A class of economic growth model with memory in the context of fractional derivatives
Main Article Content
Abstract
This article presents a model of economic growth that considers the effects of generalized power-law fading memory. To consider the memory effects in macroeconomic models, we propose a class of fractional differential equations involving a Caputo-type fractional derivative with respect to another function (or -Caputo derivative). For this purpose, the objective of this paper is to obtain the unique solution of the proposed -Caputo fractional differential equation. Behaviors of Mittag-Leffler’s functions, which characterize the growth behaviors of the solution are described. To evaluate the solution, we will compute the Mittag-Leffler functions by Matlab code. As a result, we obtain the solutions in explicit form. This work also provides a comparative analysis of solutions of our model and model without memory effect. To the best of our knowledge, the Harrod-Domar fractional differential equation with Caputo-type fractional derivative has not yet been investigated. Hence the result obtained from our study is essentially new. Moreover, this approach allows for greater flexibility in modeling economic behavior, making it possible to account for non-instantaneous effects and the cumulative impact of previous decisions.
Keywords
Analytical solution; Economic growth model; Fractional derivatives; Fractional differential equation
Article Details
Field of Economic (JEL Codes)
C00 - General - C02 - Mathematical Methods - General, E20 - General - Consumption, Saving, Production, Investment, Labor Markets, and Informal Economy
References
Garrappa, R. (2015). Numerical evaluation of two and three parameter Mittag-Leffler functions. SIAM Journal on Numerical Analysis, 53(3), 1350-1369. https://doi.org/10.1137/140971191
Gorenflo, R., Kilbas, A. A., Mainardi, F., & Rogosin, S. V. (2020). Mittag-Leffler functions, Related topics and applications (p. 540). Berlin: Springer. https://doi.org/10.1007/978-3-662-43930-2
Granger, C. W. J., & Joyeux, R. (1980). An introduction to long‐memory time series models and fractional differencing. Journal of Time Series Analysis, 1(1), 15-29. https://doi.org/10.1111/j.1467-9892.1980.tb00297.x
Kee, C. Y., Chua, C., Zubair, M., & Ang, L. K. (2022). Fractional modeling of urban growth with memory effects. Chaos: An Interdisciplinary Journal of Nonlinear Science, 32(8). https://doi.org/10.1063/5.0085933
Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and applications of fractional differential equations: Volume 204 of North-Holland Mathematics Studies. Elsevier.
Tarasov, V. E. (2020). Fractional econophysics: Market price dynamics with memory effects. Physica A: Statistical Mechanics and its Applications, 557. https://doi.org/10.1016/j.physa.2020.124865
Tarasov, V. E. (2021). Nonlinear growth model with long memory: generalization of Haavelmo model. Nonlinear Dynamics, 104(4), 4413-4425. https://doi.org/10.1007/s11071-021-06484-2
Tarasova, V. V., & Tarasov, V. E. (2016). Fractional dynamics of natural growth and memory effect in economics. European Research, 12(23), 30-37. https://doi.org/10.20861/2410-2873-2016-23-004
Tarasova, V. V., & Tarasov, V. E. (2017). Logistic map with memory from economic model. Chaos, Solitons & Fractals, 95, 84-91. https://doi.org/10.1016/j.chaos.2016.12.012
Tarasova, V. V., & Tarasov, V. E. (2018). Concept of dynamic memory in economics. Communications in Nonlinear Science and Numerical Simulation, 55, 127-145. https://doi.org/10.1016/j.cnsns.2017.06.032
Tarasova, V. V., & Tarasov, V. E. (2018). Dynamic intersectoral models with power-law memory. Communications in Nonlinear Science and Numerical Simulation, 54, 100-117. https://doi.org/10.1016/j.cnsns.2017.05.015
Tejado, I., Pérez, E., & Valério, D. (2019). Fractional calculus in economic growth modelling of the group of seven. Fractional Calculus and Applied Analysis, 22, 139-157. https://doi.org/10.1515/fca-2019-0009
Traore, A., & Sene, N. (2020). Model of economic growth in the context of fractional derivative. Alexandria Engineering Journal, 59(6), 4843-4850. https://doi.org/10.1016/j.aej.2020.08.047