Overdeterrence of repeat offenders when penalties for first-time offenders are restricted

When penalties for first-time offenders are restricted, it is typically optimal for the lawmaker to overdeter repeat offenders. First-time offenders are then deterred not only by the (restricted) fine for a first offense, but also by the prospect of a large fine for a subsequent offense. Now suppose the restriction on penalties for first-time offenders is relaxed; i.e., larger fines for a first offense become enforceable. Should overdeterrence of repeat offenders now be reduced? We show that this is the case only if the original restriction was not very strong. Otherwise, overdeterrence of repeat offenders should actually be further amplified.