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