Syllabus for MatFuII fall 2003

Teacher:

This course introduces methods to analyze dynamic systems. The course can be thought of as a methodological preparation for the macro sequence and most examples will come from macro. Dynamic considerations have, however, become very important in literally all areas of economics so the contents should be useful for everybody.

The first part of the course discusses how one can characterize the behavior over time of a dynamic systems, for example a macroeconomic model, using information about its law-of-motion. For this purpose, we will learn to solve difference and differential equations of different types, e.g., first and higher order linear equations and systems of linear equations. Phase diagrams and stability of non-linear differential equations will also be discussed.

During the second part of the course, will learn how to solve maximization problems in a dynamic context, i.e., in situations where today’s actions affect also the future so that a myopic behavior is sub-optimal. We will discuss dynamic programming in discrete time optimal control (and possibly calculus of variation) in continuous time.

Finally, we will introduce some numerical methods to solve problems that are too hard to solve analytically.

There is no single textbook that covers all the material discussed in the course. The first sections of the course as well as most of the material in the second section are covered in many books, e.g., Hammond and Sydsæter. A more complete treatment of dynamic systems can be found in de la Fuente, which also covers section 3 of the course. de la Fuente also contains important economic examples. Both books should be useful for everyone, and we recommend them for purchase although they are not absolutely necessary for this course. In engineering and mathematics courses, there is a large number of books in differential calculus that could be valuable if one wants a more comprehensive coverage, examples are Anderson and Böiers or Cullen and Zill. These can be found at the Math library at SU. However, the course will be built around my fairly detailed class-notes and the literature should be used as a guide, for proofs and for alternative explanations and examples.

Students planning to do serious quantitative macro in the future are likely to have use for Ljunqvist and Sargent and also Lucas and Stokey. A very rigorous treatment of dynamic optimization can be found in Seierstad and Sydsæter (Optimal Control), Sargent (Dynamic Programming) and Ross. An alternative book for the part on Optimal Control is Kamien and Schwartz.

Prerequisites and preparations

• MatFuI
• Elementary calculus
• Linear algebra, determinants and eigenvectors.

Literature

• Andersson, Karl Gustav, and Lars-Christer Böiers, (1992), Ordinära Differentialekvationer,  Studentlitteratur.
• Cullen, Michael R. and Dennis G. Zill, (1993), Differential equation with boundry-value problems, Boston PWS.
• de la Fuente, Angel, (1999), Mathematical Methods and Models for Economists, Cambridge University Press.
• Hammond, Peter J. and Knut Sydsæter, (1995), Mathematical Methods for Economic Analysis, Prentice Hall.
• Stokey, Nancy L. and Robert E. Jr. Lucas, (1989), Recursive Methods in Economic Dynamics, Harvard University Press.
• Kamien, Morton, and Nancy Schwartz, (1981), Dynamic Optimization: The Calculus of Variation and Optimal Control in Economics and Management, Elsevier Science Publishing Co. Inc.
• Ljungqvist , Lars and Thomas J. Sargent, (2000), Recursive Macroeconomic Theory, MIT Press.
• Ross, Sheldon, (1983), Introduction to Stochastic Dynamic Programming, Academic Press.
• Sargent, Tomas (1987), Dynamic Macroeconomic Theory, Harvard University Press, Cambridge, Massachusetts.
• Seierstad, Atle, and Knut Sydsæter, (1987), Optimal Control Theory with Economic Applications, Elsevier Science Publishers.

Syllabus

1. Introduction

• Integration, complex numbers and trigonometric functions.
• Readings: Sydsaeter, chap 10:1-3, 11:1:2 and Appendix C.

2. Dynamic Systems

• Difference Equations
• Sydsaeter, chap 20,
• Fuente Chap. 9-10.
• Differential equations
• Sydsaeter, chap 20.

3. Dynamic Optimization

• Dynamic Programming