The minimal-time growth problem and 'very strong' turnpike theorem

• Authors: Emil Panek

Abstracts

EN

This paper refers to the author's previous work, in which the ‘weak’ turnpike theorem in the stationary Gale economy was proved. This theorem states that each optimal growth process {y*(t)}t*1t=0 that leads the economy in the shortest possible time t*1 from the (initial) state of y0 to the set of target/postulated states Y1 almost always runs in the neighbourhood of the production turnpike, where the economy remains in a specific dynamic equilibrium (peak growth equilibrium). This paper presents a proof of the ‘very strong’ turnpike theorem in the stationary Gale economy, which states that if the optimal process (the solution to the minimaltime growth problem) reaches a turnpike in a certain period of time tˇ < t*1 - 1, then it stays on it everywhere else, except for, at most, final period t*1. The obtained result confirms the wellknown Samuelson hypothesis about the specific turnpike stability of optimal growth paths in multiproduct/multisectoral von Neumann-Leontief-Gale-type models, also in the case where the growth criterion is not the (normally assumed) utility of production but the time needed by the economy to achieve the postulated target level or volume of production.

Keywords

EN

stationary Gale economy; von Neumann equilibrium; minimum-time growth problem; turnpike effect

• Publisher: Główny Urząd Statystyczny
• Journal: Przegląd Statystyczny
• Year: 2021
• Volume: 68
• Issue: 4
• Pages: 45-58

Dates

published

2021

Contributors

author

Emil Panek

• University of Zielona Góra, Faculty of Economics and Management, Institute of Economics and Finance

• https://orcid.org/0000000279501689

References

• Babaei, E. (2019). Von Neumann-Gale Dynamical Systems with Applications in Economics and Finance [Doctoral dissertation]. University of Manchester. https://www.research.manchester.ac.uk /portal/files/159166728/FULL_TEXT.PDF.
• Babaei, E., Evstigneev, I. V., & Schenk-Hoppé, K. R. (2020). Log-optimal and rapid paths in von Neumann-Gale dynamical systems. Journal of Mathematical Analysis and Application, 481(2), 1-30. https://doi.org/10.1016/j.jmaa.2019.123489.
• Giorgi, G., & Zuccotti, C. (2016). Equilibrium and Optimality in Gale-von Neumann Models (DEM Working Papers No. 118). http://economiaweb.unipv.it/wp-content/uploads/2017/06/ DEMWP0119.pdf.
• Jensen, M. K. (2012). Global Stability and the “Turnpike” in Optimal Unbounded Growth Models. Journal of Economic Theory, 147(2), 802-832. https://doi.org/10.1016/j.jet.2010.07.010.
• Khan, M. A., & Piazza, A. (2011). An overview of turnpike theory: towards the disconunted deterministic case. In S. Kusuoka & T. Maruyama (Eds.), Advanced in Mathematical Economics (vol. 14, pp. 39-67). Springer. https://doi.org/10.1007/978-4-431-53883-7_3.
• Majumdar, M. (2009). Equilibrium and optimality: Some imprints of David Gale. Games and Economic Behavior, 66(2), 607-626. https://doi.org/10.1016/j.geb.2009.04.018.
• Makarov, V. L., & Rubinov, A. M. (1977). Mathematical Theory of Economic Dynamics and Equilibria. Springer-Verlag. https://doi.org/10.1007/978-1-4612-9886-1.
• McKenzie, L. W. (2005). Optimal Economic Growth, Turnpike Theorems and Comparative Dynamics. In K. J. Arrow & M. D. Intriligator (Eds.), Handbook of Mathematical Economics (2nd edition, vol. 3), 1281-1355.
• Mitra, T., & Nishimura, K. (Eds.). (2009). Equilibrium, Trade, and Growth. Selected Papers of Lionel W. McKenzie. The MIT Press. https://doi.org/10.7551/mitpress/7848.001.0001.
• Nikaido, H. (1968). Convex Structures and Economic Theory (vol. 51). Academic Press.
• Panek, E. (2003). Ekonomia matematyczna. Wydawnictwo Akademii Ekonomicznej w Poznaniu.
• Panek, E. (2011). O pewnej prostej wersji „słabego” twierdzenia o magistrali w modelu von Neumanna. Przegląd Statystyczny, 58(1-2), 75-87. https://ps.stat.gov.pl/PS/2011/1-2/2011_58 _1-2_075-087.pdf.
• Panek, E. (2014). Model gospodarki Gale'a ze zmienną technologią, rosnącą efektywnością produkcji i szczególną postacią kryterium wzrostu. „Słaby” efekt magistrali. Przegląd Statystyczny, 61(4), 325-334. https://ps.stat.gov.pl/PS/2014/4/2014_61_4_325-334.pdf.
• Panek, E. (2015). A turnpike theorem for a non-stationary Gale economy with limit technology. A particular case. Eonomics and Business Review, 15(4), 3-13. https://doi.org/10.18559/ebr.2015.4.1.
• Panek, E. (2019). A Non-stationry Gale economy with Limit Technology, Multilane Turnpike and General Form of Optimality Criterion. Argumenta Oeconomica Cracoviensia, (1), 9-22. https://doi.org/10.15678/AOC.2019.2001.
• Panek, E. (2021). The minimal-time growth problem and turnpike effect in the stationary Gale economy. Economics and Business Review, 21(1), 7-25. https://doi.org/10.18559/ebr.2021.1.2.
• Panek, E. (2022). Gale Economy with Investments and Limit Technology. Central European Journal of Economic Modelling and Econometrics, 14(1), 57-80. https://doi.org/10.24425 /cejeme.2022.140512.
• Radner, R. (1961). Paths of Economic Growth that are Optimal with Regard only to Final States: A Turnpike Theorem. Review Economic Studies, 28(2), 98-104. https://doi.org/10.2307/2295707.
• Takayama, A. (1985). Mathematical economics (2nd edition). Cambrige University Press.

Identifiers

DOI

10.5604/01.3001.0015.7799

Biblioteka Nauki

2034077