Hyperbolic Punishment Function

All models in Law and Economics use punishment functions (PF) that incorporates a trade-off between probability of detection, p, and punishment, F. Suppose society wishes to minimize the total costs of enforcement and damages from crime, T (p; F). For a given p, an optimal punishment function (OPF) determines an F that minimizes T(p; F). A popular and tractable PF is the hyperbolic punishment function (HPF). We show that the HPF is an OPF for a large class of total cost functions. Furthermore, the HPF is an upper (lower) bound for an even larger class of punishment functions. If the HPF cannot (can) deter crime then none (all) of the PF's for which the HPF is an upper (lower) bound can deter crime. Thus, if one can demonstrate that a particular policy is ineffective (effective) under the HPF, there is no need to even compute the OPF. Our results should underpin an even greater use of the HPF. We give illustrations from mainstream and behavioral economics.