Explaining Cobb-Douglas with the New Mathematics of Inteduct
- 1 Department of Business and Economics, Neu-Ulm University, Neu-Ulm, Germany
- 2 Department of Business and Economics, Neu-Ulm University, Neu-Ulm, Germany
Abstract
The Cobb-Douglas production function is used for around a hundred years and describes the macro economic reality very well. Though nobody doubts its validity, there is no rigorous derivation of it. Giving this derivation is done in this publication for the first time. We start with the most general version of a production function just assuming that there exists such function of an arbitrary number of variables (all possible input factors). We then make a Taylor expansion of this function still containing many variables. We then average over groups of variables via an inteduct. The groups can be e.g. labor and capital leading exactly the well-known version of the Cobb-Douglas production function. As we know how to derive this function, we also know the origin of the normally small deviations between Cobb-Douglas and the reality.
- Bronshtein, I. N., Semenddyayev, K. A., Musiol, G., & Muehlig, H. (2007). Handbook of Mathematics (5th English ed.). Springer.
- Cobb, C. W., & Douglas, P. H. (1928). A Theory of Production. The American Economic Review, 18, 139-165.
- Grabinski, M., & Klinkova, G. (2023). Like a Sum Is Generalized into an Integral, a Product May Be Generalized into an Inteduct. Applied Mathematics, 14, 279-289. https://doi.org/10.4236/am.2023.145017
- Grabinski, M. (1990). Explanation of the Discontinuity in the Spin-Relaxation of 3He-A1. Physical Review Letters, 63, 814. https://doi.org/10.1103/PhysRevLett.63.814
- Houthakker, H. S. (1955). The Pareto Distribution and the Cobb-Douglas Production Function in Activity Analysis. The Review of Economic Studies, 23, 27-31. https://doi.org/10.2307/2296148
- Jones, C. I. (2004). The Shape of Production Functions and the Direction of Technical Change. National Bureau of Economic Research, Working Paper 10457. https://doi.org/10.3386/w10457 https://www.nber.org/papers/w10457
- Li, C. (2023). Climate Change Impacts on Rice Production in Japan: A Cobb-Douglas and Panel Data Analysis. Ecological Indicators, 147, Article ID: 110008. https://doi.org/10.1016/j.ecolind.2023.110008
- Lunkenheimer, M., Kracklauer, A., Klinkova, G., & Grabinski, M. (2022). Homo Economicus to Model Human Behavior Is Ethically Doubtful and Mathematically Inconsistent.
- Nordhaus, W. D. (1979). The Efficient Use of Energy Resources. Yale University Press. https://cowles.yale.edu/sites/default/files/2022-09/m26-all.pdf
- Schädler, T., & Grabinski, M. (2015). Income from Speculative Financial Transactions Will Always Lead to Macro-Economic Instability. International Journal of Finance, Insurance and Risk Management, 5, 922-932. https://www.journalfirm.com/journal/128#
- Schulz, H. (2015). A Theoretical Physics Primer: Analytical Tools. Harri Deutsch.
- Simon, H. A., & Levy, F. K. (1963). A Note on the Cobb-Douglas Function. The American Economic Studies, 30, 93-93. https://doi.org/10.2307/2295806
- Tobin, J. (1980). Stabilization Policy Ten Years after. Brookings Papers on Economic Activity, 1, 19-71. https://doi.org/10.2307/2534285 https://www.brookings.edu/wp-content/uploads/1980/01/1980a_bpea_tobin.pdf
- Walsh, C. E. (2017). Monetary Theory and Policy (4th ed.). The MIT Press.