Stability and Hopf bifurcation for a business cycle model with expectation and delay

• A nonlinear business cycle model with expectation is formulated based on rational expectation hypothesis.
• Theoretical results including stability, Hopf bifurcation and its direction and stability are obtained.
• The effects of expectation and delay on nonlinear business cycle model are investigated.
• Simulations are conducted to discuss the dynamic behaviors of the business cycle model.
• Some suggestions on inhibiting the abnormal economic fluctuations are proposed.

According to rational expectation hypothesis, the government will take into account the future capital stock in the process of investment decision. By introducing anticipated capital stock into an economic model with investment delay, we construct a mixed functional differential system including delay and advanced variables. The system is converted to the one containing only delay by variable substitution. The equilibrium point of the system is obtained and its dynamical characteristics such as stability, Hopf bifurcation and its stability and direction are investigated by using the related theories of nonlinear dynamics. We carry out some numerical simulations to confirm these theoretical conclusions. The results indicate that both capital stock’s anticipation and investment lag are the certain factors leading to the occurrence of cyclical fluctuations in the macroeconomic system. Moreover, the level of economic fluctuation can be dampened to some extent if investment decisions are made by the reasonable short-term forecast on capital stock.