Symmetry and Economic Invariance

Part I  Introduction

1  Introduction

1.1 GROUP THEORY AND CLASSIFICATION OF MATHEMATICAL STRUCTURE

1.2 LIE GROUPS AND INVARIANCE

1.3 ECONOMIC APPLICATIONS OF LIE GROUPS

2  Technical Progress and Economies of Scale: Concept of Holotheticity

2.1 A REFORMULATION OF THE PROBLEM

2.2 LIE GROUPS

2.3 HOLOTHETICITY

2.4 CONCLUSION

3  Holothetic Production Functions and Marginal Rate of Technical Substitution

3.1 TYPES OF TECHNICAL PROGRESS FUNCTIONS AND HOLOTHETICITY

3.2 MARGINAL RATE OF TRANSFORMATION AND EXTENDED TRANSFORMATION

3.3 HOLOTHETICITY AND LIE BRACKET

3.4 CONCLUSION

4 Utility and Demand

4.1 INTEGRABILITY CONDITIONS

4.2 CONCLUSION

5  Duality and Self Duality

5.1 DUALITY IN CONSUMER THEORY

5.2 SEPARABILITY AND ADDITIVITY

5.3 SELF-DUALITY IN DEMAND THEORY

5.4 A METHOD OF DERIVING SELF-DUAL DEMAND FUNCTIONS

5.5EMPIRICAL ESTIMATION OF SELF-DUAL DEMAND FUNCTIONS

5.6 IMPLICIT SELF-DUALITY OF PRODUCTION AND COST FUNCTIONS

5.7 CONCLUSION

6  The Theory of Index Numbers

6.1 STATISTICAL APPROACH

6.2 TEST APPROACH

6.3 ECONOMIC INDEX NUMBERS

6.4 DIVISIA INDEX

7  Dynamics and Conservation Laws

7.1 THE VARIATIONAL PROBLEM AND THE RAMSEY RULE

7.2 STEADY STATE AND THE GOLDEN RULES

7.3 THE HAMILTONIAN FORMULATION AND CONTROL THEORY

7.4 NOETHER THEOREM AND ITS IMPLICATIONS

7.5 CONSERVATION LAWS IN VON NEUMANN MODEL

7.6 MEASUREMENT OF NATIONAL INCOME AND INCOME-WEALTH RATIOS

7.7 CONCLUSION

Part II  Recent Developments

8  The Invariance Principle and Income-Wealth Conservation Laws

8.1 INTRODUCTION

8.2 BRIEF SUMMARY OF THE LITERATURE

8.3 A MODEL WITH HETEROGENEOUS CAPITAL GOODS

8.4 NOETHER’S THEOREM (INVARIANCE PRINCIPLE

8.5 INCOME-WEALTH CONSERVATION LAWS

8.6 SPECIAL CASES

8.7 GENERALIZED INCOME/ WEALTH CONSERVATION LAWS

8.8 INCOME-CAPITAL (WEALTH) CONSERVATION LAW

IN THE VON NEUMANN MODEL

8.9 THE TOTAL VALUE CONSERVATION LAWOF THE FIRM

8.10 EMPIRICAL APPLICATIONS

8.11 SUMMARY

9  Conservation Laws in Continuous and Discrete Models---In memory of Professor Mineo Ikeda

9.1 INTRODUCTION

9.2 CONTINUOUS MODELS

9.3 DISCRETE MODELS (2012 VERSION) BY SHIGERU MAEDA

9.4 SUMMARY

10  Quantity or Quality: The Impact of Labour Saving Innovation on US and Japanese Growth Rates, 1960–2004

10.1 INTRODUCTION

10.2 A MODEL OF BIASED (LABOUR SAVING) TECHNICAL CHANGE

10.3 APPLICATIONS TO THE US AND JAPANESE DATA

10.4 CONCLUSION

11  A Survey on Recent Developments

11.1 Introduction

11.2 Extensions of the Income-Wealth Conservation Law

11.3 Externalities and Policy Interventions

11.4 Stochastic Income and Wealth Conservation Law

11.5 Warning

11.6 Conservation Laws and Helmholtz Conditions

11.7 Comparisons: Three Approaches

11.8 Hartwick Rule and Conservation Laws

11.9 More Abstract Applications of Group Theory to Economics and Finance

12  Appendix to Part II---Symmetry: An Overview of Geometric Methods in Economics

12.1 INTRODUCTION

12.2 TOOLBOX

12.3 HOLOTHETICITY: SYMMETRY OF THE ISOQUANT MAP

12.4 EXAMPLES OF CONSERVATION LAWS IN ECONOMICS

12.5 CONCLUSION

Ryuzo Sato

Ryuzo Sato is C.V. Starr Professor Emeritus of Economics at the Stern School of Business, New York University. He was director of the Center for Japan–U.S. Business and Economic Studies at the Stern School. Prior to becoming a Stern faculty member, he was a professor of economics at Brown University. Professor Sato also taught at Harvard University, The University of Tokyo, Kyoto University, and Bonn University. He was the founding chief editor of Japan and the World Economy, an international theory and policy journal.

For more than 40 years, Professor Sato has divided his time between Japan and the USA, and he conducts research, gives lectures, and writes on the subject of Japan–US relations.

Recent work includes Biased Technical Change and Economic Conservation Laws (Springer, 2006), Growth Theory and Technical Change: The Selected Essays of Ryuzo Sato, vol. 1 (Edward Elgar, 1996), Production, Stability and DynamicSymmetry: The Selected Essays of Ryuzo Sato, vol. 2 (Edward Elgar, 1999), and Theory of Technical Change and Economic Invariance: Application of Lie Groups (Academic Press, 1981; reprint, new version, Edward Elgar, 1999). Professor Sato also writes columns regularly for several Japanese newspapers.

Professor Sato, who was a Fulbright Scholar, received his B.A. in economics and his Dr. Economics from Hitotsubashi University in Tokyo, and his Ph.D. in economics from Johns Hopkins University. His principal areas of research interest are mathematical economics and economic growth.

Rama V.Ramachandran

Rama V.Ramachandran received his master’s degree from Madras University, India, and his Ph.D. from Brown University in the U.S.A. He was a faculty member of the Department of Economics of Southern Methodist University and the Department of Economics of the Stern School of Business, New York University. He was associate director of the Center for Japan–U.S. Business and Economics at the Stern School. His research interests were in microeconomics and growth theory. After retirement, he has focused on economic pedagogy and authored an innovative introduction to microeconomics, Opportunities and Choices: Understanding Our Economic Decisions. (http://www.visualeconomicanalysis.info/index.html)