Journal of Optimization Theory and Applications

, Volume 107, Issue 2, pp 275–286 | Cite as

Instabilities in Concave, Dynamic, Economic Optimization

  • G. Feichtinger
  • F. Wirl


An important and numerous literature argues that nonconcavity (often convexity with respect to the state) of the Hamiltonian leads to multiple steady states, instability, and a threshold. This threshold property provides a powerful paradigm to explain history dependency and hysteresis. This paper shows that economically relevant properties (in particular, multiple steady states and thresholds) are possible in strict concave models too. Two corresponding necessary conditions with intuitive economic interpretation are derived.

optimal control thresholds multiple equilibria instability concavity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Lewis, T. R., and Schmalensee, R., Optimal Use of Renewable Resources with Nonconvexities in Production, Essays in the Economics of Renewable Resources, Edited by L. J. Mirman and D. F. Spulber, North Holland, Amsterdam, Netherlands, pp. 95–111, 1982.Google Scholar
  2. 2.
    Brock, W. A., Pricing, Predation, and Entry Barriers in Regulated Industries, Breaking Up Bell, Edited by S. E. David, North Holland, Amsterdam, Netherlands, pp. 191–229, 1983.Google Scholar
  3. 3.
    Dechert, D. W., Has the Averch-JohnsonEffect BeenTheoretically Justified?, Journal of Economic Dynamics and Control, Vol. 8, pp. 1–17, 1984.Google Scholar
  4. 4.
    Brock, W. A., and Dechert, D. W., Dynamic Ramsey Pricing, International Economic Review, Vol. 26, pp. 569–591, 1985.Google Scholar
  5. 5.
    Krugman, P., Development, Geography, and Economic Theory, MIT Press, Cambridge, Massuchusetts, 1995.Google Scholar
  6. 6.
    Tahvonen, O., and Salo, S., Nonconvexities in Optimal Pollution Accumulation, Journal of Environmental Economics and Management, Vol. 31, pp. 160–177, 1996.Google Scholar
  7. 7.
    Tahvonen, O., and Withagen, C., Optimality of Irreversible Pollution Accumulation, Journal of Economic Dynamics and Control, Vol. 20, pp. 1775–1795, 1996.Google Scholar
  8. 8.
    Van Long, N., Nishimura, K., and Shimomura, K., Endogenous Growth, Trade, and Specialization under Variable Returns to Scale: The Case of a Small OpenEconomy, Unpublished Manuscript, 1997.Google Scholar
  9. 9.
    Ladron-De-Guevara, A., Ortigueira, S., and Santos, M. S., A Two-Sector Model of Endogenous Growth with Leisure, Review of Economic Studies, Vol. 66, pp. 609–631, 1999.Google Scholar
  10. 10.
    Santos, M. S., On Nonexistence of Continuous Markov Equilibria in Competitive Market Economies, Paper Presented at the 1999 Meeting of the Society of Economic Dynamics, Alghero, Sardinia, Italy, 1999.Google Scholar
  11. 11.
    MÄler, K. G., Nonlinear Systems in Environmental Economics, The Joseph Schumpeter Lecture, European Economic Association Annual Congress, Santiago de Compostela, Spain, September 3, 1999.Google Scholar
  12. 12.
    Lucas, R. E., Making a Miracle, Econometrica, Vol. 61, pp. 251–272, 1993.Google Scholar
  13. 13.
    Skiba, A. K., Optimal Growth with a Convex-Concave Production Function, Econometrica, Vol. 46, pp. 527–539, 1978.Google Scholar
  14. 14.
    Dechert, D. W., and Nishimura, K., Complete Characterizationof Optimal Growth Paths inanAggregative Model with a Nonconcave Production Function, Journal of Economic Theory, Vol. 31, pp. 332–354, 1983.Google Scholar
  15. 15.
    Kurz, M., The General Instability of a Class of Competitive Growth Processes, Review of Economic Studies, Vol. 35, pp. 1955–1974, 1968.Google Scholar
  16. 16.
    Liviathan, N., and Samuelson, P. A., Notes on Turnpikes: Stable and Unstable, Journal of Economic Theory, Vol. 1, pp. 454–475, 1969.Google Scholar
  17. 17.
    Ramsey, F. P., A Mathematical Theory of Saving, Economic Journal, Vol. 38, pp. 543–559, 1928.Google Scholar
  18. 18.
    Hadley, G., and Kemp, M. C., Variational Methods in Economics, North Holland, Amsterdam, Netherlands, 1971.Google Scholar
  19. 19.
    Ryder, H. E., JR., and Heal, G. M., Optimal Growth with Intertemporally Dependent Preferences, Review of Economic Studies, Vol. 40, pp. 1–31, 1973.Google Scholar
  20. 20.
    Becker, G. S., and Murphy, K. M., A Theory of Rational Addiction, Journal of Political Economy, Vol. 96, pp. 675–700, 1988.Google Scholar
  21. 21.
    Wirl, F., Pathways to Hopf Bifurcations in Dynamic, Continuous-Time Optimization Problems, Journal of Optimization Theory and Applications, Vol. 91, pp. 299–320, 1996.Google Scholar
  22. 22.
    Feichtinger, G., Novak, A., and Wirl, F., Limit Cycles inIn tertemporal Adjustment Models: Theory and Applications, Journal of Economic Dynamics and Control, Vol. 18, pp. 353–380, 1994.Google Scholar
  23. 23.
    Dockner, E., Local Stability Analysis in Optimal Control Problems with Two State Variables, Optimal Control Theory and Economic Analysis 2, Edited by G. Feichtinger, North Holland, Amsterdam, Netherlands, pp. 89–103, 1985.Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • G. Feichtinger
    • 1
  • F. Wirl
    • 2
  1. 1.Institute of Econometrics, OR, and Systems TheoryVienna University of TechnologyViennaAustria
  2. 2.Faculty of Economics and ManagementOtto von Guericke University MagdeburgMagdeburgGermany

Personalised recommendations