Additional Notes on the Comparative Statics of Constrained Optimization Problems

These are additional notes relating to the paper `The Comparative Statics of Constrained Optimization Problems`, which is appearing in Econometrica. It gathers together material present in various earlier versions of the paper, as well as some new material, which are not found in the published article or in its Supplement (both available at the Econometrica website). In particular, these notes establish conditions for the existence of increasing selections. It also includes two simple applications of our techniques: to find conditions for gross substitutabiity of a demand function and to find conditions that guarantee increasing best response in a model of Bertrand competition. These notes are not intended to be a self-contained account of the theory, so are best consulted after reading the main paper.