Fractional Dynamics of Natural Growth and Memory Effect in Economics

TL;DR

This paper introduces a fractional calculus-based model for economic growth that incorporates power-law memory effects, showing how past states influence current growth and can lead to both increases and decreases in output.

Contribution

It develops a novel fractional differential equation model for natural economic growth that explicitly accounts for memory effects, extending traditional models.

Findings

Memory effects can cause output decrease or increase.

Fractional derivatives effectively model long-term memory in economic processes.

Solutions demonstrate the impact of memory on growth dynamics.

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.