Semi-Analytical Solutions for the Diffusive Kaldor–Kalecki Business Cycle Model with a Time Delay for Gross Product and Capital Stock

H. Y. Alfifi,Baogui Xin

doi:10.1155/2021/9998756

Abstract

This paper discusses the stability and Hopf bifurcation analysis of the diffusive Kaldor–Kalecki model with a delay included in both gross product and capital stock functions. The reaction-diffusion domain is considered, and the Galerkin analytical method is used to derive the system of ordinary differential equations. The methodology used to determine the Hopf bifurcation points is discussed in detail. Furthermore, full diagrams of the Hopf bifurcation regions considered in the stability analysis are shown, and some numerical simulations of the limit cycle are used to confirm the theoretical outcomes. The delay investment parameter and diffusion coefficient can have great impacts on the Hopf bifurcations and stability of the business cycle model. The investment parameters for the gross product and capital stock as well as the adjustment coefficient of the production market are also studied. These parameters can cause instability in, and the stabilization of, the business cycle model. In addition, we point out that, as the delay investment parameter increases, the Hopf bifurcation points for the diffusion coefficient values decrease considerably. When the delay investment parameter has a very small value, the solution of the business cycle model tends to become steady.

Highlights

For a long period of time, many significant nonlinear phenomena have been modelled and described via ordinary or partial differential equations (ODEs or PDEs)
A business cycle model is utilized to explain the working of economic laws and can be utilized to predict investment status, yield, costs, and other important factors in the business economic model
It uses a nonlinear model constructed with a couple of ODEs, where the nonlinearity of the investment and saving functions lead to periodic limit cycle results

Summary

Introduction

For a long period of time, many significant nonlinear phenomena have been modelled and described via ordinary or partial differential equations (ODEs or PDEs). We construct full-map diagrams of the Hopf bifurcation points and the stability analysis (stable and unstable regions) using examples of periods in limit cycle maps. All of these aims will help us explore the stability of the business cycle and predict whether policy makers’ targets will be met.

Methodology and Theoretical Analytical Framework

Unstable region α

Stable region

Diffusion coefficient

Findings

Numerical result