Content
One definition of equilibrium is "the condition of a system in which competing influences are balanced, resulting in no net change" while a definition of fixed point is "a value which is unchanged by a function or other mapping". Equilibria are important in physics, chemistry, biology, economics and other subjects. In the early 1950's it was discovered that, unlike physical and chemical equilibria, which were often treated via maximum principles, economic equilibria are often best understood in terms of fixed points of functions or correspondences.
This module is mostly concerned with the mathematics underlying fixed point theorems, but will also discuss simple economic applications.
After a quick introduction the topics considered will be:
Basic results of analysis in Euclidean spaces
Correspondences and hemicontinuity
Simplices and convexity
Simplicial complexes
Fixed point theorems
Perron-Frobenius theory
Game theory and Nash equilibria
Walrasian equilibria
Text
There is no textbook for the course, but there are extensive notes written by David Wilkins:
Online Lectures
Due to the Covid-19 outbreak all lectures from 11 March 2020 have been canceled. I'm therefore putting the remaining lectures online. The mathematical content of the module ends with Lecture 28.
| Lecture | Date | Topic(s) | Video | Slides |
|---|---|---|---|---|
| 20 | 12 March 2020 | Kakutani Fixed Pt Thm, Perron Matrices | hi-res lo-res | |
| 21 | 13 March 2020 | Perron Matrices, Leontief Model | hi-res lo-res | |
| 22 | 16 March 2020 | Two Person Zero Sum Games, Minimax Theorem | hi-res lo-res | |
| 23 | 19 March 2020 | N Person Games, Existence of Nash Equilibria | hi-res lo-res | |
| 24 | 20 March 2020 | Examples | hi-res lo-res | |
| 25 | 23 March 2020 | Exchange Economies, Budget Correspondence | hi-res lo-res | |
| 26 | 26 March 2020 | Budget Correspondence, Individual Demand | hi-res lo-res | |
| 27 | 27 March 2020 | Aggregate Supply and Demand | hi-res lo-res | |
| 28 | 30 March 2020 | Equilibria of Exchange Economies | hi-res lo-res | |
| 29 | 17 April 2020 | Module Review, Exam Details | hi-res lo-res | |
| 30 | 21 April 2020 | Question Session | hi-res lo-res |
Exams
There will be single exam in the usual exam period, worth 100% of your module mark. Because of Covid-19 restrictions this will be take home exam to be submitted online. Further details will be posted as they become available.
I've prepared a practice exam. It is half the length of the real exam and is meant mostly to illustrate the types of questions which will appear on the real exam: unseen proofs and examples. It's not meant to be illustrative of the the material covered by the real exam. There's some overlap in topics with the real exam, but there are many topics which appear on the real exam but not the practice exam and one or two which appear on the practice exam but not the real exam.
Assignments
There are no assignments for this module.
Tutorials
There are no tutorials for this module.
Past Module Webpages
This module is taught in alternate years. It is mostly the same as the old MA 3486 which was taught in 2015-2016 and 2017-2018.