Stability and Hopf bifurcation for Kaldor-Kalecki model of business cycles with two time delays

According to rational expectation hypothesis, the government will take into account the future capital stock in the process of investment decision. Business cycle, named economic cycle, refers to the total alternate expansion and contraction in economic activities, and these cyclical changes will appear in the form of fluctuation of the comprehensive economic activity indicators such as gross national product, industrial production index, employment and income. In the early research of nonlinear business cycle theories, Kaldor [1] assumed that investment depended on gross product and capital stock and proved that, through graph analysis, time-varying nonlinear investment and saving function could result in a business cycle. Chang and Smyth [2] summarized Kaldor's theory on business cycle, established a nonlinear dynamic system which described the gross national product and capital stock changed over time, gave necessary conditions of existence of limit cycles, and provided a rigorous mathematical proof of Kaldor's model proposed in 1940. On this basis, Grasman and Wentzel [3] further improved Kaldor's business cycle model by considering capital loss speed. Along another view, according to the IS-LM model raised by J.R. Hicks and A.H. Hansen, Ackley [4] established a complete Keynes system which reflects the gross domestic product and interest rate changes over time, which is also called standard IS-LM model. Kalecki [5] first considered investment delay in the business cycle model in 1935, and he claimed capital equipment needed a conceived cycle or delay from installation to production. As the theory of delay functional differential equations gradually become more accomplished in 1990s, Krawiec and Szydlowski [6,7] first made a qualitative analysis of the impact of the investment delay on the business cycle. In addition, some other scholars investigated the dynamic properties like stability, many types of bifurcation, existence and stability of periodic solutions in Kaldor–Kalecki model with investment delay [8,9]. The record of business cycle has been kept relatively well during the last 200 years, and business cycle theory, as the core issue of macroeconomics, has been attracting the widespread interests of many economists. The modern business cycle theory can be traced back to the masterpiece of Keynes's theory "The General Theory of Employment, Interest, and Money". Keynes discussed the formation of the business cycle from the perspective of psychological factors based on national income theory. The following system was formulated by Krawiec and Szydlowski [10] who combined two basic models of business cycle: the Kaldor model and the Kalecki model.