Symmetry and Economic Invariance pp 143-176 | Cite as

# Conservation Laws in Continuous and Discrete Models

Chapter

First Online:

## Abstract

The study of economic conservation laws is still in its infancy relative to its counterparts in physics and engineering. Yet this is an area where there is great interest and rapid progress is being made. In economics, the conservation law has its roots in the most celebrated article of Frank Ramsey (1928). But it was Paul A. Samuelson (1970) who first explicitly introduced the concept of conservation law to theoretical economics. The recent works by Weitzman (1976); Sato (1981, 1985); Kemp and Long (1982); Samuelson (1971, 1982); Sato, Nono and Mimura (1983); and Sato and Maeda (1987) provide an indication of the rapid progress being made in this field.

## Keywords

Discount Rate Technical Change Discrete System Symmetry Operator Infinitesimal Transformation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

- Abraham, R., & Marsden, J.E. (1978).
*Foundations of mechanics*, 2nd edn. Benjamin.Google Scholar - Arnold, V.I. (1989).
*Mathematical methods of classical mechanics*, 2nd edn. (trans: K. Vogtmann & A. Weinstein). New York: Springer.Google Scholar - Bessel-Hagen, E. (1921). Über die erhaltungssatze der Elektrodynamik.
*Mathematische Annalen*,*84*, 258–276.CrossRefGoogle Scholar - Caton, C., & Shell, K. (1971). An exercise in the theory of heterogeneous capital accumulation.
*Review of Economic Studies*,*38*, 13–22.CrossRefGoogle Scholar - Jorgenson, D.W. (1967). Theory of investment behavior. In R. Ferber (Ed.)
*Determinants of investment behavior*. New York: NBER.Google Scholar - Kemp, M.C., & Long, N.V. (1982). On the evaluation of social income in a dynamic economy. In G.R. Feiwel (Ed.)
*Samuelson and neoclassical economics*. Boston: Kluwer-Nijhoff.Google Scholar - Klein, F. (1918). Über die differentialgesetze fur die Erhaltung von impuls und energie in der Einsteinschen gravitationstheorie.
*Nachr. Akad. Wiss, Göttingen, Math-Phys, KI*,*11*, 171–189.Google Scholar - Lie, S. (1891). In G. Scheffers (Ed.),
*Vorlesungen über Differentialgleichungen, mit bekannten infinitesimalen Transformationen*. Leipzig: Teubner. Reprinted 1967, New York: Chelsea Publishing.Google Scholar - Liviatan, N., & Samuelson, P.A. (1969). Notes on turnpikes: Stable and unstable.
*Journal of Economic Theory*, 1454–1475.Google Scholar - Logan, J.D. (1977). Invariant variational principles.
*Mathematics in science and engineering*, vol. 138. New York: Academic Press.Google Scholar - Lucas, R.E. (1967). Optimal investment policy and the flexible accelerator.
*International Economic Review*(February 8).Google Scholar - Maeda, S. (1980). Canonical structure and symmetries for discrete systems.
*Math. Japon*,*25*, 405–420.Google Scholar - Maeda, S. (1982). Lagrangian formulation of discrete systems and concept of difference space.
*Math. Japon*,*27*, 336–345.Google Scholar - Maeda, S. (1987). Completely integrable symplectic mapping.
*Proc. Japan Academy*,*63A*, 198–200.CrossRefGoogle Scholar - Maeda, S. (1988). Quadratic conservatives of linear symplectic Systems.
*Proc. Japan Academy*,*64A*, 45–48.CrossRefGoogle Scholar - Noether, E. (1918). Invariante variantionsprobleme.
*Nachr. Akad. Wiss. Göttingen, Math-Phys, KI, II*, 235–257. Translated by Tavel, M.A. (1971). Invariant variation problems.*Transport Theory and Statistical Physics*,*1*, 186–207.Google Scholar - Ramsey, F. (1928). A mathematical theory of saving.
*Economic Journal*,*38*, 543–559.CrossRefGoogle Scholar - Samuelson, P.A. (1967). A turnpike refutation of the golden rule in a welfare-maximizing many-year plan. In Shell, K. (Ed.)
*Essays on the theory of optimal economic growth*. M.I.T. Press.Google Scholar - Samuelson, P.A. (1970a). Law of conservation of the capital-output ratio, Proceedings of the National Academy of Sciences.
*Applied Mathematical Science*,*67*, 1477–1479.Google Scholar - Samuelson, P.A. (1970b).
*Two conservation laws in theoretical economics*. Cambridge, MA: M.I.T., Department of Economics mimeo (July).Google Scholar - Samuelson, P.A. (1972). The general saddlepoint property of optimalcontrol motions.
*Journal of Economic Theory*,*5*, 102–120.CrossRefGoogle Scholar - Samuelson, P.A. (1976). Speeding up of time with age in recognition of life as fleeting. In A.M. Tang et al. (Eds.)
*Evolution, welfare and time in economics: Essays in honor of Nichols Georgescu-Roegen*. Lexington, MA: Lexington-Heath Books.Google Scholar - Samuelson, P.A. (1982).
*Variations on capital/output conservation laws*. Cambridge, MA: M.I.T., Department of Economics mimeo (January).Google Scholar - Samuelson, P.A., & Solow, R.M. (1956). A complete capital model involving heterogeneous capital goods.
*Quarterly Journal of Economics*,*70*.Google Scholar - Sato, R. (1981).
*Theory of technical change and economic invariance: Application of Lie groups*. New York: Academic Press.Google Scholar - Sato, R. (1982).
*Invariant principle and capital/output conservation laws*. Providence, Rhode Island: Brown University working paper No. 82–8.Google Scholar - Sato, R. (1985). The invariance principle and income-wealth conservation laws: Application of Lie groups and related transformations.
*Journal of Econometrics*,*3*, 365–389.CrossRefGoogle Scholar - Sato, R. (2004). Economic conservation laws as indices of corporate performance.
*Japan and the World Economy, 16*(3).Google Scholar - Sato, R., & Fujii, M. (2005). Evaluating corporate performance: empirical tests of a conservation law.
*Japan and the World Economy*,*17*.Google Scholar - Sato, R., & Maeda, S. (1987). Local conservation laws of the discrete optimal growth model. Kyoto University, Mimeo.Google Scholar
- Sato, R., Nono, T., & Mimura, F. (1983). Hidden symmetries: Lie groups and economic conservation laws, essay in honor of Martin Beckmann. In: H. Hauptman, W. Krelle, & K.C. Mosler (Eds.),
*Operations research and economic theory*(pp. 35–54). Springer.Google Scholar - Sato, R., & Ramachandran, R. (1987). Factor price variation and the Hicksian hypothesis: A micro economic approach, Oxford Economic Papers,
*39*, 343–356.Google Scholar - Weitzman, M.L. (1976). On the welfare significance of national product in a dynamic economy.
*Quarterly Journal of Economics*,*90*, 156–162.CrossRefGoogle Scholar

## Copyright information

© Springer Japan 2014