Fractional Dynamics of Natural Growth and Memory Effect in Economics

📝 Abstract

A generalization of the economic model of natural growth, which takes into account the power-law memory effect, is suggested. The memory effect means the dependence of the process not only on the current state of the process, but also on the history of changes of this process in the past. For the mathematical description of the economic process with power-law memory we used the theory of derivatives of non-integer order and fractional-order differential equation. We propose equations take into account the effects of memory with one-parameter power-law damping. Solutions of these fractional differential equations are suggested. We proved that the growth and downturn of output depend on the memory effects. We demonstrate that the memory effect can lead to decrease of output instead of its growth, which is described by model without memory effect. Memory effect can lead to increase of output, rather than decrease, which is described by model without memory effect.

💡 Analysis

A generalization of the economic model of natural growth, which takes into account the power-law memory effect, is suggested. The memory effect means the dependence of the process not only on the current state of the process, but also on the history of changes of this process in the past. For the mathematical description of the economic process with power-law memory we used the theory of derivatives of non-integer order and fractional-order differential equation. We propose equations take into account the effects of memory with one-parameter power-law damping. Solutions of these fractional differential equations are suggested. We proved that the growth and downturn of output depend on the memory effects. We demonstrate that the memory effect can lead to decrease of output instead of its growth, which is described by model without memory effect. Memory effect can lead to increase of output, rather than decrease, which is described by model without memory effect.

📄 Content

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European Research. 2016. No. 12 (23). P. 30-37.
DOI: 10.20861/2410-2873-2016-23-004

Fractional Dynamics of Natural Growth and Memory Effect in Economics

Valentina V. Tarasova Higher School of Business, 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: v.v.tarasov@bk.ru; tarasov@theory.sinp.msu.ru

Abstract: A generalization of the economic model of natural growth, which takes into account the power-law memory effect, is suggested. The memory effect means the dependence of the process not only on the current state of the process, but also on the history of changes of this process in the past. For the mathematical description of the economic process with power-law memory we used the theory of derivatives of non-integer order and fractional-order differential equation. We propose equations take into account the effects of memory with one-parameter power-law damping. Solutions of these fractional differential equations are suggested. We proved that the growth and downturn of output depend on the memory effects. We demonstrate that the memory effect can lead to decrease of output instead of its growth, which is described by model without memory effect. Memory effect can lead to increase of output, rather than decrease, which is described by model without memory effect.

Keywords: model of natural growth; memory effects; fading memory; fractional derivatives

1. Introduction Natural growth models are widely used in physics, chemistry, biology and economics. The economic models of natural growth are described by equations in which the marginal output (the growth rate of output) is directly proportional to income. A more realistic model is considered to be natural growth, which marginal output depends on the profit instead of the income.
   We first describe the simplest version of the economic model of natural growth that does not take into account the effects of time delay (lag) [1] and the memory effects [2, 3, 4, 5]. Let Y(t) be a function that described the value of output at time t. We use the approximation of the unsaturated market, implying that all manufactured products are sold. We also assume that the volume of sales is not large, and, therefore, does not affect the price of the goods, that is, the price P>0 is assumed a constant. Let I(t) be a function that describes the net investment, i.e. the investment that is used to the expansion of production. In the model of natural growth is assumed that the marginal income (d (P·Y(t))/dt) and the rate of output ((dY(t))/dt) are directly proportional to the value of the net investment, and we can use the accelerator equation dY(t) dt = 1 v · I(t), (1) 2

where v is a positive constant called the investment ratio and characterizes the accelerator power, 1/v - marginal productivity of capital (rate of acceleration), and dY(t)/dt is the first order derivative of the function Y(t) with respect to time. Assuming that the value of the net investment is a fixed part of the profit, which is equal to the difference between the income of P ·Y (t) and the costs C (t), we have the equation I(t) = m · (P · Y(t) −C(t)), (2) where m is the rate of net investment (0<m<1), i.e. the share of profit, which is spent on the net investment. We assume that the costs C(t) are linearly dependent on the output Y(t), such that we have the equation C(t) = a · Y(t) + b, (3) where a is the marginal costs, and b is the independent costs, i.e. the part of the cost, which does not depend on the value of output. Substituting expressions (2) and (3) into equation (1), we obtain dY(t) dt − m·(P−a) v · Y(t) = − m·b v . (4) Differential equation (4) describes the economic model of natural growth without memory and lag.
The general solution of differential equation (4) has the form Y(t) = b (P−a) + c · exp ( m·(P−a) v · t), (5) where c is a constant. Using the initial value Y(0) of function (5) at t = 0, we obtain c=Y(0)-b/ (P-a). As a result, we have the solution Y(t) = b (P−a) (1 −exp ( m·(P−a) v · t)) + Y(0) · exp ( m·(P−a) v · t). (6) Solution (6) of equation (4) describes the dynamics of output within the natural growth model without memory effects.

In the model of natural growth, which is described by equation (4), is supposed to perform the accelerator equation (1) that connects the net investment and the marginal value of output. Equations (1) and (4) contain only the first-order derivative with respect to time. It is known that the derivative of the first order is determined by the properties of differentiable functions of time only in infinitely small neighborhood of the time point. Because of this, the natural growth model (4) involves an instantaneous change of output speed when changing the net investment. This means that the model of na

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