Abstract
The purpose of this paper is to derive conditions for the optimality of a limit cycle in a dynamic economic system and to interpret them economically. A fairly general two-state continuous-time nonlinear optimal control problem is considered. It turns out that for this class of models three different economic mechanisms can be identified as the possible source of limit cycles. One relates to an intertemporal substitution effect expressed in terms of complementarity over time, the second one is a dominating cross effect between the state variables of the system (i.e., the capital stocks in our model), and the third one is positive growth at the equilibrium.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Becker, G. S., and Murphy, K. M. (1988): “A Theory of Rational Addiction.”Journal of Political Economy 96: 675–700.
Benhabib, J. (1978): “A Note on Optimal Growth and Intertemporally Dependent Preferences.”Economics Letters 1: 321–324.
Benhabib, J., and Nishimura, K. (1979): “The Hopf Bifurcation and the Existence and Stability of Closed Orbits in Multisector Models of Optimum Economic Growth.”Journal of Economic Theory 21: 421–444.
Brock, W. A., and Scheinkman, J. A. (1976): “Global Asymptotic Stability of Optimal Control Systems with Applications to the Theory of Economic Growth.”Journal of Economic Theory 12: 164–190.
— (1977): “The Global Asymptotic Stability of Optimal Control with Applications to Dynamic Economic Theory.” InApplications of Control Theory to Economic Analysis, edited by J. D. Pitchford and S. J. Turnovsky. Amsterdam: North-Holland.
Carlson, D. A., and Haurie, A. (1987):Infinite Horizon Optimal Control. Theory and Applications. Berlin: Springer.
Cass, D., and Shell, K. (1976a): “Introduction to Hamiltonian Dynamics in Economics.”Journal of Economic Theory 12: 1–10.
— (1976b): “The Structure and Stability of Competitive Dynamical Systems.”Journal of Economic Theory 12: 31–70.
Dockner, E. J. (1985): “Local Stability Analysis in Optimal Control Problems with Two State Variables.” InOptimal Control Theory and Economic Analysis 2, edited by G. Feichtinger. Amsterdam: North-Holland.
Guckenheimer, J., and Holmes, P. (1983):Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer.
Iannaccone, L. R. (1986): “Addiction and Satiation.”Economics Letters 21: 95–99.
Kemp, M. C., Van Long, N., and Shimomura, K. (1990): “Cyclical and Non-Cyclical Redistributive Taxation.” Working paper, University of New South Wales, Australia.
Kurz, M. (1968): “The General Instability of a Class of Competitive Growth Processes.”Review of Economic Studies 35: 155–174.
Levhari, D., and Liviatan, N. (1972): “On Stability in the Saddle-Point Sense.”Journal of Economic Theory 4: 88–93.
Medio, A. (1987): “Oscillations in Optimal Growth Models.”Journal of Economic Behavior and Organization 8: 413–427.
Rockafellar, R. T. (1976): “Saddle Points of Hamiltonian Systems in Convex Lagrange Problems Having a Nonzero Discount Rate.”Journal of Economic Theory 12: 71–113.
Ryder, H. E., Jr., and Heal, G. M. (1973): “Optimal Growth with Intertemporally Dependent Preferences.”Review of Economic Studies 40: 1–31.
Scheinkman, J. A. (1976): “On Optimal Steady States ofn-Sector Growth Models when Utility Is Discounted.”Journal of Economic Theory 12: 11–30.
— (1986): “Discussion Remark to J. M. Grandmont: On Endogenous Competitive Business Cycles.” InModels of Economic Dynamics, edited by H. F. Sonnenschein. (Lecture Notes in Economics and Mathematical Systems, 264). Berlin: Springer.
Sorger, G. (1989): “On the Optimality and Stability of Competitive Paths in Continuous Time Growth Models.”Journal of Economic Theory 48: 526–547.
Wan, H. Y. (1970): “Optimal Saving Programs under Intertemporally Dependent Preferences.”International Economic Review 11: 521–547.
Wirl, F. (1990): “A New Route to Cyclical Strategies in Two Dimensional Optimal Control Models.” Working Paper, Vienna University of Technology.
Woodford, M. (1987): “Equilibrium Models of Endogenous Fluctuations: Cycles, Chaos, Indeterminancy and Sunspots.” Lecture Notes prepared for the Workshop on Alternative Approaches to Macroeconomics, Siena, November 1–8, 1987.
Author information
Authors and Affiliations
Additional information
We acknowledge the helpful comments by William A. Brock, Gerhard Sorger, Franz Wirl, and two anonymous referees. The research was partly supported by the Austrian Science Foundation under contract No. P6601.
Rights and permissions
About this article
Cite this article
Dockner, E.J., Feichtinger, G. On the optimality of limit cycles in dynamic economic systems. Zeitschr. f. Nationalökonomie 53, 31–50 (1991). https://doi.org/10.1007/BF01227014
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01227014