N.K. Obrosova, A.A. Shananin. Young duality of variational inequalities. An application for the analysis of interactions in production networks ... P. 88-105

We develop a mathematical technique of Young dual variational inequalities, which are used to model market equilibrium in a network of production clusters that are heterogeneous from a technological point of view. Two formulations of the problem are considered: for a closed system with a given constraint on resources and for an open system in which resources can be supplied from outside at given prices. A theorem is proved on the existence of a solution to the variational inequality corresponding to market equilibrium in an open system.

Keywords: production network, heterogeneity, variational inequality, Young duality, market equilibrium, resource allocation problem

Received May 11, 2023

Revised June 19, 2023

Accepted June 26, 2023

Funding Agency: This work was supported by the Russian Science Foundation (project no. 23-21-00429, https://rscf.ru/project/23-21-00429/).

Nataliia Obrosova, Cand. Sci. (Phys.-Math.), Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Moscow, 119333 Russia, e-mail: nobrosova@ya.ru

Alexander Shananin, Dr. Phys.-Math. Sci., Prof., RAS Academician, Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Moscow, 119333 Russia, e-mail: alexshan@ya.ru

REFERENCES

1.   Barro R.J., Sala-i-Martin X. Economic growth. Cambridge, London, MIT Press., 2004.

2.   Chamberlin E.H. The theory of monopolistic competition. Harvard economic studies, 1969, Cambridge, Harvard Univ. Press. Translated to Russian under the title: Teoriya monopolisticheskoi konkurentsii (Reorientatsiya teorii stoimosti), Moscow: Ekonomika, 1996, 351 p.

3.   Acemoglu D., Ozdaglar A., Tahbaz-Salehi A. The network origins of aggregate fluctuations. Econometrica, 2012, vol. 80, no. 5, pp. 1977–2016. doi: 10.3982/ECTA9623

4.   Acemoglu D., Ozdaglar A., Tahbaz-Salehi A. Networks, shocks, and systemic risk. In: The Oxford Handbook of the Economics of Networks, eds. Yann Bramoulé et al, NY: Oxford Univ. Press, 2016, pp. 569–607. doi: 10.3386/w20931

5.   Sarrazin T. Europa braucht den Euro nicht. München: Deutsche Verlags-Anstalt (DVA), 2012. Translated to Russian under the title: Evrope ne nuzhen evro, Moscow: AST Publ., 2015, 512 p.

6.   Chevènement J.-P. 1914–2014 : l’Europe sortie de l’histoire?, 2013, Editions Fayard, 350 p. Translated to Russian under the title: 1914–2014: Evropa vykhodit iz istorii, Moscow: AST Publ., 2015, 352 p.

7.   Shananin A. Young duality and aggregation of balances. Dokl. Math., 2020, vol. 102, no. 1, pp. 330–333. doi: 10.1134/S1064562420040171

8.   Shananin A. Problem of aggregating of an input-output model and duality. Comput. Math. and Math. Phys., 2021, vol. 61, no. 1, pp. 153–166. doi: 10.1134/S0965542521010085

9.   Boranbayev S., Obrosova N., Shananin A. Production network centrality in connection to economic development by the case of Kazakhstan statistics. In: Optimization and Applications: 12th Internat. Conf. (OPTIMA 2021): Proc., eds. Nicholas N. Olenev et al., 2021, Ser. Lecture Notes in Computer Science, vol. 13078, pp. 321–335. doi: 10.1007/978-3-030-91059-4_23

10.   Obrosova N., Shananin A., Spiridonov A. On the comparison of two approaches to intersectoral balance analysis. J. Physics: Conf. Ser., 2021, vol. 2131, no. 2. doi: 10.1088/1742-6596/2131/2/022112

11.   Rassokha A., Shananin A. Inverse problems of the analysis of input-output balances. Math. Models and Computer Simulations, 2021, vol. 13, no. 6, pp. 943–954. doi: 10.1134/S2070048221060193

12.   Kerimkhulle S., Obrosova N., Shananin A., Azieva G. The nonlinear model of intersectoral linkages of Kazakhstan for macroeconomic decision-making processes in sustainable supply chain management. Sustainability, 2022, vol. 14, no. 21. doi: 10.3390/su142114375

13.   Boranbayev A., Obrosova N., Shananin A. Nonlinear input-output balance and Young duality: Analysis of Covid-19 macroeconomic impact on Kazakhstan. Sib. Electron. Math. Reports, 2022, vol. 19, no. 2, pp. 835–851. doi: 10.33048/semi.2022.19.070

14.   Obrosova N., Shananin A., Spiridonov A. Nonlinear input-output model with nested CES technologies for the analysis of macroeconomic effects of a foreign trade shock. Lobachevskii J. Math., 2023, vol. 4, no. 1, pp. 401–417. doi: 10.1134/S1995080223010304

15.   Leontief W.W. The Structure of American economy, 1919–1939: An empirical application of equilibrium analysis. Oxford: Oxford Univ. Press, 1951, 264 p.

16.   Ashmanov S.A. Vvedenie v matematicheskuyu ekonomiku [Introduction to mathematical economics]. Moscow: Nauka Publ., 1984, 293 p.

17.   Acemoglu D., Akcigit U., Kerr W. Networks and the macroeconomy: An empirical exploration. NBER Macroeconomics Annual, 2016, vol. 30, no. 1, pp. 273–335. Available on: http://nrs.harvard.edu/urn-3:HUL.InstRepos:17527693 

18.   Barauskaite Kristina, Nguyen Anh D.M. Global intersectoral production network and aggregate fluctuations. Economic Modelling, 2021, vol. 102(C). doi: 10.1016/j.econmod.2021.105577

19.   Acemoglu D., Azar P.D. Endogenous production networks. Econometrica, 2020, vol. 88, no. 1, pp. 33–82. doi: 10.3982/ECTA15899

20.   Acemoglu D., Ozdaglar A., Tahbaz-Salehi A. Microeconomic origins of macroeconomic tail risks. Am. Econ. Rev., 2017, vol. 107, no. 1, pp. 54–108. doi: 10.1257/aer.20151086

21.   Acemoglu D., Ozdaglar A., Tahbaz-Salehi A. Systemic risk and stability in financial networks. Am. Econ. Rev., 2015, vol. 105, no. 2, pp. 564–608. doi: 10.1257/aer.20130456

22.   Baqaee D.R. Cascading failures in production networks. Econometrica, 2018, vol. 86, no. 5, pp. 1819–1838. doi: 10.3982/ECTA15280

23.   Aubin J.P. L’analyse Non Linéaire et ses Motivations Economiques. Paris: Masson, 1984, 214 p. Translated to Russian under the title: Nelineinyi analiz i ego ekonomicheskie prilozheniya. Moscow: Mir Publ., 1988, 264 p.

24.   Nikaido H. Convex structures and economic theory. NY: Acad. Press, 1968, 405 p. ISBN: 9781483230030. Translated to Russian under the title: Vypuklye struktury i matematicheskaya ekonomika, Moscow: Mir Publ., 1972, 520 p.

25.   Shananin A.A. Duality for generalized programming problems, and variational principles in models of economic equilibrium. Doklady Akademii Nauk, 1999, vol. 366, no. 4, pp. 462–464 (in Russian).

26.   Shananin A.A. Integrability problem and the generalized nonparametric method for the consumer demand analysis. In: Proceedings of Moscow Inst. Phys. Technol., 2009, vol. 4, no. 1, pp. 84–98 (in Russian).

27.   Miller R.E., Blair P.D. Input-output analysis: Foundations and extensions. 2nd ed, Cambridge: Cambridge Univ. Press, 2009, pp. 1–750.

28.   Wixted B., Yamano N., Webb C. Input-output analysis in an increasingly globalised world: Applications of OECDs harmonised international tables. In: OECD Science, Technology and Industry: working papers, 2006, no. 2006/07, Paris: OECD Publ., 2006. doi: 10.1787/303252313764

29.   O’Mahony Mary, Timmer Marcel P. Output, input and productivity measures at the industry level: The EU KLEMS database. Economic J., 2009, vol. 119, no. 538, pp.  F374–F403. doi 10.1111/j.1468-0297.2009.02280.x

30.   Timmer M.P. [et al.] An Illustrated user guide to the world input-output database: The Case of global automotive production. Review of International Economics, 2015, vol. 23, pp. 575–605. doi 10.1111/roie.12178

Cite this article as: N.K. Obrosova, A.A. Shananin. Young duality of variational inequalities. An application for the analysis of interactions in production networks. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 3, pp. 88–105; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 323, Suppl. 1, pp. S194–S210.