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Pre-sessional Advanced Mathematics

Module leaders

Peter Hammond and Pablo Beker

Module Aims:

The module will prepare you for later advanced courses in economic and econometric analysis, by teaching you some of the key concepts and techniques normally used in research. In particular, the module will accustom you to the language used in formal mathematical arguments that are common in much advanced economics.


1. Matrix algebra:
A. Linear systems: Gaussian elimination, triangular decomposition, echelon form, linear dependence, rank;
B. Determinants, Cramer's rule, inverses;
C. Quadratic forms: Eigenvalues, diagonalization, linear constraints;
D. Ordinary least squares estimation.
Readings: EMEA chs. 15, 16; FMEA ch.1; Notes

2. Difference and differential equations:
Readings: FMEA chs.11, 5, 6, 7.

3. Real analysis:
Metric and normed spaces, sequences, limits, open and closed sets, continuity, subsequences, compactness. Readings: Notes

4. Unconstrained optimization:
Weierstrass' theorem, first- and second-order conditions, Envelope theorems. Readings: Notes; FMEA chs. 2, 3.

5–6. Constrained optimisation:
(a) Linear programming and duality;
(b) Constraint qualifications, Kuhn/Tucker Theorem, first- and second-order conditions;
(c) Comparative statics and Berge's maximum theorem;
Readings: Notes; EMEA ch.17; FMEA chs. 3, 13; Rakesh Vohra, Advanced Mathematical Economics, ch.4

7. Fixed point theorems
(a) Contraction Mapping Theorem
(b) Brouwer's Fixed Point Theorem
(c) Kakutani's Fixed Point Theorem
Readings: chapter 3 in Stokey and Lucas, FMEA (ch. 14)

8. Optimal control

Readings: FMEA, chs. 8, 9,10.

9. Probability (brief introduction):
Measures, conditional probability and independence, random variables and moments, stochatic processes equations, the fundamental theorems of probability. Reading: Notes and Carvajal 2009.

10. Deterministic and stochastic dynamic programming:
Readings: FMEA ch.12, plus notes.


Lecture notes will be distributed, but the main textbook will be Further Mathematics for Economic Analysis (FMEA) (2nd edition) by Knut Sydsæter, Peter Hammond, Atle Seierstad, and Arne Strøm (Pearson Education, 2008). The chapter numbers above refer to this book. There are also a few references to the more elementary Essential Mathematics for Economic Analysis (EMEA) by Knut Sydsaeter and Peter Hammond, 3rd edition (Pearson Education, 2008), or 4th edition (2012).

Teaching Sessions:

Start on Monday 17th September 2018 and continue Mondays to Fridays until 28th September 2018, 10am-12noon and 2-4pm; room E0.23


All incoming MRes students are strongly encouraged to attend the above intensive short course. It is a non-examined course, offered on a voluntary basis, that provides no credit for your taught first year. Even if you do not attend, however, other MRes module leaders may still assume familiarity with the material it covers.