Multiple Equilibria in an Optimal Control Model for Law Enforcement

In this paper, Becker's (1968) economic approach to crime and punishment is extended by including intertemporal aspects. We analyze a one-state control model to determine the optimal dynamic trade-off between damages caused by offenders and law enforcement expenditures. By using Pontryagin's maximum principle we obtain interesting insight into the dynamical structure of optimal law enforcement policies. It is found that inherently multiple steady states are generated which can be saddle-points, unstable nodes or focuses and boundary solutions. Moreover, thresholds (so-called Skiba points) between the basins of attraction are discussed. A bifurcation analysis is carried out to classify the various patterns of optimal law enforcement policies.