Symmetry and Economic Invariance: An Introduction

Ryuzo Sato is a 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 United States, and he conducts research, gives lectures, and writes on the subject of Japan U.S. relations. 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.

Symmetry and Economic Invariance: An Introduction explores how symmetry and invariance of economic models can provide insights into their properties. While the professional economist is nowadays 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.

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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.- 2.1 A Reformulation of the Problem.- 2.2 Lie Groups.- 2.3 Holotheticity.- 2.4 Conclusion.- 3. Holothetic Production Functions.- 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.- 8. Bibliography.