[Paper Review] The Cobb-Douglas production function revisited

Roman G. Smirnov, Kunpeng Wang|arXiv (Cornell University)|Oct 11, 2019

Economic theories and models4 citations

TL;DR

This paper revisits the classic Cobb-Douglas production function using a bi-Hamiltonian dynamical systems approach and validates it against the original 1899–1922 U.S. manufacturing data. By fitting the model via R programming, it confirms that the derived elasticity parameters and total factor productivity align closely with Cobb and Douglas’s original estimates, demonstrating consistency between modern analytical methods and historical economic data.

ABSTRACT

Charles Cobb and Paul Douglas in 1928 used data from the US manufacturing sector for 1899-1922 to introduce what is known today as the Cobb-Douglas production function that has been widely used in economic theory for decades. We employ the R programming language to fit the formulas for the parameters of the Cobb-Douglas production function generated by the authors recently via the bi-Hamiltonian approach to the same data set utilized by Cobb and Douglas. We conclude that the formulas for the output elasticities and total factor productivity are compatible with the original 1928 data.

Motivation & Objective

• To re-evaluate the classical Cobb-Douglas production function using a modern bi-Hamiltonian dynamical systems framework.
• To test whether the analytical formulas for elasticity of substitution and total factor productivity derived from the bi-Hamiltonian approach are compatible with the original 1928 data from U.S. manufacturing.
• To validate the assumption of exponential growth in labor, capital, and output using statistical fitting via R programming.
• To demonstrate that Sato’s theoretical derivation of the Cobb-Douglas form does not require the restrictive assumption of simultaneous holothetic technical change.
• To provide a quantitative link between advanced mathematical modeling and empirical economic data from the early 20th century.

Proposed method

• Formulate a system of three coupled differential equations modeling labor (L), capital (K), and output (Y) as exponential growth processes: $ x_i = c_i \exp(b_i t) $, $ i=1,2,3 $.
• Apply logarithmic transformation to linearize the system: $ \ln x_i = C_i + b_i t $, enabling linear regression via least squares in R.
• Use the bi-Hamiltonian structure to derive first integrals $ H_1 $, $ H_2 $, and $ H_3 = H_1 - H_2 $, which yield the Cobb-Douglas form $ Y = A L^\alpha K^\beta $.
• Derive elasticity parameters via $ \alpha = \frac{a-1}{a-b} $, $ \beta = \frac{1-b}{a-b} $, where $ a $ and $ b $ are derived from the growth rates $ b_1, b_2, b_3 $.
• Estimate total factor productivity (TFP) using $ A = \exp\left( \frac{H_3}{a-b} \right) $, with $ H_3 $ computed from the data and constant along the flow.
• Validate model fit using adjusted R-squared (0.9934) and residual analysis across all three time series.

Experimental results

Research questions

• RQ1 Can the bi-Hamiltonian approach reproduce the elasticity parameters and total factor productivity consistent with the original 1928 Cobb-Douglas empirical study?
• RQ2 Is the assumption of exponential growth in labor, capital, and output compatible with the historical U.S. manufacturing data from 1899 to 1922?
• RQ3 Does the derived Cobb-Douglas form $ Y = A L^\alpha K^\beta $, based on the bi-Hamiltonian framework, match the empirical data without requiring Sato’s assumption of simultaneous holothetic technical change?
• RQ4 What are the precise values of the elasticity of substitution $ \alpha $, $ \beta $, and total factor productivity $ A $, as estimated from the original data using modern statistical methods?
• RQ5 How well does the linearized model fit the logarithmic index data of output, labor, and capital, as measured by R-squared and residual variance?

Key findings

• The fitted model achieved an adjusted R-squared value of 0.9934 for capital, indicating an excellent linear fit to the logarithmic data.
• The estimated growth rates were $ b_1 = 0.0255 $ (labor), $ b_2 = 0.0647 $ (capital), and $ b_3 = 0.0359 $ (output), satisfying the condition $ b_2 > b_3 > b_1 $, which supports the theoretical framework.
• The derived elasticity of substitution for labor was $ \alpha = 0.2659 $, and for capital $ \beta = 0.7341 $, summing to 1.0000 as required by the Cobb-Douglas form.
• Total factor productivity was estimated as $ A = 1.00997 \approx 1.01 $, closely matching the original estimate of approximately 1.00 in Cobb and Douglas’s 1928 study.
• The variance of $ H_3 $ across the data points was 0.5923, with a mean of 0.1365, confirming the constancy of the first integral and the validity of the bi-Hamiltonian derivation.
• The results confirm that the bi-Hamiltonian approach yields consistent parameter estimates without requiring Sato’s restrictive assumption of simultaneous holothetic technical change.