Concept of dynamic memory in economics

Communications in Nonlinear Science and Numerical Simulation.
2018. Vol. 55. P. 127–145. DOI: 10.1016/j.cnsns.2017.06.032

CONCEPT OF DYNAMIC MEMORY IN ECONOMICS

Valentina V. Tarasova, Lomonosov Moscow State University Business School, Lomonosov Moscow State University, Moscow 119991, Russia; E-mail: v.v.tarasova@mail.ru; Vasily E. Tarasov, Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia; E-mail: tarasov@theory.sinp.msu.ru

Abstract: In this paper we discuss a concept of dynamic memory and an application of fractional calculus to describe the dynamic memory. The concept of memory is considered from the standpoint of economic models in the framework of continuous time approach based on fractional calculus. We also describe some general restrictions that can be imposed on the structure and properties of dynamic memory. These restrictions include the following three principles: (a) the principle of fading memory; (b) the principle of memory homogeneity on time (the principle of non-aging memory); (c) the principle of memory reversibility (the principle of memory recovery). Examples of different memory functions are suggested by using the fractional calculus. To illustrate an application of the concept of dynamic memory in economics we consider a generalization of the Harrod-Domar model, where the power-law memory is taken into account. MSC: 91B02; 91B55; 34A08; 26A33 Keywords: economics; dynamic memory; power-law memory; fading memory; multiplier; accelerator; fractional derivative; fractional integral; fractional dynamics; fractional calculus

1. Introduction

Concept of memory is actively used not only in psychology, but also in the modern physics [1-13]. Economic processes with memory are actively studied in recent years (for example, see [14- 24]). Fractional calculus and fractional differential equation [25-28], which are used derivatives and integrals of non-integer orders, are convenient tools to describe processes with memory in physical sciences (for example, see [11, 12] and references therein). In economics, the memory was first 2

related to fractional differencing and integrating by Granger and Joyeux [20], and Hosking [21] in the framework of the discrete time approach [22, 23, 24]. Fractional differencing and integrating, which are suggested in these papers, are not directly connected with the fractional calculus or the well-known finite differences of non-integer orders [29]. In article [29] it was shown that the fractional differencing and integrating, which are proposed in [22, 23, 24], are the well-known Grunwald-Letnikov fractional differences. These differences have been suggested one hundred and fifty years ago. Recently fractional calculus has been used to describe economic processes with nonlocality in [30-35]. Fractional differential equations have been also applied for continuous-time finance in [36-56] by using the econophysics framework. These papers consider only the financial processes. The basic economic notions and the concepts of economic processes with memory are not considered. Economic processes with power-law memory have been considered in [57-75] in the framework of the continuous time approach. Using the fractional calculus as a mathematical tool to describe the power-law memory, we proposed generalizations of some basic economic concepts [57-75]. We have suggested the marginal value of non-integer order [59, 60, 61], the concepts of accelerator and multiplier with memory [62, 63], the elasticity [64] and the measures of risk aversion [65, 66] for processes with power-law memory, the methods of deterministic factor analysis [67]. The natural growth and logistic models, the Harrod-Domar and Keynes models have been generalized by taking account the power-law memory in [68-75]. These papers consider only simplest case of dynamic memory that is represented by the Riemann-Liouville fractional integrals and the Caputo fractional derivatives. This leads to questions about the applicability of other types of fractional derivatives and integrals to describe memory effects in economics. It is necessary to have a detailed consideration of the concept of dynamic memory for application in economics in the framework of the continuous time approach and fractional calculus. Dynamic memory can be considered as an averaged characteristic (property), which describes the dependence of the process at a given time on the states in the past. Economic process with memory is a process, where economic parameters and factors (endogenous and exogenous variables) at a given time depend on their values at previous instants of a time interval. The economic process with dynamic memory assumes awareness of economic agents about the history of this process. In such processes, the behavior of economic agents is based not only on information on the state of the process {t, X(t)} at a giv

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