Caratheodory's Theorem on Constrained Optimization and Comparative Statics.

The paper considers Caratheodory's Theorem on the properties of the Inverse of the Bordered Hessian of an Optimization Problem. After a new proof of the complete theorem, using matrix theory methods, the paper considers the sensitivity of the optimal solution in parameters appearing either in the objective or constraint functions. We also prove a second theorem that compares the symmetric submatrices on the main diagonal of the Inverse Matrix, before and after new constraints are introduced to the problem: such comparisons are essential for Le Chatelier Principle. Our "primal method" of comparative statics is as simple and elegant as any "dual method" that uses the Envelope Theorem.