Consumer memory and price fluctuations in commodity markets: An integrodifferential model

  • Jacques Bélair
  • Michael C. Mackey


A model for the dynamics of price adjustment in a single commodity market is developed. Nonlinearities in both supply and demand functions are considered explicitly, as are delays due to production lags and storage policies, to yield a nonlinear integrodifferential equation. Conditions for the local stability of the equilibrium price are derived in terms of the elasticities of supply and demand, the supply and demand relaxation times, and the equilibrium production-storage delay. The destabilizing effect of consumer memory on the equilibrium price is analyzed, and the ensuing Hopf bifurcations are described.

Key words

Commodity markets time delays stability Hopf bifurcation 

AMS (MOS) subject classifications

34K15 45J05 90A16 


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  1. an der Heiden, U. (1979). Delays in physiological systems.J. Math. Biol. 8, 345–364.Google Scholar
  2. an der Heiden, U. (1985). Stochastic properties of simple differential-delay equations. In Meinardus, G., and Nurnberger, G. (eds.),Delay Equations, Approximation and Application, Birkhauser, Stuttgart, pp. 147–164.Google Scholar
  3. an der Heiden, U., and Mackey, M. C. (1982). The dynamics of production and destruction: Analytic insight into complex behaviour.J. Math. Biol. 16, 75–101.Google Scholar
  4. an der Heiden, U., and Mackey, M. C. (1987). Mixed feedback: A paradigm for regular and irregular oscillations. In Rensing, L., an der Heiden, U., and Mackey, M. C. (eds.),Temporal Disorder in Human Oscillatory Systems, Springer-Verlag, New York.Google Scholar
  5. an der Heiden, U., and Walther, H.-O. (1983). Existence of chaos in control systems with delayed feedback.J. Diff. Eq. 47, 273–295.Google Scholar
  6. an der Heiden, U., Mackey, M. C., and Walther, H. O. (1981). Complex oscillations in a simple deterministic neuronal network.Lect. Appl. Math. 19, 355–360.Google Scholar
  7. Bélair, J. (1987). Stability of a differential-delay equation with two time lags.CMS Proc. 8, 305–315.Google Scholar
  8. Bélair, J., and Mackey, M. C. (1987). A model for the regulation of mammalian platelet production.Ann. N.Y. Acad. Sci. 504, 280–282.Google Scholar
  9. Braddock, R., and van den Driessche, P. (1983). On a two lag differential delay equation.J. Aust. Math. Soc. 24, 292–317.Google Scholar
  10. Cooke, K. (1985). Stability of delay differential equations with applications in biology and medicine. In Capasso, V., Grosso, E., and Paveri-Fontana, S. L. (eds.),Mathematics in biology and medicine (Lecture Notes in Biomathematics, 57), Springer-Verlag, Berlin.Google Scholar
  11. Driver, R. (1963a). A functional differential system of neutral type arising in a two-body problem of classical electrodynamics. InNonlinear Differential Equations and Nonlinear Mechanics, Academic Press, New York, pp. 474–484.Google Scholar
  12. Driver, R. (1963b). Existence theory for a delay-differential system.Contrib. Diff. Eq. 1, 317–336.Google Scholar
  13. Ezekiel, M. (1938). The cobweb theorem.Q. J. Econ. 52, 255–280.Google Scholar
  14. Fargre, D. (1973). Réductibilité des systèmes héréditaires à des systèmes dynamiques (régis par des équations différentielles ou aux dérivées partielles).C.R. Acad. Sci. 277, 471–473.Google Scholar
  15. Gabisch, G., and Lorenz, H.-W. (1987).Business Cycle Theory (Lecture Notes in Economies and Mathematical Systems, 283), Springer-Verlag, Berlin.Google Scholar
  16. Glass, L., and Mackey, M. C. (1979). Pathological conditions resulting from instabilities in physiological control systems.Ann. N.Y. Acad Sci. 316, 214–235.Google Scholar
  17. Glass, L., Mackey, M. C. (1988).From Clocks to Chaos: The Rhythms of Life, Princeton University Press, Princeton, N.J.Google Scholar
  18. Goodwin, R. M. (1951). The nonlinear accelerator and the persistence of business cycles.Econometrica 19, 1–17.Google Scholar
  19. Goodwin, R. M., Kruger, M., and Vercelli, A. (eds.) (1984).Nonlinear Models of Fluctuating Growth (Lecture Notes in Economics and Mathematical Systems, 228), Springer-Verlag, Berlin.Google Scholar
  20. Grandmont, J.-M., and Malgrange, P. (1986). Nonlinear economic dynamics: Introduction.J. Econ. Theory 40, 3–12.Google Scholar
  21. Guckenheimer, J., and Holmes, P. (1983).Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York.Google Scholar
  22. Hadeler, K. P. (1976). On the stability of the stationary state of a population growth equation with time-lag.J. Math. Biol. 3, 197–201.Google Scholar
  23. Haldane, J. B. S. (1933). A contribution to the theory of price fluctuations.Rev. Econ. Stud. 1, 186–195.Google Scholar
  24. Hayes, N. D. (1950). Roots of the transcendental equation associated with a certain difference-differential equation.J. Lond. Math. Soc. 25, 226–232.Google Scholar
  25. Howroyd, T. D., and Russell, A. M. (1984). Cournot oligopoly models with time delays.J. Math. Econ. 13, 97–103.Google Scholar
  26. Kaczmarek, L. K., and Babloyantz, A. (1977). Spatiotemporal patterns in epileptic seizures.Biol. Cybern. 26, 199–208.Google Scholar
  27. Kaldor, N. (1933). A classificatory note on the determinateness of equilibrium.Rev. Econ. Stud. 1, 122–136.Google Scholar
  28. Kalecki, M. (1935). A macroeconomic theory of the business cycle.Econometrica 3, 327–344.Google Scholar
  29. Kalecki, M. (1937). A theory of the business cycle.Rev. Econ. Stud. 4, 77–97.Google Scholar
  30. Kalecki, M. (1943).Studies in Economic Dynamics, Allen-Unwin, London.Google Scholar
  31. Kalecki, M. (1952).Theory of Economic Dynamics, Unwin University Books, London.Google Scholar
  32. Kalecki, M. (1972). A theory of the business cycle (1939). In Kalecki, M. (ed.),Essays in the Theory of Economic Fluctuation, Allen-Unwin, London.Google Scholar
  33. Kolmanovskii, V. B., and Nosov, V. R. (1986). Stability of functional differential equations. InMathematics in Science and Engineering, Vol. 180, Academic Press, London.Google Scholar
  34. Larson, A. B. (1964). The hog cycle as harmonic motion.J. Farm Econ. 46, 375–386.Google Scholar
  35. Lasota, A. (1977). Ergodic problems in biology.Astérisque 50, 239–250.Google Scholar
  36. Lasota, A., and Mackey, M. C. (1985).Probabilistic Properties of Deterministic Systems, Cambridge University Press, New York.Google Scholar
  37. Lasota, A., and Traple, J. (1986). Differential equations with dynamical perturbations.J. Diff. Eq. 63, 406–417.Google Scholar
  38. Leontief, W. W. (1934). Verzogarte Angebotsanpassung und Partielles Gleichgewicht.Z. Nationallokon. 5, 670–676.Google Scholar
  39. Mackey, M. C., and an der Heiden, U. (1984). The dynamics of recurrent inhibition.J. Math. Biol. 19, 211–225.Google Scholar
  40. Mackey, M. C., and Glass, L. (1977). Oscillation and chaos in physiological control systems.Science 197, 287–289.Google Scholar
  41. Mallet-Paret, J., and Nussbaum, R. D. (1986). A bifurcation gap for a singularly perturbed delay equation. In Barnsley, M., and Demko, S. (eds.),Chaotic Dynamics and Fractals, Academic Press, Orlando, FL.Google Scholar
  42. Mates, J. W. B., and Horowitz, J. M. (1976). Instability in a hippocampal network.Comp. Prog. Biomed. 6, 74–84.Google Scholar
  43. Myshkis, A. (1977). On certain problems in the theory of differential equations with deviating arguments.Russ. Math. Surv. 32, 181–213.Google Scholar
  44. Nisbet, R., and Gurney, W. S. C. (1976). Population dynamics in a periodically varying environment.J. Theor. Biol. 56, 459–475.Google Scholar
  45. Nussbaum, R. D. (1974). Periodic solutions of some nonlinear autonomous functional differential equations.Ann. Mat. Pura Appl. 101, 263–306.Google Scholar
  46. Nussbaum, R. (1975). Differential delay equations with two time lags.Mem. Am. Math. Soc. 205, 1–62.Google Scholar
  47. Peters, H. (1980). Globales Losungsverhalten zeitverzogerter Differentialgleichungen am Beispiel von Modellfunktionen. Dissertation, University of Bremen, Bremen, FRG.Google Scholar
  48. Ricci, U. (1930). Die “Synthetische Ökonomie” von Henry Ludwell Moore.Z. Nationalökon. 1, 649–668.Google Scholar
  49. Saupe, D. (1982). Beschleunigte PL-Kontinuitätsmethoden und periodische Losungen parametrisierter Differentialgleichungen mit Zeitverzogerung. Dissertation, University of Bremen, Bremen, FRG.Google Scholar
  50. Schultz, H. (1930). Der Sin der Statistischen Nachfragen.Veroeff. Frankf. Ges. Konjunkturforsch. 10, Kurt Schroeder Verlag, Bonn, FRG.Google Scholar
  51. Slutzky, E. (1937). The summation of random causes as the source of cyclic processes.Econometrica 5, 105–146.Google Scholar
  52. Stech, H. (1978). The effects of time lags on the stability of the equilibrium state of a population growth equation.J. Math. Biol. 5, 115–120.Google Scholar
  53. Stech, H. (1985). Hopf bifurcation calculations for functional differential equations.J. Math. Anal. Appl. 109, 472–491.Google Scholar
  54. Sugie, J. (1988). Oscillating solutions of scalar delay-differential equations with state dependence.Appl. Anal. 27, 217–227.Google Scholar
  55. Tinbergen, J. (1930). Bestimmung und Deutung von Angebotskurven.Z. Nationalökon. 1, 669–679.Google Scholar
  56. Walther, H. O. (1976). On a transcendental equation in the stability analysis of a population growth model.J. Math. Biol. 3, 187–195.Google Scholar
  57. Walther, H. O. (1981). Homoclinic solution and chaos inx(t)=f(x t−1).J. Nonlinear Anal. 5, 775–788.Google Scholar
  58. Walther, H. O. (1985). Bifurcations from a heteroclinic solution in differential delay equations.Trans. Am. Math. Soc. 290, 213–233.Google Scholar
  59. Waugh, F. V. (1964). Cobweb models.J. Farm Econ. 46, 732–750.Google Scholar
  60. Wazewska-Czyzewska, M., and Lasota, A. (1976). Matematyyczme problemy dynamiki ukladu krwinek czerwonych, Rocznijki Polskiego Towarzystwa Mathematcycznego. Seria III.Mat. Stosowana 6, 23–47.Google Scholar
  61. Weidenbaum, M. L., and Vogt, S. C. (1988). Are economic forecasts any good?Math. Comput. Modelling 11, 1–5.Google Scholar
  62. Winston, E. (1974). Asymptotic stability for ordinary differential equations with delayed perturbations.SIAM J. Math. Anal. 5, 303–308.Google Scholar
  63. Zarnowitz, V. (1985). Recent work on business cycles in historical perspective: A review of theories and evidence.J. Econ. Liter. 23, 523–580.Google Scholar

Copyright information

© Plenum Publishing Corporation 1989

Authors and Affiliations

  • Jacques Bélair
    • 1
  • Michael C. Mackey
    • 2
  1. 1.Département de Mathématiques et de Statistique and Centre de Recherches MathématiquesUniversité de MontréalMontréalCanada
  2. 2.Department of PhysiologyMcGill UniversityMontrealCanada

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