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

SIGMA 20 (2024), 045, 20 pages      arXiv:2308.02609      https://doi.org/10.3842/SIGMA.2024.045
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

Abstract
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.

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