# Course Materials

## EC9A0 course outline and notes

Greek alphabet (for mathematically inclined economists)

**1. Matrix algebra:** (Peter J. Hammond)

A. Linear systems: Gaussian elimination, 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

Matrix algebra slides, Part A

Matrix algebra slides, Part B

Matrix algebra slides, Part C

Matrix algebra slides, Part D

**2. **** Difference and differential equations:** (Peter J. Hammond)

Readings: FMEA chs. 11, 5, 6, 7.

Difference equation slides, Part A

Difference equation slides, Part B

Difference equation slides, Part C

Differential equation slides

**3. Real Analysis: **(Pablo Beker)

Metric and normed spaces, sequences, limits, open and closed sets, continuity, subsequences, compactness.

Readings: lecture notes, FMEA ch. 13.

Slides: Real Analysis

Lecture Notes: Real Analysis

**4. Unconstrained optimization: **(Pablo Beker)

Concave and convex functions, Weierstrass' theorem, first- and second-order conditions, envelope theorems.

Readings: lecture notes; FMEA chs. 2, 3. Simon and Blume (ch. 18)

Slides: Unconstrained Optimisation

Lecture Notes: Unconstrained Optimisation

**5. Constrained Optimisation: **(Pablo Beker)

(a) Constraint qualifications,

(b) Kuhn/Tucker Theorem, first- and second-order conditions;

Readings: lecture notes; EMEA, Simon and Blume (chs. 19, 30) and Mas-Colell et al (Section M of Mathematical Appendix)

Slides: Constrained Optimisation with Equality Constraints.

Slides: Constrained Optimisation with Inequality Constraints.

Lecture Notes: Constrained Optimisation I

Lecture Notes: Constrained Optimisation II

**6. Theorem of the Maximum and Envelope Theorem **(Pablo Beker)

(a) Correspondences: upper and lower hemi-continuity

(b) Comparative statics and Berge's maximum theorem

Readings: chapter 3 in Stokey and Lucas.

Slides: Theorem of the Maximum and Envelope Theorem>

Lecture Notes: Theorem of the Maximum and Envelope Theorem

**7. Fixed Point Theorems:** (Pablo Beker)

(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)

Slides: Fixed Point Theorems

Lecture Notes: Fixed Point Theorems

**8. Calculus of Variations and Optimal Control:** (Peter J. Hammond)

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

Slides on calculus of variations

Slides on optimal control

**9. Probability:** (Peter J. Hammond)

Integral and measure, conditional probability and independence, random variables and moments, laws of large numbers, central limit theorem.

Readings: Notes.

Slides on Probability

10. **Deterministic and stochastic dynamic programming**: (Peter J. Hammond)

Readings: FMEA ch. 12, plus notes.

Slides on dynamic programming