A dynamic IS-LM model with delayed taxation revenues

Some recent contributions to Economic Dynamics have shown a new interest for delay differential equations. In line with these approaches, we re-proposed the problem of the existence of a finite lag between the accrual and the payment of taxes in a framework where never this type of lag has been considered: the well known IS-LM model. The qualitative study of the system of functional (delay) differential equations shows that the finite lag may give rise to a wide variety of dynamic behaviours. Specifically, varying the length of the lag and applying the 'stability switch criteria', we prove that the equilibrium point may lose or gain its local stability, so that a sequence of alternated stability/instability regions can be observed if some conditions hold. An important scenario arising from the analysis is the existence of limit cycles generated by sub-critical and supercritical Hopf bifurcations. As numerical simulations confirm, if multiple cycles exist, the so called 'crater bifurcation' can also be detected. Economic considerations about a stylized policy analysis stand by qualitative and numerical results in the paper