Abstract
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.
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an der Heiden, U. (1979). Delays in physiological systems.J. Math. Biol. 8, 345–364.
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.
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.
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.
an der Heiden, U., and Walther, H.-O. (1983). Existence of chaos in control systems with delayed feedback.J. Diff. Eq. 47, 273–295.
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.
Bélair, J. (1987). Stability of a differential-delay equation with two time lags.CMS Proc. 8, 305–315.
Bélair, J., and Mackey, M. C. (1987). A model for the regulation of mammalian platelet production.Ann. N.Y. Acad. Sci. 504, 280–282.
Braddock, R., and van den Driessche, P. (1983). On a two lag differential delay equation.J. Aust. Math. Soc. 24, 292–317.
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.
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.
Driver, R. (1963b). Existence theory for a delay-differential system.Contrib. Diff. Eq. 1, 317–336.
Ezekiel, M. (1938). The cobweb theorem.Q. J. Econ. 52, 255–280.
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.
Gabisch, G., and Lorenz, H.-W. (1987).Business Cycle Theory (Lecture Notes in Economies and Mathematical Systems, 283), Springer-Verlag, Berlin.
Glass, L., and Mackey, M. C. (1979). Pathological conditions resulting from instabilities in physiological control systems.Ann. N.Y. Acad Sci. 316, 214–235.
Glass, L., Mackey, M. C. (1988).From Clocks to Chaos: The Rhythms of Life, Princeton University Press, Princeton, N.J.
Goodwin, R. M. (1951). The nonlinear accelerator and the persistence of business cycles.Econometrica 19, 1–17.
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.
Grandmont, J.-M., and Malgrange, P. (1986). Nonlinear economic dynamics: Introduction.J. Econ. Theory 40, 3–12.
Guckenheimer, J., and Holmes, P. (1983).Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York.
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.
Haldane, J. B. S. (1933). A contribution to the theory of price fluctuations.Rev. Econ. Stud. 1, 186–195.
Hayes, N. D. (1950). Roots of the transcendental equation associated with a certain difference-differential equation.J. Lond. Math. Soc. 25, 226–232.
Howroyd, T. D., and Russell, A. M. (1984). Cournot oligopoly models with time delays.J. Math. Econ. 13, 97–103.
Kaczmarek, L. K., and Babloyantz, A. (1977). Spatiotemporal patterns in epileptic seizures.Biol. Cybern. 26, 199–208.
Kaldor, N. (1933). A classificatory note on the determinateness of equilibrium.Rev. Econ. Stud. 1, 122–136.
Kalecki, M. (1935). A macroeconomic theory of the business cycle.Econometrica 3, 327–344.
Kalecki, M. (1937). A theory of the business cycle.Rev. Econ. Stud. 4, 77–97.
Kalecki, M. (1943).Studies in Economic Dynamics, Allen-Unwin, London.
Kalecki, M. (1952).Theory of Economic Dynamics, Unwin University Books, London.
Kalecki, M. (1972). A theory of the business cycle (1939). In Kalecki, M. (ed.),Essays in the Theory of Economic Fluctuation, Allen-Unwin, London.
Kolmanovskii, V. B., and Nosov, V. R. (1986). Stability of functional differential equations. InMathematics in Science and Engineering, Vol. 180, Academic Press, London.
Larson, A. B. (1964). The hog cycle as harmonic motion.J. Farm Econ. 46, 375–386.
Lasota, A. (1977). Ergodic problems in biology.Astérisque 50, 239–250.
Lasota, A., and Mackey, M. C. (1985).Probabilistic Properties of Deterministic Systems, Cambridge University Press, New York.
Lasota, A., and Traple, J. (1986). Differential equations with dynamical perturbations.J. Diff. Eq. 63, 406–417.
Leontief, W. W. (1934). Verzogarte Angebotsanpassung und Partielles Gleichgewicht.Z. Nationallokon. 5, 670–676.
Mackey, M. C., and an der Heiden, U. (1984). The dynamics of recurrent inhibition.J. Math. Biol. 19, 211–225.
Mackey, M. C., and Glass, L. (1977). Oscillation and chaos in physiological control systems.Science 197, 287–289.
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.
Mates, J. W. B., and Horowitz, J. M. (1976). Instability in a hippocampal network.Comp. Prog. Biomed. 6, 74–84.
Myshkis, A. (1977). On certain problems in the theory of differential equations with deviating arguments.Russ. Math. Surv. 32, 181–213.
Nisbet, R., and Gurney, W. S. C. (1976). Population dynamics in a periodically varying environment.J. Theor. Biol. 56, 459–475.
Nussbaum, R. D. (1974). Periodic solutions of some nonlinear autonomous functional differential equations.Ann. Mat. Pura Appl. 101, 263–306.
Nussbaum, R. (1975). Differential delay equations with two time lags.Mem. Am. Math. Soc. 205, 1–62.
Peters, H. (1980). Globales Losungsverhalten zeitverzogerter Differentialgleichungen am Beispiel von Modellfunktionen. Dissertation, University of Bremen, Bremen, FRG.
Ricci, U. (1930). Die “Synthetische Ökonomie” von Henry Ludwell Moore.Z. Nationalökon. 1, 649–668.
Saupe, D. (1982). Beschleunigte PL-Kontinuitätsmethoden und periodische Losungen parametrisierter Differentialgleichungen mit Zeitverzogerung. Dissertation, University of Bremen, Bremen, FRG.
Schultz, H. (1930). Der Sin der Statistischen Nachfragen.Veroeff. Frankf. Ges. Konjunkturforsch. 10, Kurt Schroeder Verlag, Bonn, FRG.
Slutzky, E. (1937). The summation of random causes as the source of cyclic processes.Econometrica 5, 105–146.
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.
Stech, H. (1985). Hopf bifurcation calculations for functional differential equations.J. Math. Anal. Appl. 109, 472–491.
Sugie, J. (1988). Oscillating solutions of scalar delay-differential equations with state dependence.Appl. Anal. 27, 217–227.
Tinbergen, J. (1930). Bestimmung und Deutung von Angebotskurven.Z. Nationalökon. 1, 669–679.
Walther, H. O. (1976). On a transcendental equation in the stability analysis of a population growth model.J. Math. Biol. 3, 187–195.
Walther, H. O. (1981). Homoclinic solution and chaos inx(t)=f(x t−1).J. Nonlinear Anal. 5, 775–788.
Walther, H. O. (1985). Bifurcations from a heteroclinic solution in differential delay equations.Trans. Am. Math. Soc. 290, 213–233.
Waugh, F. V. (1964). Cobweb models.J. Farm Econ. 46, 732–750.
Wazewska-Czyzewska, M., and Lasota, A. (1976). Matematyyczme problemy dynamiki ukladu krwinek czerwonych, Rocznijki Polskiego Towarzystwa Mathematcycznego. Seria III.Mat. Stosowana 6, 23–47.
Weidenbaum, M. L., and Vogt, S. C. (1988). Are economic forecasts any good?Math. Comput. Modelling 11, 1–5.
Winston, E. (1974). Asymptotic stability for ordinary differential equations with delayed perturbations.SIAM J. Math. Anal. 5, 303–308.
Zarnowitz, V. (1985). Recent work on business cycles in historical perspective: A review of theories and evidence.J. Econ. Liter. 23, 523–580.
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Bélair, J., Mackey, M.C. Consumer memory and price fluctuations in commodity markets: An integrodifferential model. J Dyn Diff Equat 1, 299–325 (1989). https://doi.org/10.1007/BF01053930
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DOI: https://doi.org/10.1007/BF01053930