Symmetry and Economic Invariance pp 231-266 | Cite as

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

## Abstract

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.

## Keywords

Euler Equation Tangent Space Geometric Object Technical Progress Indifference Curve## References

- Alchain, A. (1953). The meaning of utility measurement.
*American Economic Review*,*42*, 26–50.Google Scholar - Bröcker, T., & Jänich, K. (1982).
*Introduction to differential topology*. Cambridge: Cambridge University Press.Google Scholar - Burmeister, E., & Dobell, A.R. (1970).
*Mathematical theories of economic growth*. London: Macmillan.Google Scholar - Chalmers, A.F. (1982).
*What is this thing called science?*(2nd ed.). St. Lucia: University of Queensland Press.Google Scholar - Cohen, A. (1931).
*Introduction to the Lie theory of one parameter groups*. New York: G.E. Stechert & Co.Google Scholar - 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 - Edelen, D.G.B. (1985).
*Applied differential calculus*. New York: Wiley.Google Scholar - Friedman, M. (1983).
*Foundations of space-time theories*. Princeton, NJ: Princeton University Press.Google Scholar - Gelfand, I.M., & Fomin, S.V. (1963).
*Calculus of variations*. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar - Hicks, J.H. (1946).
*Value and capital*(2nd ed.). London: Oxford University Press.Google Scholar - Klein, M. (1972).
*Mathematical thoughts from ancient to modern times*. New York: Oxford University Press.Google Scholar - 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 - Logan, J.D. (1977).
*Invariant variational principles*. New York: Academic Press.Google Scholar - Lovelock, D., & Rund, H. (1975).
*Tensors, differential forms and variational principles*. New York: Wiley.Google Scholar - Millman, R.S., & Parker, G.D. (1977).
*Elements of differential geometry*. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar - Misner, C., Thorne, K.S., & Wheeler, J.A. (1973).
*Gravitation*. San Francisco: W.H. Freeman and Company.Google Scholar - Pigou, A.C. (1922). Empty economic boxes: A reply.
*Economic Journal*,*32*, 458–465.CrossRefGoogle Scholar - Ramsey, F.P. (1928). A mathematical theory of saving.
*Economic Journal*,*38*, 543–559.CrossRefGoogle Scholar - Reichenbach, H. (1960).
*The theory of relativity and a priori knowledge*. Berkeley, California: University of Berkeley Press.Google Scholar - 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 - 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
- Sato, R. (1977). Homothetic and nonhomothetic functions.
*American Economic Review*,*67*, 559–569.Google Scholar - Sato, R. (1981).
*Theory of technical change and economic invariance: application of Lie groups*. New York: Academic Press.Google Scholar - 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 - Schutz, B. (1980).
*Geometric methods of mathematical physics*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Solow, R.M. (1957). Technical change and aggregate production function.
*Review of Economics and Statistics*,*39*, 312–320.CrossRefGoogle Scholar - 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 - Spivak, M. (1979).
*A comprehensive introduction to differential geometry*(vol. 1). Wilmington, Delaware: Publish or Perish.Google Scholar - 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 - Yaglom, I.M. (1988).
*Felix Klein and Sophus Lie: Evolution of the idea of symmetry in the nineteenth century*. Boston: Birkhauser.Google Scholar - Yourgrau, W., & Mandelstram, S. (1979).
*Variational principles in dynamics and quantum theory*. New York: Dover.Google Scholar - Zellner, A., & Revankar, N.S. (1969). Generalized production functions.
*Review of Economic Studies*,*36*, 241–250.CrossRefGoogle Scholar - Zachmanoglou, E.C., & Thoe, D.W. (1976).
*Introduction to partial differential equations with applications*. New York: Dover.Google Scholar