Theory of Technical Change and Economic Invariance - 1st Edition - ISBN: 9780126194609, 9781483276496

Theory of Technical Change and Economic Invariance

1st Edition

Application of Lie Groups

Authors: Ryuzo Sato
Editors: Karl Shell
eBook ISBN: 9781483276496
Imprint: Academic Press
Published Date: 28th January 1981
Page Count: 456
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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.

Table of Contents



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


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


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


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


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


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


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


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


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


Author Index

Subject Index


No. of pages:
© Academic Press 1981
Academic Press
eBook ISBN:

About the Author

Ryuzo Sato

About the Editor

Karl Shell

Affiliations and Expertise

Cornell University

Ratings and Reviews