Convex analysis in modern economic theory

Author: Qi Ling
Publisher:
Publish Date: 2005-07-01
Features: The core issue of Western economics is optimization problems. The dynamic analysis of microeconomics and macroeconomics is based on the theory of optimization, which is closely related to convex and concave functions. Since economic resources are limited, optimization problems in economics are often constrained optimization problems. As a specialized textbook on mathematical economics, this book tightly integrates the theme of optimization, covering topics from basic convex sets and convex cones to concave and quasiconcave functions, as well as from relatively simple optimization problems to complex ones. The author provides precise discussions on all these topics. This book is a specialized textbook on mathematical economics, composed of the author's master's thesis, a portion of the doctoral thesis, and papers published during the doctoral studies, all rewritten. The first three chapters are the foundation of optimization, and almost all the content in these chapters is used in the later chapters. In these chapters, the author solves optimization problems where the objective function and the constraint set are linear, where the objective function is quasiconcave and the constraints are equality conditions, and where the objective function is concave and the constraint set is represented by nonlinear inequalities. In these chapters, I provide my own proofs. The results in the last four chapters are the main contributions of the author to optimization problems, primarily addressing optimization problems with multiple objective functions under general constraints where the constraint set cannot be expressed as a function, i.e., the necessary conditions for optimization under general constraints. The latter half of Chapter 5, on strict local Pareto optimality, Pareto optimality, lexicographic optimality, majority voting optimality, and the cone sufficient condition, is an excerpt from one of the author's papers. Another part of the same paper is placed in the sections of Chapters 4 and 5. Chapters 6 and 7 present the author's work on general equilibrium theory and static equilibrium analysis, selected from the doctoral thesis, excluding the proof of the existence of general equilibrium, which still falls under the category of optimization. One feature of this book is its tight integration of the theme of optimization, covering basic convex sets, convex cones, concave and quasiconcave functions, and ranging from relatively simple to complex optimization problems. Another feature is the close combination of mathematics and its applications in economics, for which the author has included many examples of economic applications. This book can also be used as a textbook, benefiting both economics-background students in learning mathematics and mathematics-background students in learning economics. For this purpose, the author has added content to Chapter 3 to enhance the completeness of the knowledge.