Publicado

2017-01-01

The Cobb-Douglas function for a continuum model

La función Cobb-Douglas para un modelo continuo

A função Cobb-Douglas para um modelo contínuo

Palabras clave:

CES function, Cobb-Douglas function, continuum, product integral, functional derivative. (en)
función CES, función Cobb-Douglas, continuo, integral producto, derivada funcional (es)
Função CES, função Cobb-Douglas, contínuo, integral produto, derivada funcional (pt)

Descargas

Autores/as

  • Javier Humberto Ospina Holguín Universidad del Valle
This paper introduces two formal equivalent definitions of the Cobb-Douglas function for a continuum model based on a generalization of the Constant Elasticity of Substitution (CES) function for a continuum under not necessarily constant returns to scale and based on principles of product calculus. New properties are developed, and to illustrate the potential of using the product integral and its functional derivative, it is shown how the profit maximization problem of a single competitive firm using a continuum of factors of production can be solved in a manner that is completely analogous to the one used in the discrete case.
Este artículo introduce dos definiciones formales equivalentes de la función Cobb-Douglas para un modelo continuo basadas en una generalización de la función de elasticidad de sustitución constante (CES) para un continuo bajo rendimientos a escala no necesariamente constantes y con base en principios del cálculo de producto. Se desarrollan propiedades nuevas; y para ilustrar el potencial de usar la integral producto y su derivada funcional, se muestra cómo el problema de maximización de beneficios de una única empresa competitiva que usa un continuo de factores de producción se puede solucionar de una manera completamente análoga a la utilizada en el caso discreto.
Este artigo introduz duas definições formais equivalentes da função Cobb-Douglas para um modelo contínuo baseadas em uma generalização da função de elasticidade de substituição constante (CES) para um contínuo baixo rendimento a escala não necessariamente constante e com base em princípios do cálculo de produto. São desenvolvidas propriedades novas e, para ilustrar o potencial de usar a integral produto e sua derivada funcional, mostra-se como o problema de maximização de benefícios de uma única empresa competitiva que utiliza um contínuo de fatores de produção pode ser solucionado de maneira completamente análoga à utilizada no caso discreto.

Descargas

Los datos de descargas todavía no están disponibles.

Citas

Arrow, K. J., Chenery, H. B., Minhas, B. S., & Solow, R. M. (1961). Capital-Labor substitution and economic efficiency. The Review of Economics and Statistics, 43(3), 225-250.

Bartle, R. G. (2001). A modern theory of integration (Vol. 32). Providence, R. I.: American Mathematical Society.

Cheng, M. L., & Han, Y. (2014). A modified Cobb-Douglas production function model and its application. IMA Journal of Management Mathematics, 25(3), 353-365.

Cobb, C. W., & Douglas, P. H. (1928). A theory of production. The American Economic Review, 18(1), 139-165.

Dixit, A. K., & Stiglitz, J. E. (2004). Monopolistic competition and optimum product diversity (May 1974). In S. Brakman & B. J. Heijdra (Eds.), The monopolistic competition revolution in retrospect (pp. 70-88). New York: Cambridge University Press.

Dornbusch, R., Fischer, S., & Samuelson, P. A. (1977). Comparative advantage, trade, and payments in a Ricardian model with a continuum of goods. The American Economic Review, 67(5), 823-839.

Felipe, J., & McCombie, J. S. L. (2005). How sound are the foundations of the aggregate production function? Eastern Economic Journal, 31(3), 467-488.

Fioretti, G. (2007). The production function. Physica A: Statistical Mechanics and its Applications, 374(2), 707-714.

Gelfand, I. M., & Fomin, S. V. (1963). Calculus of variations. Englewood Cliffs, N. J.: Prentice-Hall, Inc.

Greiner, W., & Reinhardt, J. (1996). Field quantization. Berlin: Springer- Verlag.

Ilca, M., & Popa, D. (2014). On approximate Cobb-Douglas production functions. Carpathian Journal of Mathematics, 30(1), 87-92.

Jablecki, J. (2007, August 23). How many traders can you fit into a model? [Online article]. Retrieved from https://mises.org/library/how-many-traders-can-you-fit-model. [URL]

Jehle, G. A., & Reny, P. J. (2011). Advanced microeconomic theory (3rd ed.). Harlow, England: Financial Times Prentice Hall.

Krein, S. G. (2011). Product integral. In Encyclopedia of mathematics. Retrieved from http://www.encyclopediaofmath.org/index.php?title=Product_integral&oldid=17399. [URL]

Manturov, O. V. (1991). The product integral. Journal of Soviet Mathematics, 55(5), 2042-2076.

Mas-Colell, A., Whinston, M. D., & Green, J. R. (1995). Microeconomic theory. New York: Oxford University Press.

McFadden, D. (1963). Constant elasticity of substitution production functions. The Review of Economic Studies, 30(2), 73-83.

Mimkes, J., Fruend, T., & Willis, G. (2002). Lagrange statistics in systems (markets) with price constraints: Analysis of property, car sales, marriage and job markets by the Boltzmann function and the Pareto distribution. Preprint. (arXiv:cond-mat/0204234v1 [cond-mat.stat-mech]).

Mishra, S. K. (2007). A brief history of production functions (MPRA Paper 5254). Munich: Munich University Library.

Nironi, F. (2011). Taylor series and polynomials [Lecture notes]. Retrieved from http://www.math.columbia.edu/~nironi/taylor2.pdf. [URL]

Qi, F. (1998). Generalized weighted mean values with two parameters. Proceedings: Mathematical, Physical and Engineering Sciences, 454(1978), 2723-2732.

Saito, T. (2012). How do we get Cobb-Douglas and Leontief functions from CES function: A lecture note on discrete and continuum differentiated object models. Journal of Industrial Organization Education, 6(1), 1-12.

Shigemoto, K. (2003). Weber-Fechner's law and demand function. Preprint. (arXiv:physics/0303118v2 [physics.gen-ph]).

Thomson, B. S., Bruckner, J. B., & Bruckner, A. M. (2008). Elementary real analysis (2nd ed.). Retrieved from http://www.classicalrealanalysis.com. [URL]

Uzawa, H. (1962). Production functions with constant elasticities of substitution. The Review of Economic Studies, 29(4), 291-299.

Vîlcu, A. D., & Vîlcu, G. E. (2011). On some geometric properties of the generalized CES production functions. Applied Mathematics and Computation, 218(1), 124-129.

Vîlcu, G. E. (2011). A geometric perspective on the generalized Cobb- Douglas production functions. Applied Mathematics Letters, 24(5), 777-783.

Wang, X., & Fu, Y. (2013). Some characterizations of the Cobb-Douglas and CES production functions in microeconomics. Abstract and Applied Analysis, 2013, Article ID 761832, 1-6.

Licencia

Derechos de autor 2017 Cuadernos de Economía

Cuadernos de Economía a través de la División de Bibliotecas de la Universidad Nacional de Colombia promueve y garantiza el acceso abierto de todos sus contenidos. Los artículos publicados por la revista se encuentran disponibles globalmente con acceso abierto y licenciados bajo los términos de Creative Commons Atribución-No_Comercial-Sin_Derivadas 4.0 Internacional (CC BY-NC-ND 4.0), lo que implica lo siguiente: