## Introduction

This course covers basic mathematical methods to study dynamical systems, in discrete time and in continuous time. Dynamical systems are systems that involve more than one time-period; they are prevalent in macroeconomics. We first discuss dynamic programming, which is a method to solve dynamic optimization problems in discrete time. We then turn to optimal control, which is a method to solve dynamic optimization problems in continuous time. Finally, we show how to solve differential equations, which are used to describe continuous-time dynamical systems.

## Dynamic programming

This section starts by introducing the key concepts of dynamic programming in a simple, deterministic consumption-saving problem. It then introduces randomness into the consumption-saving problem and solve this stochastic problem with dynamic programming. Finally, to illustrate how to use dynamic programming in macroeconomics, we solve a Real Business-Cycle model using dynamic programming.

## Optimal control

This section starts by formulating the consumption-saving problem in continuous time. We solve this continuous-time problem first by using a present-value Hamiltonian, and then by using a current-value Hamiltonian. (Both approaches are equivalent.) Then we discuss optimality conditions for general optimization problems solved by optimal control. To conclude, we discuss the Hamilton-Jacobi-Bellman equation.

## Differential equations

In this section we first solve linear first-order differential equations. We then move to linear systems of first-order differential equations. Next we discuss how we can derive the properties of a linear system of first-order differential equations by drawing its phase diagram. Finally, we turn to nonlinear systems of first-order differential equations—which are common in macroeconomics. Although we cannot solve them explicitly, we characterize their properties by constructing their phase diagrams.

## Conclusion

To conclude, a little more practice and an application. The problem set below brings together all the material from the course. And the paper below applies the course’s techniques to analyze the New Keynesian model in normal times and at the zero lower bound. The analysis covers in particular the response to recessionary shocks, the effects of forward guidance, and the effects of government spending.