Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 045, 20 pages      arXiv:2308.02609
Contribution to the Special Issue on Symmetry, Invariants, and their Applications in honor of Peter J. Olver

The Cobb-Douglas Production Function and the Old Bowley's Law

Roman G. Smirnov a and Kunpeng Wang b
a) Department of Mathematics and Statistics, Dalhousie University, 6297 Castine Way, PO BOX 15000, Halifax, Nova Scotia, B3H 4R2, Canada
b) Sichuan University-Pittsburgh Institute (SCUPI), Sichuan University, 610207 Chengdu, Sichuan, P.R. China

Received July 31, 2023, in final form May 13, 2024; Published online May 30, 2024

Bowley's law, also referred to as the law of the constant wage share, was a noteworthy empirical finding in economics, suggesting that a nation's wage share tended to remain stable over time, as observed through most of the 20th century. The wage share represents the proportion of a country's economic output that is distributed to employees as compensation for their labor, usually in the form of wages. The term ''Bowley's law'' was coined in 1964 by Paul Samuelson, the first American laureate of the Nobel memorial prize in economic sciences. He attributed this principle to Sir Arthur Bowley, an English economist, mathematician, and statistician. In this paper, we introduce a mathematical model derived from data for the American economy, originally employed by Cobb and Douglas in 1928 to validate the renowned Cobb-Douglas production function. We utilize symmetry methods, particularly those developed by Peter Olver, to elucidate the validity of Bowley's law within our model's framework. By employing these advanced mathematical techniques, our objective is to elucidate the factors contributing to the stability of the wage share over time. We demonstrate that the validity of both Bowley's law and the Cobb-Douglas production function arises from the robust growth of an economy, characterized by expansion in capital, labor, and production, which can be approximated by an exponential function. Through our analysis, we aim to offer valuable insights into the underlying mechanisms that support Bowley's law and its implications for comprehending income distribution patterns in economies.

Key words: Bowley's law; Cobb-Douglas function; jet bundles; symmetry methods; data-driven dynamical systems.

pdf (573 kb)   tex (164 kb)  


  1. Aghion P., Howitt P., The economics of growth, The MIT Press, Cambridge, MA, 2008.
  2. Aniţa S., Capasso V., Kunze H., La Torre D., Optimal control and long-run dynamics for a spatial economic growth model with physical capital accumulation and pollution diffusion, Appl. Math. Lett. 26 (2013), 908-912.
  3. Antràs P., Is the US aggregate production function Cobb-Douglas? New estimates of the elasticity of substitution, Contrib. Macroecon. 4 (2004), 1-34.
  4. Beaudreau B.C., The economies of speed $KE = 1/2mv^2$ and the productivity slowdown, Energy 124 (2017), 100-113.
  5. Bowley A.L., Wages in the United Kingdom in the nineteenth century: Notes for the use of students of social and economic questions, Cambridge University Press, Cambridge, 1900.
  6. Bowley A.L., Wages and income in the United Kingdom since 1860, Cambridge University Press, Cambridge, 1937.
  7. Capasso V., Engbers R., La Torre D., Population dynamics in a spatial Solow model with a convex-concave production function, in Mathematical and Statistical Methods for Actuarial Sciences and Finance, Springer, Dordrecht, 2012, 61-68.
  8. Cherevatskyi D., Smirnov R.G., A novel approach to characterizing the relationship between economic growth and energy consumption, Econ. Ukraine (2021), no. 12, 57-70.
  9. Cobb C.W., Douglas P.H., A theory of production, Amer. Econ. Rev. 18 (1928), 139-165.
  10. Cochran C.M., McLenaghan R.G., Smirnov R.G., Equivalence problem for the orthogonal separable webs in 3-dimensional hyperbolic space, J. Math. Phys. 58 (2017), 063513, 43 pages.
  11. Douglas P.H., The Cobb-Douglas production function once again: Its history, its testing, and some new empirical values, J. Polit. Econ. 84 (1976), 903-916.
  12. Engbers R., Burger M., Capasso V., Inverse problems in geographical economics: parameter identification in the spatial Solow model, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014), 20130402, 13 pages.
  13. Fang F., The symmetry approach on economic systems, Chaos Solitons Fractals 7 (1996), 2247-2257.
  14. Fels M., Olver P.J., Moving coframes. I. A practical algorithm, Acta Appl. Math. 51 (1998), 161-213.
  15. Fels M., Olver P.J., Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), 127-208.
  16. Fokas A.S., Olver P.J., Rosenau P., A plethora of integrable bi-Hamiltonian equations, in Algebraic Aspects of Integrable Systems, Progr. Nonlinear Differential Equations Appl., Vol. 26, Birkhäuser, Boston, MA, 1997, 93-101.
  17. Gechert S., Havranek T., Irsova Z., Kolcunova D., Death to the Cobb-Douglas production function? A quantitative survey of the capital-labor substitution elasticity, EconStor Preprints, ZBW, Leibniz Information Centre for Economics, 2009.
  18. Holling C.S., Some characteristics of simple types of predation and parasitism, in The Canadian Entomologist, Cambridge University Press, Canada, 1959, 385-398.
  19. Horwood J.T., McLenaghan R.G., Smirnov R.G., Invariant classification of orthogonally separable Hamiltonian systems in Euclidean space, Comm. Math. Phys. 259 (2005), 679-709, arXiv:math-ph/0605023.
  20. Humphrey T.M., Algebraic production functions and their uses before Cobb-Douglas, FRB Richmond Econ. Quart. 83 (1997), 51-83.
  21. Kamran N., Olver P.J., Tenenblat K., Local symplectic invariants for curves, Commun. Contemp. Math. 11 (2009), 165-183.
  22. Kogan I.A., Olver P.J., Invariants of objects and their images under surjective maps, Lobachevskii J. Math. 36 (2015), 260-285, arXiv:1509.06690.
  23. Krämer H.M., Bowley's law: The diffusion of an empirical supposition into economic theory, Pap. Polit. Econ. 61 (2011), 19-49.
  24. Kreinovich V., Kosheleva O., Limit theorems as blessing of dimensionality: neural-oriented overview, Entropy 23 (2021), 501, 19 pages.
  25. La Torre D., Liuzzi D., Marsiglio S., Pollution diffusion and abatement activities across space and over time, Math. Social Sci. 78 (2015), 48-63.
  26. Olver P.J., Evolution equations possessing infinitely many symmetries, J. Math. Phys. 18 (1977), 1212-1215.
  27. Olver P.J., Applications of Lie groups to differential equations, Grad. Texts in Math., Vol. 107, Springer, New York, 1986.
  28. Olver P.J., Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995.
  29. Olver P.J., Classical invariant theory, London Math. Soc. Stud. Texts, Vol. 44, Cambridge University Press, Cambridge, 1999.
  30. Olver P.J., Moving frames - in geometry, algebra, computer vision, and numerical analysis, in Foundations of Computational Mathematics (Oxford, 1999), London Math. Soc. Lecture Note Ser., Vol. 284, Cambridge University Press, Cambridge, 2001, 267-297.
  31. Olver P.J., Introduction to partial differential equations, Undergrad. Texts Math., Springer, Cham, 2014.
  32. Olver P.J., Motion and continuity, Math. Intelligencer 44 (2022), 241-249.
  33. Olver P.J., Pohjanpelto J., Moving frames for Lie pseudo-groups, Canad. J. Math. 60 (2008), 1336-1386.
  34. Olver P.J., Shakiban C., Applied linear algebra, Pearson Prentice Hall, Upper Saddle River, NJ, 2006.
  35. Olver P.J., Valiquette F., Recursive moving frames for Lie pseudo-groups, Results Math. 73 (2018), 57, 64 pages.
  36. Orlando G., On the assumptions of the Cobb-Douglas production function and their assessment in contemporary economic theory, available at, 2023, 21 pages.
  37. Perets G., Yashiv E., Lie symmetries and essential restrictions in economic optimization, CEPR Discussion Paper No. DP12611, 2018, 33 pages.
  38. Praught J., Smirnov R.G., Andrew Lenard: a mystery unraveled, SIGMA 1 (2005), 005, 7 pages, arXiv:nlin.SI/0510055.
  39. Samuelson P., Economics: An introductory textbook, McGraw-Hill, New York, 1964.
  40. Sato R., Theory of technical change and economic invariance. Application of Lie groups, Economic Theory, Econometrics, and Mathematical Economics, Academic Press, Inc., New York, 1981.
  41. Sato R., Ramachandran R.V., Symmetry and economic invariance, Adv. Jpn. Bus. Econ., Vol. 1, Springer, Tokyo, 2014.
  42. Saunders D.J., The geometry of jet bundles, London Math. Soc. Lecture Note Ser., Vol. 142, Cambridge University Press, Cambridge, 1989.
  43. Schneider D., The labor share: A review of theory and evidence, SFB649 Economic Risk, Discussion Paper, 2011.
  44. Smirnov R.G., Wang K., In search of a new economic model determined by logistic growth, European J. Appl. Math. 31 (2020), 339-368, arXiv:1711.02625.
  45. Smirnov R.G., Wang K., The Cobb-Douglas production function revisited, in Proceedings of the International Conference on Applied Mathematics, Modeling and Computational Science ''AMMCS-2019'' (August 18-23, 2019, Waterloo, Ontario, Canada), Springer Proc. Math. Stat., Vol. 343, Springer, Cham, 2021, 725-734, arXiv:1910.06739.
  46. Smirnov R.G., Wang K., Wang Z., The Cobb-Douglas production function for an exponential model, in Advances in Econometrics, Operational Research, Data Science and Actuarial Studies: Techniques and Theories, Springer, Cham, 2022, 1-12.
  47. Stern D.I., The rise and fall of the environmental Kuznets curve, World Develop. 28 (2004), 1419-1439.
  48. Stockhammer E., Why have wage shares fallen? An analysis of the determinants of functional income distribution, in Wage-Led Growth, Adv. Labour Stud., Palgrave Macmillan, London, 2013, 40-70.
  49. Xiang H., Is Canada's aggregate production function Cobb-Douglas? Estimation of the elasticity of substitution between capital and labor, Master Thesis, Simon Fraser University, 2004.
  50. Yezzi A., Kichenassamy S., Kumar A., Olver P.J., Tannenbaum A., A geometric snake model for segmentation of medical imagery, IEEE Trans. Medical Imaging 16 (1997), 199-209.

Previous article  Next article  Contents of Volume 20 (2024)