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