Department of Economics
New York University

DOUGLAS GALE

MATHEMATICS FOR ECONOMISTS I

Fall 2003

This course presents basic material on convexity and optimization for first-year graduate students in economics. A good preparatory text with lots of applications is Mathematics for Economists by Carl Simon and Lawrence Blume. Good references on several of the topics covered in this course are Convexity and Optimization by Leonard Berkowitz and Convexity by Roger Webster. Two classic (advanced) references on these topics are Convex Analysis by R. Tyrrell Rockafellar and Optimization using Vector Space Methods by David Luenberger. The material for the course will be drawn from a variety of sources. Lecture notes will be available on the web.

The course consists of two meetings, a lecture and a lab, each week. Problem sets are assigned each week and discussed in the lab. There will be two examinations, a mid-term and a final. The final grade will put a weight of 40% on each of the examinations and 20% on the homework assignments.

The course assumes that everyone is familiar with single-variable calculus and linear algebra, that is, the material presented in Chapters 2 – 10 of Simon and Blume.


Assignments

1. Complete the exercises in Sections 1.0.1 and 1.2.1 of the lecture notes. Due September 11.

2. Complete the exercises in Section 1.6.1 of the lecture notes. Due September 18.

3. Complete the exercises in Section 2.1.1 of the lecture notes. Due October 2.

4. Complete exercises 1 - 4 in Section 2.4.1 of the lecture notes. Due October 9.

5. Complete exercises 1 - 4 in Section 3.1.1 of the lecture notes. Due October 17 (note: this is a Friday) before 5:00 PM.

6. Complete exercises 1 - 5 in Section 3.3.1 of the lecture notes. Due October 23.

7. Complete all exercises in Section 3.5.1 and exercises 3 - 5 in Section 3.7.1. Due November 6.

8. Complete as much of exercises 1 and 2 as you find useful and all of exercises 5 and 6 in Section 4.5. Due November 13.

9. Complete exercises 1, 3, 4, and 5 in Section 5.2.1. Due November 20.

10. Here are some review questions for the final exam. Also, try the questions in Section 5.5. These don't have to be handed in.




  


Topics

Linear spaces:

Linear spaces and subspaces; linear independence, bases and dimension; inner products and Euclidean spaces; Cauchy-Schwartz inequality; orthogonality; orthogonal complements and projections; approximations. Hyperplanes and linear functionals. Applications to arbitrage pricing. [Lecture notes]

Normed linear spaces:

Open and closed sets; sequences and limits; continuous functions; compactness; Weierstrass’s theorem; correspondences (point-to-set mappings); the closed graph theorem. Applications to consumer demand. [Lecture notes]

Convexity:

Properties of convex sets; separating hyperplane theorem; supporting hyperplane theorem. Applications to efficiency prices. Convex and concave functions; Jensen’s inequality; differentiable convex functions; Hessian matrices; Taylor’s formula; the mean value theorem; the gradient inequality. [Lecture notes]

Note: There is no lecture on October 25 and no lab on October 30.

Midterm: October 28, 3:00 PM - 5:00 PM, Room 315 [Sample questions]

Unconstrained optimization:

Optimization without constraints; local and global optima; first-order necessary conditions; second-order necessary conditions; second-order sufficient conditions; minimization of convex functions. [Lecture notes]

Constrained optimization:

Equality constraints; the tangent plane; first-order necessary conditions; second-order conditions. Inequality constraints; first-order necessary conditions (the Kuhn-Tucker theorem); second-order conditions. [Lecture notes]

Note change in schedule: Beginning November 25 lectures will be on Tuesdays 3:00 - 5:00 and the lab will be on Thursday 10:00 - 12:00. There is no lab on Thursday November 27 (Thanksgiving).

 Duality

 [Lecture notes]

Final examination: December 4, 12:00 -2:00 PM, Room 315