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

A dynamic extension of the classical model of production prices determination

Juan Gabriel Brida 1, , Gaston Cayssials 1,, , Oscar Córdoba Rodríguez 2, and  Martín Puchet Anyul 2,

1. 

Departamento de Métodos Cuantitativos - Facultad de Ciencias Económicas y de Administración, Universidad de la República (Uruguay)
2. 

Facultad de Economia, Universidad Nacional Autónoma de México

* 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.

Keywords: Dynamics in discrete time, non linear dynamical systems, theory of price, production prices, implicit function theorem.
Mathematics Subject Classification: Primary: 91B55, 93C55.
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 & 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.  Google Scholar

[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. Google Scholar

[3]

J. Eatwell, The irrelevance of returns to scale in Sraffa's analysis, Journal of Economic Literature, 15 (1977), 61-68.   Google Scholar

[4]

C. H. Edwards, Advanced Calculus of Several Variables, Courier Corporation, Massachusetts, 2012. doi: 10.1007/978-0-8176-8412-9.  Google Scholar

[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.  Google Scholar

[6]

K. Jittorntrum, An implicit function theorem, Journal of Optimization Theory and Applications, 25 (1978), 575-577.  doi: 10.1007/BF00933522.  Google Scholar

[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.  Google Scholar

[8] H. D. Kurz and N. Salvadori, Theory of Production: A Long-Period Analysis, Cambridge University Press, 1997.  doi: 10.1017/CBO9780511625770.  Google Scholar
[9]

R. Solow, On the structure of linear models, Econometrica: Journal of the Econometric Society, 20 (1952), 29-46.  doi: 10.2307/1907805.  Google Scholar

[10] P. Sraffa, Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory, Cambridge University Press, 1960.   Google Scholar

show all references

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.  Google Scholar

[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. Google Scholar

[3]

J. Eatwell, The irrelevance of returns to scale in Sraffa's analysis, Journal of Economic Literature, 15 (1977), 61-68.   Google Scholar

[4]

C. H. Edwards, Advanced Calculus of Several Variables, Courier Corporation, Massachusetts, 2012. doi: 10.1007/978-0-8176-8412-9.  Google Scholar

[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.  Google Scholar

[6]

K. Jittorntrum, An implicit function theorem, Journal of Optimization Theory and Applications, 25 (1978), 575-577.  doi: 10.1007/BF00933522.  Google Scholar

[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.  Google Scholar

[8] H. D. Kurz and N. Salvadori, Theory of Production: A Long-Period Analysis, Cambridge University Press, 1997.  doi: 10.1017/CBO9780511625770.  Google Scholar
[9]

R. Solow, On the structure of linear models, Econometrica: Journal of the Econometric Society, 20 (1952), 29-46.  doi: 10.2307/1907805.  Google Scholar

[10] P. Sraffa, Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory, Cambridge University Press, 1960.   Google Scholar
Table 1.   
$ 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)
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$ 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)
Table 2.   
type of equilibriums frequency percent
repulsor $ 18 $ $ 0.011 $
saddle point $ 16711 $ $ 0.972 $
attractor $ 29 $ $ 0.017 $
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type of equilibriums frequency percent
repulsor $ 18 $ $ 0.011 $
saddle point $ 16711 $ $ 0.972 $
attractor $ 29 $ $ 0.017 $
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