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A dynamic extension of the classical model of production prices determination

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

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

    Citation:

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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 $
     | Show Table
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  • [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. SalvadoriTheory 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. SraffaProduction of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory, Cambridge University Press, 1960. 
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