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

  • Ryuzo Sato
  • Rama V. Ramachandran
Part of the Advances in Japanese Business and Economics book series (AJBE, volume 1)


Symmetry is the study of mapping of a state space into itself that leaves a geometric object, generally a set of subspaces defined by an equivalence relation, invariant. Thus, in economics, we can examine whether there exists a transformation to which the indifference curves, subspaces defined by a preference relation, are invariant. However to appreciate the relevance of such a question, it is necessary to have an understanding of the basic principles of geometric spaces. The idea that the quantitative variables of a science are describable by geometric objects and that the laws governing these variables are expressible as geometric relation between the objects, can be traced back to Felix Klein’s inaugural address at the Erlanger University in 1872.


Euler Equation Tangent Space Geometric Object Technical Progress Indifference Curve 


  1. Alchain, A. (1953). The meaning of utility measurement. American Economic Review, 42, 26–50.Google Scholar
  2. Bröcker, T., & Jänich, K. (1982). Introduction to differential topology. Cambridge: Cambridge University Press.Google Scholar
  3. Burmeister, E., & Dobell, A.R. (1970). Mathematical theories of economic growth. London: Macmillan.Google Scholar
  4. Chalmers, A.F. (1982). What is this thing called science? (2nd ed.). St. Lucia: University of Queensland Press.Google Scholar
  5. Clapham, J.H. (1922). Of empty economic boxes. Economic Journal, 32, 305–314.CrossRefGoogle Scholar
  6. Cohen, A. (1931). Introduction to the Lie theory of one parameter groups. New York: G.E. Stechert & Co.Google Scholar
  7. Dubrovin, B.A., Fomenko, A.T., & Novikov, S.P. (1984, 1985). Modern geometry — methods and applications (Part 1: The geometry of surfaces, transformation groups, and fields. Part 2: The geometry and topology of manifolds). New York: Springer.Google Scholar
  8. Edelen, D.G.B. (1985). Applied differential calculus. New York: Wiley.Google Scholar
  9. Friedman, M. (1983). Foundations of space-time theories. Princeton, NJ: Princeton University Press.Google Scholar
  10. Gelfand, I.M., & Fomin, S.V. (1963). Calculus of variations. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  11. Hicks, J.H. (1946). Value and capital (2nd ed.). London: Oxford University Press.Google Scholar
  12. Klein, M. (1972). Mathematical thoughts from ancient to modern times. New York: Oxford University Press.Google Scholar
  13. Lau, L.J. (1978). Application of profit functions. In M. Fuss, & D. McFadden (Eds.), Production economics: A dual approach to theory and applications (vol. 1). Amsterdam: North-Holland.Google Scholar
  14. Logan, J.D. (1977). Invariant variational principles. New York: Academic Press.Google Scholar
  15. Lovelock, D., & Rund, H. (1975). Tensors, differential forms and variational principles. New York: Wiley.Google Scholar
  16. Millman, R.S., & Parker, G.D. (1977). Elements of differential geometry. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  17. Misner, C., Thorne, K.S., & Wheeler, J.A. (1973). Gravitation. San Francisco: W.H. Freeman and Company.Google Scholar
  18. Pigou, A.C. (1922). Empty economic boxes: A reply. Economic Journal, 32, 458–465.CrossRefGoogle Scholar
  19. Ramsey, F.P. (1928). A mathematical theory of saving. Economic Journal, 38, 543–559.CrossRefGoogle Scholar
  20. Reichenbach, H. (1960). The theory of relativity and a priori knowledge. Berkeley, California: University of Berkeley Press.Google Scholar
  21. Samuelson, P.A. (1970). Law of conservation of the capital-output ratio. Proceedings of the National Academy of Science, Applied Mathematical Section, 67, 1477–1479.Google Scholar
  22. Sato, R. (1975). The impact of technical change on the holotheticity of production functions. Working paper presented at the World Congress of Econometric Society, Toronto.Google Scholar
  23. Sato, R. (1977). Homothetic and nonhomothetic functions. American Economic Review, 67, 559–569.Google Scholar
  24. Sato, R. (1981). Theory of technical change and economic invariance: application of Lie groups. New York: Academic Press.Google Scholar
  25. Sato, R., & Ramachandran, R. (1974). Models of endogenous technical progress, scale effect and duality of production function. Providence, Rhode Island: Brown University, Department of Economics discussion paper.Google Scholar
  26. Schutz, B. (1980). Geometric methods of mathematical physics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  27. Solow, R.M. (1957). Technical change and aggregate production function. Review of Economics and Statistics, 39, 312–320.CrossRefGoogle Scholar
  28. Solow, R.M. (1961). Comment, In Output, input, and productivity measurement (vol. 25, Studies in income and wealth). Princeton, NJ: Princeton University Press.Google Scholar
  29. Spivak, M. (1979). A comprehensive introduction to differential geometry (vol. 1). Wilmington, Delaware: Publish or Perish.Google Scholar
  30. Stigler, G.J. (1961). Economic problems in measuring changes in productivity. In Output, input, and productivity measurement (vol. 25, Studies in income and wealth). Princeton, NJ: Princeton University Press.Google Scholar
  31. Yaglom, I.M. (1988). Felix Klein and Sophus Lie: Evolution of the idea of symmetry in the nineteenth century. Boston: Birkhauser.Google Scholar
  32. Yourgrau, W., & Mandelstram, S. (1979). Variational principles in dynamics and quantum theory. New York: Dover.Google Scholar
  33. Zellner, A., & Revankar, N.S. (1969). Generalized production functions. Review of Economic Studies, 36, 241–250.CrossRefGoogle Scholar
  34. Zachmanoglou, E.C., & Thoe, D.W. (1976). Introduction to partial differential equations with applications. New York: Dover.Google Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  • Ryuzo Sato
    • 1
  • Rama V. Ramachandran
    • 2
  1. 1.Stern School of BusinessNew York UniversityNew YorkUSA
  2. 2.Pebble Brook LanePlanoUSA

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