## Friday, September 17, 2010

### Introductory Maths and Stats: Static Unconstrained Optimization of N-Variable Function...

The title is a mouthful, but the lecture itself is fairly straightforward (aside from the notation being a bit complex)...

Rules for Single Variable Optimization:
• If df/dx=0 and d^2f/dx^2<0 at any point x0, then x0 is a local max
• If df/dx=0 and d^2f/dx^2>0 at any point x0, then x0 is a local min
• If df/dx=0 and d^2f/dx^2=0 at any point x0, then necessary but not sufficient conditions exist for x0 to be an inflexion point...
N-variable case is similar, but the notation becomes more complex (uses vectors instead of single variables etc.)

The Two Variable Case: The discussion in the lecture notes of unconstrained optimization with two variables is good.  I particularly like how emphasis is placed on using Taylor expansions in the argument.  I would only recommend that more pictures be included.  Anytime a Taylor expansion is used, it just screams DRAW A PICTURE!!!

Quadratic Form, Definite Matrices and Hessians: I would like to see this discussion moved up a bit.  Hessians should be introduced in lecture 2 on multi-variable calculus.  The maths notes should link more closely with the stats notes (particularly the linear algebra parts).   Definite matrices should be emphasized in both the maths and stats, and a solid amount of lecture and tutorial time should be spent on the concept.  Definite matrices provide the coat-hanger on which much of the linear algebra that is used in microeconomics and QM hangs...

Concavity and Convexity: Again draws pictures.  Emphasize that the definitions are almost identical to the single variate case.  Only difference is that we are dealing with vectors now and not scalars in the argument of the function.

Chain Rule and the Envelope Theorem:  Material on the Envelope Theorem is scattered across three lectures.  I think the best think to do is devote an entire lecture to the envelope theorem after all of the necessary maths have been developed.  This would serve as a useful mid-course refresher for the students, and I think would make the theorem more understandable.  It is important, and thus I think it should get its own lecture...

Economic Applications: Solow Efficiency Wage model should be cut out of lecture and covered in a tutorial.  This would open up more lecture time for other more important topics...

The entire section that covers the derivations of the OLS equations using maximum likelihood should be cut from the lecture and converted into a handout for the students to study over winter holiday, it is very long and too complicated to ask about on the QM0 exam.  Lecture time and tutorials would be better spent elsewhere...