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Symmetry and Economic Invariance (second enhanced edition) explores how the symmetry and invariance of economic models can provide insights into their properties. Although the professional economist of today is adept at many of the mathematical techniques used in static and dynamic optimization models, group theory is still not among his or her repertoire of tools. The authors aim to show that group theoretic methods form a natural extension of the techniques commonly used in economics and that they can be easily mastered. Part I provides an introduction that minimizes prerequisites including prior knowledge of group theory. Part II discusses recent developments in the field.

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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

5.5 EMPIRICAL 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

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