American Institute of Mathematical Sciences

July  2020, 7(3): 185-196. doi: 10.3934/jdg.2020013

A dynamic extension of the classical model of production prices determination

* Corresponding author: gcayssials@ccee.edu.uy

Received  February 2020 Revised  April 2020 Published  July 2020

Fund Project: Our research was supported by CSIC-UDELAR (project "Grupo de investigación en Dinámica Económica"; ID 881928)

This paper generalizes the classical model of determination of production prices for two commodities by introducing a dynamics generated by the possibility that the profit rate can be computed using prices of different stages. In this theoretical framework, the prices show a codependency between the two sectors, given by the rate of profit, and inter-industry transactions. In this setup and using discrete time, the general model can be represented by a nonlinear two dimensional dynamical system of difference equations of second order. The study shows that the dynamical system admits a unique solution for any initial condition and that there is a unique nontrivial equilibrium. In addition, it can be shown that locally the dynamical system can be represented in the canonical form $x_{t+1} = f(x_{t})$ and that the stability of the equilibrium depends on the parameters of the production process. Future research includes the extension of the model to the case of several commodities and the closed solution of the model.

Citation: Juan Gabriel Brida, Gaston Cayssials, Oscar Córdoba Rodríguez, Martín Puchet Anyul. A dynamic extension of the classical model of production prices determination. Journal of Dynamics and Games, 2020, 7 (3) : 185-196. doi: 10.3934/jdg.2020013
References:
 [1] A. Brauer, Limits for the characteristic roots of a matrix, Duke Mathematical Journal, 13 (1946), 387-395.  doi: 10.1215/S0012-7094-46-01333-6. [2] O. Córdoba Rodriguez, Dinámica de Precios en una Economía de dos Bienes, Master's thesis, Universidad Nacional Autónoma de México, México, 2015. Available from https://ru.dgb.unam.mx/handle/DGB_UNAM/TES01000724337. [3] J. Eatwell, The irrelevance of returns to scale in Sraffa's analysis, Journal of Economic Literature, 15 (1977), 61-68. [4] C. H. Edwards, Advanced Calculus of Several Variables, Courier Corporation, Massachusetts, 2012. doi: 10.1007/978-0-8176-8412-9. [5] D. Hawkins and H. A. Simon, Note: Some conditions of macroeconomic stability, Econometrica, Journal of the Econometric Society, (1949), 245–248. doi: 10.2307/1905526. [6] K. Jittorntrum, An implicit function theorem, Journal of Optimization Theory and Applications, 25 (1978), 575-577.  doi: 10.1007/BF00933522. [7] S. G. Krantz and H. R. Parks, Implicit Function Theorem: History, Theory, and Applications, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4614-5981-1. [8] H. D. Kurz and N. Salvadori, Theory of Production: A Long-Period Analysis, Cambridge University Press, 1997.  doi: 10.1017/CBO9780511625770. [9] R. Solow, On the structure of linear models, Econometrica: Journal of the Econometric Society, 20 (1952), 29-46.  doi: 10.2307/1907805. [10] P. Sraffa, Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory, Cambridge University Press, 1960.

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References:
 [1] A. Brauer, Limits for the characteristic roots of a matrix, Duke Mathematical Journal, 13 (1946), 387-395.  doi: 10.1215/S0012-7094-46-01333-6. [2] O. Córdoba Rodriguez, Dinámica de Precios en una Economía de dos Bienes, Master's thesis, Universidad Nacional Autónoma de México, México, 2015. Available from https://ru.dgb.unam.mx/handle/DGB_UNAM/TES01000724337. [3] J. Eatwell, The irrelevance of returns to scale in Sraffa's analysis, Journal of Economic Literature, 15 (1977), 61-68. [4] C. H. Edwards, Advanced Calculus of Several Variables, Courier Corporation, Massachusetts, 2012. doi: 10.1007/978-0-8176-8412-9. [5] D. Hawkins and H. A. Simon, Note: Some conditions of macroeconomic stability, Econometrica, Journal of the Econometric Society, (1949), 245–248. doi: 10.2307/1905526. [6] K. Jittorntrum, An implicit function theorem, Journal of Optimization Theory and Applications, 25 (1978), 575-577.  doi: 10.1007/BF00933522. [7] S. G. Krantz and H. R. Parks, Implicit Function Theorem: History, Theory, and Applications, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4614-5981-1. [8] H. D. Kurz and N. Salvadori, Theory of Production: A Long-Period Analysis, Cambridge University Press, 1997.  doi: 10.1017/CBO9780511625770. [9] R. Solow, On the structure of linear models, Econometrica: Journal of the Econometric Society, 20 (1952), 29-46.  doi: 10.2307/1907805. [10] P. Sraffa, Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory, Cambridge University Press, 1960.
 $a_{11}$ $a_{12}$ $a_{21}$ $a_{22}$ $l_{1}$ $l_{2}$ Qty. of eigenvalues with modulus smaller than one $0.96$ $0.02$ $0.01$ $0.94$ $0.09$ $0,91$ zero (repulsor) $0.89$ $0.06$ $0.06$ $0.75$ $0.53$ $0.47$ one (saddle point) $0.17$ $0.23$ $0.8$ $0.17$ $0.19$ $0.81$ two (saddle point) $0.04$ $0.18$ $0.02$ $0.27$ $0.99$ $0.01$ three (saddle point) $0.69$ $0.28$ $0.14$ $0.49$ $0.29$ $0.71$ four (attractor)
 $a_{11}$ $a_{12}$ $a_{21}$ $a_{22}$ $l_{1}$ $l_{2}$ Qty. of eigenvalues with modulus smaller than one $0.96$ $0.02$ $0.01$ $0.94$ $0.09$ $0,91$ zero (repulsor) $0.89$ $0.06$ $0.06$ $0.75$ $0.53$ $0.47$ one (saddle point) $0.17$ $0.23$ $0.8$ $0.17$ $0.19$ $0.81$ two (saddle point) $0.04$ $0.18$ $0.02$ $0.27$ $0.99$ $0.01$ three (saddle point) $0.69$ $0.28$ $0.14$ $0.49$ $0.29$ $0.71$ four (attractor)
 type of equilibriums frequency percent repulsor $18$ $0.011$ saddle point $16711$ $0.972$ attractor $29$ $0.017$
 type of equilibriums frequency percent repulsor $18$ $0.011$ saddle point $16711$ $0.972$ attractor $29$ $0.017$
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