Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-23T00:29:36.254Z Has data issue: false hasContentIssue false
  • English
  • Français

HOPF CYCLES IN ONE-SECTOR OPTIMAL GROWTH MODELS WITH TIME DELAY

Published online by Cambridge University Press:  13 September 2016

Hitay Özbay ,
Hüseyin Çağrı Sağlam  and
Mustafa Kerem Yüksel
Hitay Özbay
Affiliation:
Bilkent University
Hüseyin Çağrı Sağlam
Affiliation:
Bilkent University
Mustafa Kerem Yüksel*
Affiliation:
University of Turkish Aeronautical Association and Bilkent University
*
Address correspondence to: Mustafa Kerem Yüksel, Department of International Trade and Finance, University of Turkish Aeronautical Association, and Department of Economics, Bilkent University, Ankara, Turkey; e-mail: mkerem@bilkent.edu.tr.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper analyzes the existence of Hopf bifurcation and establishes the conditions under which the equilibrium path converges toward periodic solutions in a one-sector optimal growth model with delay. We establish the limits and the possibilities of nonlinear dynamics (i.e., cycles) vis-à-vis delays. In particular, we formulate a new method to further comprehend the root distribution of the characteristic equation of a standard optimal growth model with delayed investment structure. We show that nonmonotonic dynamics (limit cycles, persistent oscillations) occurs when the delayed investment causes permanent adjustment failures among the economic variables in the economy.

Keywords

Optimal Growth ModelsTime DelayHopf Bifurcation
Type
Articles
Information
Macroeconomic Dynamics , Volume 21 , Issue 8 , December 2017 , pp. 1887 - 1901
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

For many helpful discussions and comments, we thank Stefano Bosi, Paolo Brito, and Omar Licandro, as well as seminar participants at COIA in Ankara, Turkey. The usual disclaimer applies.

References

REFERENCES

Asea, Patrick K. and Zak, Paul J. (1999) Time-to-build and cycles. Journal of Economic Dynamics and Control 23, 1155–1175.CrossRefGoogle Scholar
Bambi, Mauro (2008) Endogenous growth and time-to-build: The AK case. Journal of Economic Dynamics and Control 32, 1015–1040.Google Scholar
Bambi, Mauro and Gori, Franco (2013) Unifying time-to-build theory. Macroeconomic Dynamics 18, 1713–1725.Google Scholar
Barnett, William A. and Chen, Guo (2015) Bifurcation of macroeconometric models and robustness of dynamical inferences. Foundations and Trends in Econometrics 8 (1–2), 1–144.Google Scholar
Barnett, William A., Serletis, Apostolos, and Serletis, Demitre (2015) Nonlinear and complex dynamics in economics. Macroeconomic Dynamics 19, 1749–1779.CrossRefGoogle Scholar
Bellman, Richard E. and Cooke, Kenneth L. (1963) Differential–Delay Equations. New York: Academic Press.Google Scholar
Besomi, Daniele (2006) Formal modelling vs. insight in Kalecki's theory of the business cycle. Research in the History of Economic Thought and Methodology 24, 1–48.CrossRefGoogle Scholar
Brandt-Pollmann, Ulrich, Winkler, Ralph, Sager, Sebastian, Moslener, Ulf, and Schlöder, Johannes P. (2008) Numerical solution of optimal control problems with constant control delays. Computational Economics 31, 181–206.Google Scholar
Collard, Fabrice, Licandro, Omar, and Puch, Luis A. (2006) Time-to-Build Echoes. Working paper 16, Fundación de Estudios de Economía Aplicada (FEDEA).Google Scholar
Collard, Fabrice, Licandro, Omar, and Puch, Luis A. (2008) The short-run dynamics of optimal growth models with delays. Annales d'Economie et de Statistique 90, 127–143.CrossRefGoogle Scholar
Farkas, Miklós (1994) Periodic Motions. New York: Springer-Verlag.Google Scholar
Frisch, Ragnar and Holme, Harold (1935) The characteristic solutions of a mixed difference and differential equation occuring in economic dynamics. Econometrica 3, 225–239.Google Scholar
Guckenheimer, John and Holmes, Philip (1983) Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. New York: Springer-Verlag.Google Scholar
Hale, Jack (1977) Theory of Functional Differential Equations. New York: Springer-Verlag.CrossRefGoogle Scholar
Hale, Jack and Koçak, Hüseyin (1991) Dynamics and Bifurcations. New York: Springer-Verlag.Google Scholar
Hayes, N.D. (1950) Roots of the transcendental equation associated with a certain difference–differential equation. Journal of the London Mathematical Society 25, 226–232.CrossRefGoogle Scholar
Hopf, Eberhard (1942) Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Berichten der Mathematisch-Physischen Klasse der Sächsischen Akademie der Wissenschaften zu Leipzig 94, 1–22.Google Scholar
James, R.W. and Belz, M.H. (1938) The significance of the characteristic solutions of mixed difference and differential equations. Econometrica 6, 326–343.CrossRefGoogle Scholar
Kaldor, Nicholas (1940) A model of the trade cycle. Economic Journal 50, 78–92.Google Scholar
Kalecki, Michał (1935) A macrodynamic theory of the business cycle. Econometrica 3, 327–344.CrossRefGoogle Scholar
Kind, Christoph (1999) Remarks on the economic interpretation of Hopf bifurcations. Economic Letters 62, 147–154.Google Scholar
Kolmanovskii, Vladimir Borisovich and Myshkis, Anatolii Dmitrievich (1992) Applied Theory of Functional Differential Equations. Dordrecht, Netherlands: Academic Publishers.CrossRefGoogle Scholar
Krawiec, Adam and Szydłowski, Marek (1999) The Kaldor–Kalecki business cycle model. Annals of Operational Research 89, 89–100.Google Scholar
Kydland, Finn E. and Prescott, Edward C. (1982) Time to build and aggregate fluctuations. Econometrica 50 (6), 1345–1370.Google Scholar
Long, John B. Jr. and Plosser, Charles I. (1983) Real business cycles. Journal of Political Economy 91 (1), 39–69.Google Scholar
Marsden, Jerrold E. and McCracken, Marjorie (1976) The Hopf Bifurcation and Its Applications. New York: Springer-Verlag.Google Scholar
Medio, Alfredo (1998) Nonlinear dynamics and chaos: I. A geometrical approach. Macroeconomic Dynamics 2, 505–532.Google Scholar
Michiels, Wim and Niculescu, Silviu-Iulian (2007) Stability and Stabilization of Time Delay Systems: An Eigenvalue-based Approach. Philadelphia: SIAM.CrossRefGoogle Scholar
Özbay, Hitay (2000) Introduction to Feedback Control Theory. Boca Raton, FL: CRC Press LLC.Google Scholar
Ramsey, F.P. (1927) A contribution to the theory of taxation. Economic Journal 37 (145), 47–61.Google Scholar
Solow, Robert (1956) A contribution to the theory of economic growth. Quarterly Journal of Economics 70, 65–94.Google Scholar
Szydłowski, Marek (2002) Time-to-build in dynamics of economic Models: I. Kalecki's model. Chaos, Solitons and Fractals 14, 697–703.Google Scholar
Winkler, Ralph (2009) Time-to-Build in the AK Growth Model: Short-Run Dynamics and Response to Exogenous Shocks. Mimeo, CER-ETH, Center for Economic Research at Eidgenössische Technische Hochschule (ETH) Zurich.Google Scholar
Winkler, Ralph (2011) A note on the optimal control stocks accumulating with a delay. Macroeconomic Dynamics 15, 565–578.Google Scholar
Winkler, Ralph, Brandt-Pollmann, Ulrich, Moslener, Ulf, and Schlöder, Johannes P. (2004) Time lags in capital accumulation. In Ahr, D., Fahrion, R., Oswald, M., and Reinelt, G. (eds.), Operations Research Proceedings 2003, pp. 451–458. Heidelberg, Germany: Springer.Google Scholar
Winkler, Ralph, Brandt-Pollmann, Ulrich, Moslener, Ulf, and Schlöder, Johannes P. (2005) On the Transition from Instantaneous to Time-Lagged Capital Accumulation: The Case of Leontief-Type Production Functions. Discussion paper 05-30, ZEW–Zentrum für Europäische Wirtschaftsforschung/Center for European Economic Research.CrossRefGoogle Scholar
Yüksel, Mustafa Kerem (2011) Capital dependent population growth induces cycles. Chaos, Solitons & Fractals 44 (9), 759–763.Google Scholar
Zak, Paul (1999) Kaleckian lags in general equilibrium. Review of Political Economy 11, 321–330.CrossRefGoogle Scholar