Simple-zero and double-zero singularities of a Kaldor-Kalecki model of business cycles with delay
The main aim of the paper is to study a variant of the Kaldor-Kalecki model of business cycles with delay in both the gross product and the capital stock. This work concerns the broadly discussed issue of delayed differential equations in the economic context of business cycles models. The investigated model is one of the widely recognized models of business cycles. It incorporates the ideas of Kaldor (a model of business cycles), Kalecki (the delay between an investment and a business decision) and Krawiec and Szydowski (on whose work the author bases much of his discussion), who proposed to study the ideas of Kaldor and Kalecki as one model, allowing for further modifications. The paper is organized as follows. In Section 1, an introduction is given, with particular emphasis on the economic context. The equations of Kaldor and Kalecki are written out explicitly and several references are given that discuss some of its properties. Then, a modification of Kaddar and Talibi Alaoui is presented. This modification introduces delays in both the gross product and capital stock. Kaddar and Talibi Alaoui studied the characteristic equation of the linear part of the Kalecki-Kaldor system of equations with the proposed modification (further we refer to it as the characteristic equation) at an equilibrium point and used the delay as a bifurcation parameter. In this way, they showed that a Hopf bifurcation may occur under certain conditions, when the delay passes some critical values. Their work was further extended by Wang and Wu. However, neither of the authors discussed before the issue that the characteristic equation can have a simple-zero root, a double-zero root or a simple-zero root and a pair of purely imaginary roots. The analysis of this issue is the main motivation of the described paper by Wu. Section 2 regards a detailed presentation of the distribution of eigenvalues of the linear part of the modified Kaldor-Kalecki model at an equilibrium point. The bifurcation parameters are the delay (denoted by tau) and a newly defined parameter (denoted by k), which is related to the derivative of the investment function. Several properties of the characteristic equation are discussed and conditions for the occurence of simple-zero roots, double-zero roots etc. are given in a series of lemmas. In Section 3, the author briefly summarizes the theory of center manifold reduction for general delayed differential equations. This theory serves further to perform an analysis of the simple-zero singularities (Section 4) and double-zero singularities (Section 5). The normal forms of these singularities are obtained on the center manifold. The normal forms for the double-zero singularities are used in Section 6 to construct the bifurcation diagrams of the considered system of differential equations. The curves on which the system undergoes different types of bifurcations (fold, Hopf, pitchfork, homoclinic bifurcations) are derived. A detailed discussion is provided. Section 7 presents the results of numerical simulations with the aim of verifying the theoretical results on the bifurcation diagrams, derived in Section 6. Several of these theoretical expectations are confirmed. The paper provides a list of 24 references with most of the relevant work in the area of delayed differential equations in the context of business cycles.
scientific article; zbMATH DE number 5701476