We derive the sentence--choosing among the sanctions of prison, parole, and probation--that achieves a target level of deterrence at least cost. Potential offenders discount the future disutility of sanctions and the state discounts the future costs of sanctions. Prison has higher disutility and higher cost per unit time than parole and probation, but the cost of prison per unit of disutility can be lower or higher than the cost of parole and probation per unit of disutility. The optimal order of sanctions depends on the relative discount rates of offenders and the state, and the optimal duration of sanctions depends on the relative costs per unit of disutility among the sanctions and on the target level of deterrence. In the case we focus on, in which potential offenders discount the disutility of sanctions at a higher rate than the state discounts the costs of sanctions, we demonstrate that (a) it is optimal for prison to precede parole if both sanctions are used; and (b) it may be optimal to employ prison even if prison has a higher cost per unit of disutility than parole and probation and even if prison is not needed to achieve the deterrence target.
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