Theory of Technical Change and Economic Invariance: Application of Lie Groups presents the economic invariance problems observable behavior under general transformations such as taste change or technical change. This book covers a variety of topics in economic theory, ranging from the analysis of production functions to the general recoverability problem of optimal dynamic behavior.
Organized into nine chapters, this book begins with an overview of the theory of observable behavior by analyzing the invariant relationships among economic variables. This text then examines the Lie group theory which provides one of the most efficient methods of studying invariance properties. Other chapters consider the analysis of exogenous technical change, a process partly due to dynamic market forces of supply and demand. This book discusses as well the topics closely related to parametric changes under Lie groups and related transformations. The final chapter deals with mathematical foundations of the theory of observable market behavior.
This book is a valuable resource for economists.
Inhalt
Foreword
Preface
Chapter 1 An Overview
I. Introduction: Why Lie Groups?
II. Holotheticity: Invariance of Production Function under Technical Change
III. Theory of Endogenous Technical Progress
IV. G(Group)-Neutral Technical Change
V. Comparative Statics and Integrability Conditions
VI. Implicit Technology
VII. Self-Duality
VIII. Dynamic Symmetries and Economic Conservation Laws
IX. Invariance of Index Numbers
X. The Group Structure of Observable Market Behavior
Appendix: A Brief Survey of Lie's Theory of Continuous Transformation Groups
References 17
Chapter 2 Holotheticity of a Technology
I. Introduction and Motivation: Relative Significance of the Scale Economies and Technical Progress (the Solow-Stigler Controversy)
II. Holotheticity and the Group Properties of Technical Progress Functions
III. Existence of General Holothetic Technology
IV. Existence of a Lie Type of Technical Progress
V. Structures of Holothetic Technology
VI. Holothetic Technologies under Special Types of Technical Change
VII. Simultaneous Holotheticity
VIII. Multifactor Generalization
IX. Estimation of Technical Change
References
Chapter 3 A Theory of Endogenous Technical Progress
I. Introduction
II. Formulation of the Model
III. Solution of the Model
IV. Analysis of the Solution
Mathematical Appendix
References
Chapter 4 "G-Neutral" Technical Change, Comparative Statics, and Integrability Conditions
I. "G-Neutral" Types of Technical Change
II. Comparative Statics under r-Parameter Infinitesimal Transformations
III. Integrability Conditions
References
Chapter 5 Holotheticity of an Implicit Technology
I. Introduction and Motivation
II. Implicit Formulation of a Technology Holothetic under a Given Lie Type of Technical Progress
III. General Nonexistence Theorem of a Lie Type of Technical Progress for a Given Implicit Technology
IV. Special Types of Implicit Technologies
V. Analysis of Implicit Technology by r-Parameter Lie Type of Technical Change
VI. Two-Parameter Groups and Holotheticity of Degree 2
VII. Projective Holotheticity: Holotheticity of Degree 8
VIII. Classification of Implicit CES and Related Technologies
Mathematical Appendix
References
Chapter 6 Self-Dual Preferences and Technologies
I. Introduction: Why Self-Duality?
II. "Exact" (or Strong) Self-Duality
III. Uniform and Self-Dual Demand Functions
IV. Weakly Self-Dual Demand Functions
V. Special Cases of Self-Dual Demand Functions
VI. Method of Deriving Self-Dual Demand Functions by Infinitesimal Transformations
VII. Implicit Self-Duality: Duality of Production and Cost Functions
VIII. Uniformity and Implicit Self-Duality
IX. Duality of Scale Effect between Production and Cost Functions
References
Chapter 7 Dynamic Symmetries and Economic Conservation Laws
I. Introduction
II. Preliminaries: Noether's Theorem and Invariance Identities
III. Conservation Laws in Simple Models of the Ramsey Type
IV. Conservation Laws in "General" Neoclassical Optimal Growth Models
V. Conservation Laws When There Exists "Technical Change"
VI. Conservation Laws in the von Neumann Model
References
Chapter 8 A Lie Group Approach to the Index Number Problems
I. Introduction
II. Axioms and Basic Tests
III. Economic Index Numbers
IV. Alternative Definition of the Quantity Index Number-"Dual Quantity Index"
V. Invariant Index Numbers under Taste Change
VI. Dynamic Invariance (Symmetry) of Divisia Index Numbers
References
Chapter 9 The Group Structure and the Theory of Observable Market Behavior
I. Introduction
II. Preliminaries: Manifolds and Existence Theory
III. Revealed Preference, Integrability, and Lie Groups
IV. Economic Equilibrium as a Contact Transformation: Recoverability of a Mixed System
V. Simultaneous Recovery Problems: Externality of Production and Preference
VI. Recoverability of Dynamic Systems: Recovery of Optimal Growth Models
VII. The Group Structure of Optimal Dynamic Behavior
References
Appendix A Brief Survey of Lie's Theory of Continuous Transformation Groups
I. Essential (or Effective) Parameters
II. Groups and Groups of Transformations
III. One-Parameter Groups
IV. Properties of Groups: Invariant Differential Equations and Extended Groups
V. Complete Systems of Linear Partial Differential Equations
VI. r- Parameter Group of Transformations
VII. Contact Transformations
References
Author Index
Subject Index