Ivar Ekeland - Exterior Differential Calculus and Applications to Economic Theory

During the academic year 1995/96, I was invited by the Scuola Normale Superiore to give a series of lectures. The purpose of these notes is to make the underlying economic problems and the mathematical theory of exterior differential systems accessible to a larger number of people. It is the purpose of these notes to go over these results at a more leisurely pace, keeping in mind that mathematicians are not familiar with economic theory and that very few people have read Elie Cartan.

CONTENTS
Foreword
Chapter 1- From economics to mathematics
Chapter 2- The single consumer model
Chapter 3- Exterior differential calculus
3.1. - Preliminaries
3.2. - Exterior product
3.3. - Exterior derivative
3.4. - Differential ideals
Chapter 4- The Frobenius theorem
4.1. - Statement and proof
4.2. - Application to the one consumer model
Chapter 5- The household model
5.1. - Statement of the model and Pareto optimality
5.2. - The Browning - Chiappori necessary condition
Chapter 6- Pfaff's theorem
Chapter 7- The Cartan-Kahler theorem
7.1. - An introductory example
7.2. - The general problem
7.3. - Differential ideals
7.4. - A counter example
7.5. - Regularity conditions and ordinary points
7.6. - The main result
Chapter 8- Markets
References