# Lecture Notes for Math Camp

### Economics graduate students have to go through a math camp in the first summer of their PhD. Here are some lecture notes for the macro part of the camp.

Economics graduate students have to attend a math camp in the first summer of their PhD. These math camps are generally divided into three parts: mathematical tools for microeconomics, mathematical tools for econometrics, and mathematical tools for macroeconomics. This post is for students who are about to attend or currently attending math camp, and who are looking for additional resources or for a different take on the material.

I used to teach the part of math camp on mathematical methods for macroeconomics when I was at the London School of Economics. My lecture notes are available online at https://pascalmichaillat.org/c3/. The notes covers both dynamic optimization and differential equations.

### Dynamic programming

The first part of the notes covers dynamic programming, which is a method to solve dynamic optimization problems in discrete time. The key concepts of dynamic programming are introduced in a simple, deterministic consumption-saving problem. Then randomness is introduced into the consumption-saving problem, and the stochastic problem is solved with dynamic programming. As a further application, a Real Business-Cycle model is solved with dynamic programming. Finally, the notes discuss how to solve general problems with dynamic programming.

This first section also shows that optimizing via dynamic programming yields the same results as optimizing via the Lagrangian approach. Macro folks are obsessed with dynamic programming, but sometimes you will find that it is easier and more intuitive to use a Lagrangian.

### Optimal control

The next part of the notes studies optimal control, which is a method to solve dynamic optimization problems in continuous time. This section starts by formulating the consumption-saving problem in continuous time. The continuous-time problem is solved first with a present-value Hamiltonian, then with a current-value Hamiltonian. (Both approaches are equivalent.) Then the notes discuss the optimality conditions for general optimization problems solved by optimal control. They also establish the connection between these optimality conditions and the optimality conditions obtained via dynamic programming. To conclude, the notes derive and discuss the Hamilton-Jacobi-Bellman equation.

### Differential equations

The third and final part of the notes introduces differential equations, which are used to describe continuous-time dynamical systems. This section first solve linear first-order differential equations. The notes then move to linear systems of first-order differential equations. Next, they show how to derive the properties of a linear system of first-order differential equations by drawing its phase diagram. Finally, the notes turn to nonlinear systems of first-order differential equations—which are common in macroeconomics. Although such system cannot be solved explicitly, its properties can be characterized by constructing its phase diagram.

Outside of growth theory, optimal control and differential equations are not used very much. Yet they are powerful tools to obtain theoretical results not only in long-run macro but also in short-run macro. For instance, you can use them to analyze the New Keynesian model. It is easy to set up the model in continuous time, solve the household’s problem with optimal control, and study the model’s properties in normal times and at the zero lower bound using phase diagrams. The analysis is short and simple, and it offers many insights that are difficult to obtain in discrete time. For instance, with the phase diagrams, it is easy to understand where the anomalies of the New Keynesian model at the zero lower bound come from—the collapse of output and inflation collapse to implausibly low levels, and the implausibly large effects of government spending and forward guidance. It is also easy to understand how to resolve these anomalies.

### Homework

To practice, the course website provides four problem sets: one for each part of the course, and one cumulative.

### References

The material covered in the course should be helpful to get through math camp, but it remains pretty basic. If you would like to learn more, a lot of great resources are available. My favorites are the following:

Chapter 6 of Acemoglu’s textbook for dynamic programming

Chapter 7 of Acemoglu’s textbook for optimal control

Chapters 1–6 of Hirsch, Smale, and Devaney’s textbook for differential equations