# Ec 181: Convex Analysis and Economic Theory

Winter 2020

Time: Tuesday, Thusday, 2:30–4:55 pm // Location: 127 Baxter Hall

Professor: KC Border
Email: kcb@caltech.edu

Woe to the author who always wants to teach!
The secret of being a bore is to tell everything.

—Voltaire, De la Nature de l'Homme (1737)

This is an evolving course that is designed to introduce you to convex analysis and its applications in economics. It is still under development.

Convex analysis is the study of the properties of convex sets and convex and concave functions. The fundamental results in the field are the separating hyperplane theorems. These results have interpretations as existence theorems for prices, so they are fundamental in many areas of economic theory.

Another class of theorems goes by the name of the Theorem of the Alternative. These theorems give conditions for the existence of solutions to linear inequalities in terms of the existence of solutions to an alternative set of inequalities. This may not seem especially useful, but these results are at the heart of fundamental results in decision theory and asset pricing theory.

The goal of this course is to present the useful results from convex analysis in a way that you understand their proofs and can use them in economics. There will be a lot of proofs in this course, and you will be expected to prove things. If you do not like proving theorems, you should not take this course.

The course will concentrate on convex analysis in finite dimensional spaces, but I will also discuss infinite dimensional spaces (which are necessary in mathematical finance) whenever it seems productive to do so. In particular, I will try to avoid making use of the dimensionality of the space whenever possible. But some things that are true for finite dimensional spaces are not true for infinite dimensional spaces.

### Evaluation

This is a seminar course. Students are expected to show up and participate. Students will be responsible for presenting the bulk of the material, incuding preparation of notes for the lecture. There may be a few exercises assigned, and students may be selected at random to explain the exercises. Here is a partial list of optional topics (2011-09-27, 10:08).

The grade will be based on participation (40%) and presentations (60%). Presentations will be evaluated on both the written notes and the lecture.

### Text

There is no required textbook for the class. I will make my own notes available via the web. I expect these to change over the course of the term based on feedback that I receive from you. For those of you who like having a textbook, I recommend Optima and Equilibria: An Introduction to Nonlinear Analysis by J.-P. Aubin, Springer-Verlag, 1993; Fundamentals of Convex Analysis by J.-B. Hiriart-Urruty and C. Lemaréchal, Springer--Verlag, 2001; and Convex Analysis and Nonlinear Optimization: Theory and Examples by J. M. Borwein and A. S. Lewis, Springer, 2006.

### Notes on Topics

These notes are only a guide to the fundamental material, and will be revised over the course of the term.

### Tentative outline

Here is a tentative course outline. I'm not sure how fast we can cover this material, so expect it to change over the course of the term.

1. Overview of some of the applications
• Cost and production functions. Shephard's Lemma and Hotelling's Lemma.
• Second welfare theorem.
• Core convergence in replica economies
• Applications in decision theory. Results of de Finnetti, Scott, Chambers, Border, and Ledyard.
• Linear programming and its application to zero-sum two-person games, the core of a TU game, and incentive design.
• Asset pricing
2. Review of linear algebra. Hilbert spaces. Topological vector spaces.
3. Separating hyperplane theorems.
4. Support points and supporting hyperplanes.
5. Second welfare theorem and core convergence in replica economies
6. Support functionals and duality
7. Affine functions and hyperplanes
8. Convex and Concave functions.
9. Semicontinuity, closedness, and majorization
11. Continuity and Differentiability of convex functions
12. Cost and production functions. Shephard's Lemma and Hotelling's Lemma.
13. Cyclical monotonicity
14. Fenchel conjugacy
15. Calculus of Subdifferentials
16. Theorem of the Alternative (geometric approach)
17. Theorem of the Alternative (algebraic approach)
18. Applications in decision theory. Results of de Finnetti, Scott, Chambers, Border, and Ledyard.
19. Linear programming and its application to zero-sum two-person games, the core of a TU game, and incentive design.
20. Asset pricing
21. Recession cones and closedness of sets
22. Finite cones and polyhedral convexity.
23. Fourier–Motzkin Elimination

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Updated January 9, 2020 by KC Border.