On the optimal control on an infinite planning horizon of consumption, pollution, population and natural resource use

  • Alain Haurie
  • Michael P. Polis
  • Pierre Yansouni
Environmental Systems
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4)


In this paper a five state variable economic planning model is presented. A Malthusian hypothesis is employed which gives rise to a zero-growth argument. The asymptotic behavior of the model is studied. Classical results involving Turnpike theory are used in conjunction with recently published results on the infinite time optimal control problem to show the convergence towards a Von Neumann point of the optimal trajectory on an infinite planning horizon.


Optimal Control Problem Capital Stock Optimal Trajectory Mathematical Programming Problem Infinite Time Horizon 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    FORRESTER, J.W., World Dynamics, Wright-Allen, Cambridge, Mass., 1971.Google Scholar
  2. [2]
    WARFIELD, J.N., Book Review — World Dynamics, IEEE Transactions on Systems, Man and Cybernetics, Vol. Smc-2, No. 4, Sept. 1972, pp. 558–559.Google Scholar
  3. [3]
    BOYD, R., "World Dynamics: A Note", Science, Vol 177, August 1972, pp. 516–519.Google Scholar
  4. [4]
    RAMSEY, F., A Mathematical Theory of Saving, Economic Journal, Dec. 1928, pp. 543–559.Google Scholar
  5. [5]
    KOOPMANS, T.C., "On the Concept of Optimal Economic Growth" in "The Econometric Approach to Development and Planning", North Holland; 1965.Google Scholar
  6. [6]
    KOOPMANS, T.C., "Objectives, Constraints and Outcomes in Optimal Growth models", Econometrica 35, 1, Jan. 1967, pp. 1, 15.Google Scholar
  7. [7]
    ARROW, K.J., "Applications of Control Theory to Economic Growth" in Mathematics of the Decision Sciences, G.B. Dantzig and A.F. Veinott edit., A.M.S. 1968.Google Scholar
  8. [8]
    ARROW, K.J. and KURZ, M., Public Investment, the Rate of Return, and Optimal Fiscal Policy, The Johns Hopkins Press, 1970.Google Scholar
  9. [9]
    MEADOWS, D.L., The Limits to Growth, Universe Books 381, Park Avenue, N.Y.C. 10016.Google Scholar
  10. [10]
    HALKIN, H., "Necessary Conditions for Optimal Control Problems with Infinite Horizon", Core-Discussion Paper. 1971, Core, De Croylaan 54, Heverlee, Belgium.Google Scholar
  11. [11]
    McKENZIE, "Accumulation Programms of Maximum Utility and the Von Neumann Facet" in Value, Capital and Growth (J.N. Wolfe edit.) Chicago Aldine, 1968, pp. 353–383.Google Scholar
  12. [12]
    BURMEISTER, E. and DOBELL, A.R., "Mathematical Theories of Economic Growth", Macmillan, N.Y., 1970.Google Scholar
  13. [13]
    LOTKA, A.J., Elements of Mathematical Biology, Dover 1956.Google Scholar
  14. [14]
    SMITH, V.L., "The Economics of Production from Natural Resources", American Economic Review, Vol. LVIII (June 1968) pp. 409–431.Google Scholar
  15. [15]
    PLOURDE, C.G., "A Simple Model of Replenishable Natural Resource Exploitation" Amer. Econ. Rev., June 1970, pp. 518–522.Google Scholar
  16. [16]
    KOLM, S.C., "La Croissance et la Qualité de l'Environnement", Analyse et Prévision, VIII, 1969, pp. 445–452.Google Scholar
  17. [17]
    KEELER, E., SPENCE, M., ZECHAUSER, R., "The Optimal Control of Pollution" Journal of Economic Theory 4, 1971, pp. 19–34.Google Scholar
  18. [18]
    HAURIE, A., POLIS, M. and YANSOUNI, P., "On optimal Pollution and Consumption Control in a Macro-economic System", Proceeding 1972, IEEE Conference on Decision and Control.Google Scholar
  19. [19]
    HAURIE, A., POLIS, M. and YANSOUNI, P., "On Optimal Convergence to a Steady-State-Zero-Growth Economy with Pollution and Consumption Control", to appear Proceeding 1973, IFAC/IFORS Conference on Dynamic Modelling and Control of National Economies.Google Scholar
  20. [20]
    PITCHFORD, J., "Population and Optimal Growth", Econometrica, Vol 40, No 1 (January 1972).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1973

Authors and Affiliations

  • Alain Haurie
    • 1
  • Michael P. Polis
    • 2
  • Pierre Yansouni
    • 2
  1. 1.Ecole des Hautes Etudes Commerciales and Ecole PolytechniqueMontrealCanada
  2. 2.Ecole PolytechniqueMontrealCanada

Personalised recommendations