Solvability of the Economic Input-Output Equation by Time Irreversibility
Author(s)
Shinji Miura
ABSTRACT
This paper reinterprets the economic input-output equation as a description of a realized situation without considering decision making. This paper uses the equation that the self-sufficiency rate is added to the Leontief type, and discusses its solvability. The equation has a unique solution if and only if each part of the relevant society satisfies the space-time openness condition. This condition means that commodities which a part of the relevant society possesses are not all inputted to its inside. Moreover, if the process of input and output is time irreversible, each part of the relevant society satisfies the space-time openness condition. Therefore, the solvability of the equation is guaranteed by time irreversibility. This proposition seems to be relevant to the grandfather paradox which is a type of time paradox.
Cite this paper
Miura, S. (2014) Solvability of the Economic Input-Output Equation by Time Irreversibility. Advances in Linear Algebra & Matrix Theory, 4, 143-155. doi: 10.4236/alamt.2014.43013.
Miura, S. (2014) Solvability of the Economic Input-Output Equation by Time Irreversibility. Advances in Linear Algebra & Matrix Theory, 4, 143-155. doi: 10.4236/alamt.2014.43013.
References
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http://dx.doi.org/10.4236/alamt.2014.42009 [5] Hawkins, D. and Simon, H.A. (1949) Note: Some Conditions of Macroeconomic Stability. Econometrica, 17, 245-248.
http://dx.doi.org/10.2307/1905526 [6] Ostrowski, A. (1937-38) über die Determinanten mit überwiegender Hauptdiagonale. Commentarii Mathematici Helvetici, 10, 69-96.
http://dx.doi.org/10.1007/BF01214284 [7] Berman, A. and Plemmons, R.J. (1979) Nonnegative Matrices in the Mathematical Sciences. Academic Press, Cambridge. [8] Plemmons, R.J. (1977) M-Matrix Characterizations.I—Nonsingular M-Matrices. Linear Algebra and Its Applications, 18, 175-188. [9] Bidard, C. (2007) The Weak Hawkins-Simon Condition. Electronic Journal of Linear Algebra, 16, 44-59.
http://hermite.cii.fc.ul.pt/iic/ela/ela-articles/articles/vol16_pp44-59.pdf [10] Mori, K. (2008) Maurice Potron’s Linear Economic Model: A De Facto Proof of “Fundamental Marxian Theorem”. Metroeconomica, 59, 511-529.
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http://dx.doi.org/10.1016/0164-0704(82)90095-7 [14] Jeong, K. (1984) The Relation between Two Different Notions of Direct and Indirect Requirements. Journal of Macroeconomics, 6, 473-476.
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http://dx.doi.org/10.1016/0164-0704(91)90062-Y [16] Dasgupta, D. (1992) Using the Correct Economic Interpretation to Prove the Hawkins-Simon-Nikaido Theorem: One More Note. Journal of Macroeconomics, 14, 755-761.
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http://dx.doi.org/10.1016/S0164-0704(98)00054-8 [18] Fujita, Y. (2006) A Reconsideration of a Correct Economic Interpretation of the Hawkins-Simon Condition. CAES Working Paper Series.
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http://dx.doi.org/10.1137/0713013 [22] Neumann, M. (1979) A Note on Generalizations of Strict Diagonal Dominance for Real Matrices. Linear Algebra and Its Applications, 26, 3-14.
http://dx.doi.org/10.1016/0024-3795(79)90168-X [23] Varshney, K.R. (2013) Opinion Dynamics with Bounded Confidence in the Bayes Risk Error Divergence Sense. IEEE Conference on Acoustics, Speech and Signal Processing, Vancouver, 26-31 May 2013, 6600-6604.
http://dx.doi.org/10.1109/ICASSP.2013.6638938 [24] Shang, Y. (2013) Deffuant Model with General Opinion Distributions: First Impression and Critical Confidence Bound. Complexity, 19, 38-49.
http://dx.doi.org/10.1002/cplx.21465 [25] Shang, Y. (2014) An Agent Based Model for Opinion Dynamics with Random Confidence Threshold. Communications in Nonlinear Science and Numerical Simulation, 19, 3766-3777.
http://dx.doi.org/10.1016/j.cnsns.2014.03.033 [26] Oaklander, L.N. (2008) Introduction. In: Oaklander, L.N., Ed., The Philosophy of Time: Critical Concepts in Phylosophy, Volume 4: Time and Physics, Routledge, London, 1-5.
[1] Leontief, W. (1953) The Structure of the American Economy, 1919-1939: An Empirical Application of Equilibrium Analysis. 2nd Edition, Oxford University Press, Oxford. [2] Administrative Management Agency and Other Six Authorities of Japan (1970) The Work Report of Making the Modified Input-Output Table in 1960 and the Price Evaluation Table in 1965. Hitachi Kosoku Insatsu K.K., Hitachi. (In Japanese) [3] Leontief, W. (1986) Input-Output Economics. 2nd Edition, Oxford University Press, Oxford. [4] Miura, S. (2014) Non-Singularity Conditions for Two Z-Matrix Types. Advances in Linear Algebra & Matrix Theory, 4, 109-119.
http://dx.doi.org/10.4236/alamt.2014.42009 [5] Hawkins, D. and Simon, H.A. (1949) Note: Some Conditions of Macroeconomic Stability. Econometrica, 17, 245-248.
http://dx.doi.org/10.2307/1905526 [6] Ostrowski, A. (1937-38) über die Determinanten mit überwiegender Hauptdiagonale. Commentarii Mathematici Helvetici, 10, 69-96.
http://dx.doi.org/10.1007/BF01214284 [7] Berman, A. and Plemmons, R.J. (1979) Nonnegative Matrices in the Mathematical Sciences. Academic Press, Cambridge. [8] Plemmons, R.J. (1977) M-Matrix Characterizations.I—Nonsingular M-Matrices. Linear Algebra and Its Applications, 18, 175-188. [9] Bidard, C. (2007) The Weak Hawkins-Simon Condition. Electronic Journal of Linear Algebra, 16, 44-59.
http://hermite.cii.fc.ul.pt/iic/ela/ela-articles/articles/vol16_pp44-59.pdf [10] Mori, K. (2008) Maurice Potron’s Linear Economic Model: A De Facto Proof of “Fundamental Marxian Theorem”. Metroeconomica, 59, 511-529.
http://dx.doi.org/10.1111/j.1467-999X.2008.00315.x [11] Gale, D. (1960) The Theory of Linear Economic Models. McGraw-Hill Book Company, Hoboken. [12] Dorfman, R., Samuelson, P.A. and Solow, R.M. (1958) Linear Programming and Economic Analysis. McGraw-Hill, Hoboken. [13] Jeong, K. (1982) Direct and Indirect Requirements: A Correct Economic Interpretation of the Hawkins-Simon Conditions. Journal of Macroeconomics, 4, 349-356.
http://dx.doi.org/10.1016/0164-0704(82)90095-7 [14] Jeong, K. (1984) The Relation between Two Different Notions of Direct and Indirect Requirements. Journal of Macroeconomics, 6, 473-476.
http://dx.doi.org/10.1016/0164-0704(84)90043-0 [15] Fujita, Y. (1991) A Further Note on a Correct Economic Interpretation of the Hawkins-Simon Conditions. Journal of Macroeconomics, 13, 199-208.
http://dx.doi.org/10.1016/0164-0704(91)90062-Y [16] Dasgupta, D. (1992) Using the Correct Economic Interpretation to Prove the Hawkins-Simon-Nikaido Theorem: One More Note. Journal of Macroeconomics, 14, 755-761.
http://dx.doi.org/10.1016/0164-0704(92)90010-6 [17] Gim, H. and Kim, K. (1998) The General Relation between Two Different Notions of Direct and Indirect Input Requirements. Journal of Macroeconomics, 20, 199-208.
http://dx.doi.org/10.1016/S0164-0704(98)00054-8 [18] Fujita, Y. (2006) A Reconsideration of a Correct Economic Interpretation of the Hawkins-Simon Condition. CAES Working Paper Series.
http://www.econ.fukuoka-u.ac.jp/english/researchcenter/workingpapers/WP-2006-001.pdf [19] Morishima, M. (1973) Marx’s Economics: A Dual Theory of Value and Growth. Cambridge University Press, Cambridge. [20] Nikaido, H. (1970) Introduction to Sets and Mappings in Modern Economics. Translated by Sato, K., North-Holland Publishing Company, Amsterdam. [21] Beauwens, R. (1976) Semistrict Diagonal Dominance. SIAM Journal on Numerical Analysis, 13, 109-112.
http://dx.doi.org/10.1137/0713013 [22] Neumann, M. (1979) A Note on Generalizations of Strict Diagonal Dominance for Real Matrices. Linear Algebra and Its Applications, 26, 3-14.
http://dx.doi.org/10.1016/0024-3795(79)90168-X [23] Varshney, K.R. (2013) Opinion Dynamics with Bounded Confidence in the Bayes Risk Error Divergence Sense. IEEE Conference on Acoustics, Speech and Signal Processing, Vancouver, 26-31 May 2013, 6600-6604.
http://dx.doi.org/10.1109/ICASSP.2013.6638938 [24] Shang, Y. (2013) Deffuant Model with General Opinion Distributions: First Impression and Critical Confidence Bound. Complexity, 19, 38-49.
http://dx.doi.org/10.1002/cplx.21465 [25] Shang, Y. (2014) An Agent Based Model for Opinion Dynamics with Random Confidence Threshold. Communications in Nonlinear Science and Numerical Simulation, 19, 3766-3777.
http://dx.doi.org/10.1016/j.cnsns.2014.03.033 [26] Oaklander, L.N. (2008) Introduction. In: Oaklander, L.N., Ed., The Philosophy of Time: Critical Concepts in Phylosophy, Volume 4: Time and Physics, Routledge, London, 1-5.