# 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 & 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
 $a_{11}$ $a_{12}$ $a_{21}$ $a_{22}$ $l_{1}$ $l_{2}$ 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 $29$ $0.017$
 [1] Felix X.-F. Ye, Hong Qian. Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4341-4366. doi: 10.3934/dcdsb.2019122 [2] Byungik Kahng, Miguel Mendes. The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems. Conference Publications, 2013, 2013 (special) : 393-406. doi: 10.3934/proc.2013.2013.393 [3] David Iglesias-Ponte, Juan Carlos Marrero, David Martín de Diego, Edith Padrón. Discrete dynamics in implicit form. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1117-1135. doi: 10.3934/dcds.2013.33.1117 [4] Armengol Gasull, Víctor Mañosa. Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 651-670. doi: 10.3934/dcdsb.2019259 [5] Karl P. Hadeler. Quiescent phases and stability in discrete time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 129-152. doi: 10.3934/dcdsb.2015.20.129 [6] Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1039-1056. doi: 10.3934/dcds.2008.20.1039 [7] C. D. Ahlbrandt, A. C. Peterson. A general reduction of order theorem for discrete linear symplectic systems. Conference Publications, 1998, 1998 (Special) : 7-18. doi: 10.3934/proc.1998.1998.7 [8] Dariusz Idczak. A global implicit function theorem and its applications to functional equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2549-2556. doi: 10.3934/dcdsb.2014.19.2549 [9] Jingang Zhao, Chi Zhang. Finite-horizon optimal control of discrete-time linear systems with completely unknown dynamics using Q-learning. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020030 [10] Mohammadreza Molaei. Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds. Electronic Research Announcements, 2018, 25: 8-15. doi: 10.3934/era.2018.25.002 [11] Stefan Siegmund, Petr Stehlík. Time scale-induced asynchronous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020151 [12] David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499 [13] Mădălina Roxana Buneci. Morphisms of discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 91-107. doi: 10.3934/dcds.2011.29.91 [14] Peng Cui, Hongguo Zhao, Jun-e Feng. State estimation for discrete linear systems with observation time-delayed noise. Journal of Industrial & Management Optimization, 2011, 7 (1) : 79-85. doi: 10.3934/jimo.2011.7.79 [15] Le Viet Cuong, Thai Son Doan. Assignability of dichotomy spectra for discrete time-varying linear control systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3597-3607. doi: 10.3934/dcdsb.2020074 [16] Vladimir Răsvan. On the central stability zone for linear discrete-time Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 734-741. doi: 10.3934/proc.2003.2003.734 [17] Weigu Li, Kening Lu. Takens theorem for random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3191-3207. doi: 10.3934/dcdsb.2016093 [18] J. C. Robinson. A topological time-delay embedding theorem for infinite-dimensional cocycle dynamical systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 731-741. doi: 10.3934/dcdsb.2008.9.731 [19] Robert Baier, Thuy T. T. Le. Construction of the minimum time function for linear systems via higher-order set-valued methods. Mathematical Control & Related Fields, 2019, 9 (2) : 223-255. doi: 10.3934/mcrf.2019012 [20] Chuandong Li, Fali Ma, Tingwen Huang. 2-D analysis based iterative learning control for linear discrete-time systems with time delay. Journal of Industrial & Management Optimization, 2011, 7 (1) : 175-181. doi: 10.3934/jimo.2011.7.175

Impact Factor:

## Tools

Article outline

Figures and Tables