there is a nasty exception that brakes the generality of the result), and eventually presents the non-intuitive version of the proof and how it deals with that nasty exception.
Math Camp can also be difficult if you forgot some important results from your undergraduate classes. The “Economists’ Mathematical Manual” by Berck, Strom and Sydsaeter is a useful reference book to deal with that problem as it gathers pretty much all the mathematical formulas that economists generally use.
Yes, you will be optimizing a lot of stuff during your academic career. And although it will certainly not be the major part of what you learn in Math Camp, you will start learning about Lagrangians and Hamiltonians as soon as you start the first macro sequence of the program.
The best reference for optimization is certainly the mathematical appendix in the book “Economic Growth” by Robert Barro and Xavier Salaa-i-Martin. The appendix has, in particular, a useful heuristic derivation of the first-order conditions in continuous time problems using the Lagrangian formulation, which helps understand the link between Lagrangians and Hamiltonians. If you’re sill perplexed about that link, the “Guide for the Perplexed” by Maurice Obstfeld is a useful reference. For a more general and systematic treatment of optimization, the book “Optimization in Economic Theory” by Avinash Dixit is a useful reference.
If you remain perplexed and do not like the black magic of Kuhn-Tucker conditions, Nolan Miller has a useful note called “You, the Kuhn-Tucker Conditions, and You” that help you remember that that the KT conditions are nothing more than a short cut for writing down the first-order conditions for a constrained optimization problem when there are non-negativity constraints on the variables over which you are maximizing.
Difference and differential equations
As pointed by Steve Shnider, applications of differential equations are now used in modeling motion and change in all areas of science. The theory of differential equations has become an essential tool of economic analysis particularly since computer has become commonly available.
Let’s start with some background in linear algebra since the applications you will be working with will in general be multidimensional. The lecture notes by Leo Simon are useful to deepen your understanding about what the things you’ve been computing really “mean”. This approach applies nicely to linear algebra, as we all know what a matrix is and how to manipulate it, but few of us understand well what it graphically does to the vector it is applied to.
The appendix in “Economic Growth” by Robert Barro and Xavier Salaa-i-Martin has a clear and concise treatment of differential equations and phase diagrams. The detour discussing a diagonal system of ordinary differential equations (combined with the discussion of what a matrix does by Leo Simon) is particularly useful before moving to the more general case of non-diagonal systems. The supplement to Chapter 2 in “Foundations of International Macroeconomics” by Maurice Obstfeld and Kenneth Rogoff exposits a unified methodology for solving linear equations involving lagged as well as current values of relevant endogenous variables, which is also useful to deepen your understanding of such systems.
The dynamic systems we derive in economics are, in general, non-linear. But non-linear systems are hard to analyze. For small perturbations (you should worry here about whether it makes sense to think of macro shocks as small perturbations), linear approximations capture the dynamics of the model well and are easy to analyze. So, in general, we linearize the system around its steady state and study its properties using the linearized version of the model in the neighborhood of the steady state.
The appendix of Miles Kimball’s book “Real Business Cycle Theory: A Semiparametric Approach” provides simple proofs for various log-linearization relationships, as well as a list of useful rules that one can refer to. Harald Uhlig’s “Toolkit for Analyzing Nonlinear Dynamic Stochastic Models Easily” often comes handy as it shows how to log-linearize the nonlinear equations without the need for explicit differentiation, how to use the method of undetermined coefficients for models with a vector of endogenous state variables to provide a general solution by characterizing the solution with a matrix and how to solve it. Since the method is an Euler-equation based approach rather than an approach based on solving a social planner’s problem, models with externalities or distortionary taxation do not pose additional problems. MATLAB programs to carry out the calculations in this paper are made available.
Completing the guide
I’ve certainly missed some good references in what I presented above. I can think for example of the appendix of Mas-Collel, which has intuitive presentations of some of the results that are typically derived in Math Camp (Implicit Function Theorem, upper and lower hemicontinuous correspondences, and Fixed-Point theorems), as well as the popular book by Simon and Blume, which is always worth checking when one starts studying a new topic.
Feel free to leave comments below with the appropriate references and why you found them useful in your graduate career, as it will certainly be also useful for new admits.