Nonlinear multiplier–accelerator model with investment and consumption delays

• One single delay, either a consumption delay or an investment delay, is harmless.
• In case of the two delays, the stability switching curves are analytically constructed.
• It is numerically confirmed that the delay dynamical system can produce limit cycles and the stability switch repeatedly occurs.

This paper shows how cyclic dynamics of national income can emerge in the multiplier–accelerator model with continuous time scale when delays in investment and consumption are present. An S-shaped functional form of investment and a linear consumption function are adopted to illustrate the phenomenon and to compute the stability-switching curves on which a stability gain or loss occurs. Assuming that the equilibrium national income is locally stable if there are no delays, it is demonstrated that one delay is harmless and with two delays, the system can produce limit cycles and the stability switch repeatedly occurs when one of the delays increases and the other is kept to be positive constant.