Dynamical Models for Prices with Distributed Delays
AbstractIn the present paper we study some models for the price dynamics of a single commodity market. The quantities of supplied and demanded are regarded as a function of time. Nonlinearities in both supply and demand functions are considered. The inventory and the level of inventory are taken into consideration. Due to the fact that the consumer behavior affects commodity demand, and the behavior is influenced not only by the instantaneous price, but also by the weighted past prices, the distributed time delay is introduced. The following kernels are taken into consideration: demand price weak kernel and demand price Dirac kernel. Only one positive equilibrium point is found and its stability analysis is presented. When the demand price kernel is weak, under some conditions of the parameters, the equilibrium point is locally asymptotically stable. When the demand price kernel is Dirac, the existence of the local oscillations is investigated. A change in local stability of the equilibrium point, from stable to unstable, implies a Hopf bifurcation. A family of periodic orbits bifurcates from the positive equilibrium point when the time delay passes through a critical value. The last part contains some numerical simulations to illustrate the effectiveness of our results and conclusions.
Belair, J. & Mackey, M.C. (1989). Consummer memory and price fluctuations in commodity markers: An integrodifferential model. Journal of dynamics and differential equation, 1(3), 299-325.
Hale, J.K. & Lunel, S. V. (1993). Introduction to functional differential equations. Springer Verlag, New York.
Hasssard, B.D., Kazarinoff, N.D. & Wan Y.H. (1981). Theory and applications of Hopf bifurcation, vol. 41 of London Mathematical Society. Lecture Notes Series, Cambridge University Press, Cambridge, UK.
Huang, C., Peng, C., Chen, X. & Wen, F. (2013). Dynamics Analysis of a Class of Delayed Economic Model. Abstract and Applied Analysis, 2013, Article ID962738, http://dx.doi.org/10.1155/2013/962738
Kutznenetsov, Y.A. (1995). Elements of applied bifurcation theory, Springer Verlag, New York
Li, X., Ruin, S. & Wei, J. (1999). Stability and bifurcation in delay-differential equation with two delays. Journal of Math. Analysis and Applications, 236(2), 254-280.
Mircea, G., Neamtu, M. & Opris, D. (2011). Uncertain, stochastic and fractional dynamic systems with delay. Applications, Lambert Academic Publishing, Germany
Mircea, G., Neamtu, M., Bundau, O. & Opris, D. (2012). Uncertain and stochastic financial models with multiple delays. International journal of bifurcation and chaos, 22(6), 1250131-1-19, DOI: 10.1142/s0218127412501313
Sirghi, N. & Neamtu, M. (2013). Deterministic and stochastic advertising diffusion model with delay. WSEAS Transaction on system and control, 4(8), 141-150.
Zhai, Y., Bai, H., Xiong, Y., & Ma X. (2013). Hopf bifurcation analysis for the modified Rayleight price model with time delay. Abstract and applied analysis, volume 2013, article ID 290497, http://dx.doi.org/10.1155/2013/290497
Weidenbaum, M.L. & Vogt, S.C. (1988). Are economic forecast any good? Mathematical and Computing Modelling, 11, 1-5.
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