Abstract
In this paper, we consider a model of economic growth with a distributed time-delay investment function, where the time-delay parameter is a mean time delay of the gamma distribution. Using the linear chain trick technique, we transform the delay differential equation system into an equivalent one of ordinary differential equations (ODEs). Since we are dealing with weak and strong kernels, our system will be reduced to a three- and four-dimensional ODE system, respectively. The occurrence of Hopf bifurcation is investigated with respect to the following two parameters: time-delay parameter and rate of growth parameter. Sufficient criteria on the existence and stability of a limit cycle solution through the Hopf bifurcation are presented in case of time-delay parameter. Numerical studies with the Dana and Malgrange investment function show the emergence of two Hopf bifurcations with respect to the rate growth parameter. In this case, we have been able to detect the existence of stable long-period cycles in the economy. According to the time-delay and adjustment speed parameters, the range of admissible values of the rate of growth parameter breaks down into three intervals. First, we have stable focus, then the limit cycle and finally again the stable solution with two Hopf bifurcations. Such behavior appears for some middle interval of the admissible range of values of the rate of growth parameter.
Highlights
In economics, many processes depend on past events, so it is natural to use time-delay differential equations to model economic phenomena
In [38], we proposed an economic growth model where the average time of investment completion is replaced by a distributed time length of investment
We consider two simplest cases of three- and four-dimensional dynamical systems obtained through the linear chain trick from the Kaldor–Kalecki growth model with the distributed delay, corresponding to the weak and strong kernels, respectively
Summary
Many processes depend on past events, so it is natural to use time-delay differential equations to model economic phenomena. We consider two simplest cases of three- and four-dimensional dynamical systems obtained through the linear chain trick from the Kaldor–Kalecki growth model with the distributed delay, corresponding to the weak and strong kernels, respectively. For both models, we establish conditions for the existence of Hopf bifurcation with respect to the time-delay parameter. In Economic growth cycles driven by investment delay, Krawiec and Szydłowski [26] formulated the model based on the Kaldor business cycle model with two modifications: exponential growth introduced by Dana and Malgrange [27] and Kaleckian investment time delay [18]. We will analyze the stability and Hopf bifurcation of systems (7)–(9) and (10)–(13) by determining eigenvalues of linearized systems around the critical point (y∗, y∗, k∗) and (y∗, y∗, y∗, k∗), respectively
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