A dynamic extension of the classical model of production prices determination

Juan Gabriel Brida; Gaston Cayssials; Oscar Córdoba Rodríguez; Martín Puchet Anyul

doi:10.3934/jdg.2020013

Article Contents

2020, Volume 7, Issue 3: 185-196. Doi: 10.3934/jdg.2020013

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A dynamic extension of the classical model of production prices determination

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
^* Corresponding author: gcayssials@ccee.edu.uy

Received: February 2020

Revised: April 2020

Published: July 2020

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 91B55, 93C55.

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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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Table 2.

type of equilibriums	frequency	percent
repulsor	$ 18 $	$ 0.011 $
saddle point	$ 16711 $	$ 0.972 $
attractor	$ 29 $	$ 0.017 $

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References

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