**Syllabus for MatFuII fall 2003**

- John
Hassler, (john.hassler@iies.su.se),
room A842 at SU, phone: 162070 (work), 0708-117263 (mobile).

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.

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

- 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.

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

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

**Dynamic Programming**- Readings: Fuente Chap. 12.1.
**Optimal Control**- Readings: Fuente Chap. 12.2.

**Numerical solutions to equations****Numerical maximization**- Readings: Classnotes.