Numerical Analysis of Modified Kaldor-Kalecki Model with Couplings and Delays

Abstract

Modified Kaldor-Kalecki-type model of business cycles with delays are considered. Unidirectional and bidirectional couplings are introduced to investigate relationships between three "global" markets and two "local" markets. Selected results of an extensive numerical analysis are presented. DOI: 10.12693/APhysPolA.129.1008 PACS/topics: 89.65. Gh, 88.05.Lg, 02.60.Lj, 05.45.Xt Investigations of macroeconomic cycles have a quite long and interesting history, see [1][2][3] for details. It seems that especially fruitful are various versions and generalizations of Kaldor-Kalecki models [4][5][6][7]. Quite recently we have proposed another generalization that takes into account the interactions between two economies [8]. Such interactions are inevitable in real life situations and in the present paper we develop a more realistic model of three "global" and two "local" economies mutually interacting in a way resembling actual economical influences and relationships between countries on the global market.Let us consider five "economies", each characterized by a pair (y i , y i+1 ), where the first element y i stands for the gross domestic product (GDP), the second element y i+1 is the capital stock, and i=1,3,5,7,9. Three of these systems, namely (y 1 , y 2 ), (y 7 , y 8 ), and (y 9 , y 10 ) represent "global" economies, say the European Union, China, and the USA, respectively. The remaining two systems: (y 3 , y 4 ) and (y 5 , y 6 ) represent "local" economies, say, Poland and Germany. Later on we will justify such a choice by a proper adjustment of relevant parameters.Following our previous arguments [8] we construct a set of ten equations describing five mutually interacting economies with delays and both unidirectional and bidirectional couplings:−s 5 (y 9 (t) − y 1 (t)) + s 6 (y 7 (t) − y 1 (t)),−y 3 (t)) + s 10 (y 7 (t) − y 3 (t)) − s 12 (y 5 (t) − y 3 (t)), y 5 = α 3 (F 3 (t) − δ 3 y 6 (t) − γ 3 y 5 (t)),−y 5 (t)) + s 9 (y 7 (t) − y 5 (t)) − s 13 (y 3 (t) − y 5 (t)),ẏ 7 = α 4 (F 4 (t) − δ 4 y 8 (t) − γ 4 y 7 (t)),−y 7 (t)) − s 11 (y 1 (t) − y 7 (t)), y 9 = α 5 (F 5 (t) − δ 5 y 10 (t) − γ 5 y 9 (t)),−y 9 (t)) + s 7 (y 7 (t) − y 9 (t)).Here F i (t) (i = 1, . . . , 5) are the investment functions, s j (j = 2, . . . , 13) are the coupling coefficients, α k (k = 1, . . . , 5) are the adjustment coefficients in the good market, δ ∈ (0, 1) is the depreciation rate of capital stock, γ l (l = 1, . . . , 5) and δ m (m = 1, . . . , 5) are constants, and τ is a time delay fixed at τ = 3.We assume that the investment functions for global markets such as the European Union, Chinese, and American economy have logistic character, but the investment function for local markets have trigonometric characteristics, i.e., for Poland we use a sine and for Germany we have a hyperbolic tangent. Thus F 1,4,5 (t) = e y1,7,9(t) /(1 + e y1,7,9(t) ), F 2 (t) = 0.8 sin(y 3 (t)), and F 3 = 0.5 tgh(y 5 (t)). Being global or local economy is determined by respective values of α k parameters, for which we make a natural assumption that this corr...