On an anomaly of the deterrent effect of punishment

A two period choice problem is presented. An individual has to choose in each period between legal and illegal activities. Two punishment policies are compared: (1) The maximal punishment is imposed even for a first offense. (2) A new offender gets a reduced punishment. It is proved that for a given policy of type 1 there exist a utility function and a policy of type 2 which yields a better deterrent on individual characterized by this utility function.