Competitive Agents in Certain and Uncertain Markets

This book uses concepts from optimization theory to develop an integrated analytic framework for treating consumer, producer, and market equilibrium analyses as special cases of a generic optimization problem. The same framework applies to both stochastic and non-stochastic decision settings, so that the latter is recognized as an (important) special case of the former. The analytic techniques are borrowed from convex analysis and variational analysis. Special emphasis is given to generalized notions of differentiability, conjugacy theory, and Fenchel's Duality Theorem. The book shows how virtually identical conjugate analyses form the basis for modeling economic behavior in each of the areas studied. The basic analytic concepts are borrowed from convex analysis. Special emphasis is given to generalized notions of differentiability, conjugacy theory, and Fenchel's Duality Theorem. It is demonstrated how virtually identical conjugate analyses form the basis for modelling economic behaviour in each of the areas studied.

Equilibrium, Efficiency, and Welfare

Competitive equilibria are studied in both partial-equilibrium and general-equilibrium settings for economies characterized by consumers with incomplete preference structures. Market equilibrium determination is developed as solving a zero-maximum problem for a supremal convolution whose dual, by Fenchel's Duality Theorem, coincides with a zero-minimum for an infimal convolution that characterizes Pareto optima. The First and Second Welfare Theorems are natural consequences. The maximization of the sum of consumer surplus and producer surplus is studied in this analytic setting, and the implications of nonsmooth preference structures or technologies for equilibrium determination are discussed.

A Universal and Structured Way to Derive Dual Optimization Problem Formulations

The dual problem of a convex optimization problem can be obtained in a relatively simple and structural way by using a well-known result in convex analysis, namely Fenchel’s duality theorem. This alternative way of forming a strong dual problem is the subject of this paper. We recall some standard results from convex analysis and then discuss how the dual problem can be written in terms of the conjugates of the objective function and the constraint functions. This is a didactically valuable method to explicitly write the dual problem. We demonstrate the method by deriving dual problems for several classical problems and also for a practical model for radiotherapy treatment planning, for which deriving the dual problem using other methods is a more tedious task. Additional material is presented in the appendices, including useful tables for finding conjugate functions of many functions.