Zeno points in optimal control models with endogenous regime switching

Abstract

The present paper considers a type of multi-stage problem where a decision maker can optimally decide to switch between two alternating stages (regimes). In such a problem, either one of the two stages dominates in the long run or it is optimal to infinitely switch between stages. A point where such a phenomenon occurs with optimal switching time zero is called Zeno point.By a reformulation of the multi-stage problem using an additional binary control to account for the alternating stages, the necessary matching conditions for optimal switching are derived. A relaxation with a linear control is considered to analyze the implications of the occurrence of a singular solution. In case of the control becoming singular, the matching conditions are fulfilled, i.e. it is optimal to switch between the stages. If the time derivative of the matching conditions is zero, it is shown that trajectories approaching this point are tangential to the switching curve. Furthermore, if the trajectories move into opposing directions, this implies Zeno behavior.A two-stage version of a standard capital accumulation problem is analyzed and a Zeno point is found analytically. Numerical calculations are performed to visualize the Zeno phenomenon. The economic relevance of the phenomenon is discussed.