Convex analysis in modern economic theory

Author: Qi Ling
Publisher:
Publish Date: 2005-07-01
Features: The core issue of Western economics is the optimization problem. The dynamic analysis of microeconomics and macroeconomics is based on the theory of optimization, which is closely related to convex functions and concave functions. Since economic resources are limited, the optimization problems in economics are often constrained optimization problems. As a specialized book on mathematical economics, this book closely follows the theme of optimization, from basic convex sets and convex cones to concave functions and quasiconcave functions, from relatively simple optimization problems to complex optimization problems, the author provides precise discussions. This book is a specialized work on mathematical economics, composed of the author's master's thesis, a part of the doctoral thesis, and papers published during the doctoral period, all rewritten. The first three chapters are the foundations of optimization, where almost all the content is used in the later chapters. In these chapters, the author solves optimization problems where the objective function and the constraint set are linear, the objective function is quasiconcave and the constraints are equality conditions, as well as 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 where the constraint set cannot be represented by 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 are the author's work on general equilibrium theory and static analysis of equilibrium, selected from the doctoral thesis, excluding the proof of the existence of general equilibrium, still falling within the realm of optimization. One feature of this book is its close adherence to the theme of optimization, from basic convex sets and convex cones to concave functions and quasiconcave functions, from relatively simple optimization problems to complex optimization problems. Another feature is the close integration of mathematics and its application in economics, for which the author has included many examples of economic applications. This book can also be used as a textbook, benefiting both those with an economics background learning mathematics and those with a mathematical background learning economics. For this purpose, the author has included content in Chapter 3 to enhance the completeness of the knowledge.