Optimal Imprisonment with General Enforcement of Law

1 Introduction

Becker (1968) famously showed that it is optimal to deter crime by combining the lowest possible probability (to economize on enforcement costs) with the severest possible penalty (to maintain adequate deterrence). This proposition remains the basic tenet in the optimal law enforcement literature, although several papers have identified specific circumstances in which this combination of instrument levels is not optimal (Garoupa 1997).

In an important contribution, Shavell (1991) argues that acknowledging the relevance of general instead of specific law enforcement can align prescriptions from theory with the real-world observation of non-maximal sanctions. According to Shavell (1991), specific enforcement concerns apprehending and penalizing individuals who have committed a single kind of criminal act as identified by its level of social harm (e.g. enforcers who write tickets for overtime parking). In contrast, general enforcement enables apprehending and penalizing individuals who have committed any of a range of acts (e.g. a police officer on patrol may assist in the conviction of perpetrators of minor or major crimes).^[1] Shavell (1991) reports that the maximum imprisonment term is socially optimal when enforcement is specific. For each act in isolation, the Becker proposition applies, that is, any expected sanction should comprise the severest sanction possible in order to economize on the costs of enforcement. In contrast, he finds that, if enforcement efforts are general, optimal imprisonment rises with the act’s harmfulness and is equal to the maximum term only for the most harmful acts. The intuition is that, under general enforcement, it is impossible to tailor the detection effort to the different acts, creating a need to adjust the sanction to the acts’ severity. These results have become a corner stone in the analysis of optimal law enforcement.^[2]

This paper challenges this conventional wisdom. We show that optimal imprisonment is either zero or maximal when the distribution of criminal benefits exhibits an increasing hazard rate, that is, fulfills the monotone hazard rate (MHR) assumption. Not only is this property of the distribution of a random variable usually assumed to hold in many fields of applied microeconomics (including law and economics), it is also fulfilled for a wide range of actual distribution functions.^[3] The intuition for the extreme-imprisonment result is that welfare evolves in a U shaped manner with the imprisonment term when MHR applies, signifying that the interior solution minimizes welfare.

With the MHR assumption, the optimal pattern of enforcement and imprisonment is as follows: (i) no enforcement and any imprisonment term at all harms level, or (ii) positive enforcement and no (maximum) imprisonment for acts that cause harm below (above) a critical harm level. As a result, both specific and general enforcement of law are associated with the use of only extreme sanctions.

Whereas the result by Shavell (1991) that imprisonment increases in the harmfulness of the act remains conventional wisdom to this day, Kaplow (1990) already mentioned that interior solutions may require possibly strong assumptions. Our paper adds to Kaplow (1990), who analyzes only a single act (i.e. specific enforcement), by identifying the precise characteristics of the distribution functions that induce boundary solutions, by highlighting that these characteristics are taken for granted in many strands of the literature, and by describing the optimal regime when enforcement is general. Whereas Kaplow (1990, p. 247) states that the “most salient factor relevant to whether an intermediate sanction is optimal is the level of the social cost” of punishment, we suggest that the shape of the distribution function is the decisive factor.^[4]

The present contribution is complementary to D’Antoni, Friehe, and Tabbach (2023), where we describe how the unobservability of wealth changes the optimal combination of fines and imprisonment when enforcement is specific. In the present paper, wealth levels are irrelevant as we focus on imprisonment as the only kind of sanction in order to understand optimal imprisonment when enforcement is general (i.e. when there is a range of acts to consider).

2 Model

We draw on the model suggested by Shavell (1991) in which risk-neutral individuals choose whether to commit harmful acts by comparing the expected sanction with the criminal benefit. Individuals differ regarding this benefit b and the act’s harmfulness denoted h. The benefit is distributed on the support B ⊆ [0, ∞) according to the cumulative distribution function F(b) and density F′(b) = f(b). We also refer to the hazard rate function

and assume that it is non-decreasing. This is the widely used monotone hazard rate assumption (see, for example, Tirole 1994, p. 156). The harm is distributed on the support [0, ∞) according to G(h) as the cumulative distribution function and g(h) as density. We assume that the distribution of benefits is the same for different harm levels.

If an individual commits an act, she will be detected with probability p and receive an imprisonment term z. We focus on general enforcement, so p cannot depend on h. Let y denote the costs of enforcement and assume that p′(y) > 0 and p″(y) ≤ 0, with p(0) = 0. The imprisonment term can depend on the level of harm (i.e. the kind of act) so we write z(h). Imprisonment cannot exceed an upper limit z ̄ , which may reflect physical limits or moral constraints and is the same for all individuals.

Imprisonment is costly for the convicted offender and for society. The cost per-unit amount to σ > 1. They comprise the individual’s per-unit cost (equal to 1) and the additional social costs per unit (σ − 1 > 0).

3 Analysis

An individual will commit an act if and only if her benefit exceeds the expected sanction, that is, if b > p(y)z(h), where z(h) is the imprisonment term as a function of harm. Understanding this decision criterion, the social planner chooses enforcement efforts y and imprisonment z(h) to maximize social welfare W defined to be the benefits individuals obtain from committing acts, less the harm done, less the social costs of imposing imprisonment, less the cost of enforcement. Formally,

where

is social welfare at each level of harm.

When enforcement efforts are positive, the marginal welfare effect from a longer imprisonment term results as

where the first term reflects the marginal benefit from deterring the marginal offender, and the second term reflects the marginal cost from increasing imprisonment on those who will continue to offend even at the higher sanction level. A higher imprisonment term increases the term h + pz(σ − 1) in the marginal benefits and lowers the marginal costs of imprisonment as it reduces the population of undeterred individuals.

Shavell (1991) considers an interior solution for the level of imprisonment (see equation (A17) in Shavell 1991, p. 1105). Assuming that the marginal welfare effect in (3) is equal to zero, we can also state the necessary requirement as

using the hazard rate H. When we derive this term with respect to z for the second-order condition, we obtain (after dividing by p)

which is positive as long as the hazard rate function is non-decreasing (i.e. H′(b) ≥ 0). A non-decreasing monotone hazard rate indicates that the ratio between the marginal probability of deterrence (captured by the criminal benefit density function f(b)) to the crime rate (captured by the survival function 1 − F(b)) is non-decreasing. A very intuitive understanding results from noticing that the hazard rate is equal to the absolute value of the elasticity of the crime rate with respect to the expected sanction S divided by the expected sanction S , that is,

Assuming H′(b) ≥ 0, our result rules out that an interior solution represents a global maximum, and points to boundary solutions. Namely, z(h) is either zero or z ̄ , with z(h) = 0 if and only if

After establishing that optimal imprisonment results in a corner solution under the MHR assumption, we are now ready to characterize the general pattern of imprisonment ^[5]

4 Discussion and Conclusion

According to the conventional wisdom, optimal imprisonment increases with the harmfulness of the act when enforcement effort is general, whereas optimal imprisonment is necessarily maximal when enforcement effort is specific. For the case of general enforcement, this paper shows that optimal imprisonment is either zero or maximal when the distribution of benefits fulfills a widely used and intuitive characteristic that is also fulfilled by many regularly used distribution functions. Our result implies that general as opposed to specific enforcement cannot unconditionally improve the alignment of theoretically prescribed and practically observed sanction regimes.^[6]

Although our conclusions do not invalidate widely accepted results, they put them into perspective by pointing out that their validity relies on hypotheses that are implicit, unconventional, and could be easily violated. As clarified by our analysis, reliance on internal solutions in our context requires a discussion of the implied shape of the hazard rate of the criminal benefits.

This raises the issue of identifying circumstances under which the hazard rate decreases, such that an interior solution may result. We suggest some possibilities that could be further explored.

A first possibility is assuming a distribution of benefits that is “fat-tailed”, for example, because it is a Pareto distribution.^[7] The Pareto distribution is used in finance, to deal with catastrophic events, and in the analysis of income and wealth distributions, as it is consistent with empirical data for that context.^[8] It can be debated in which contexts a fat-tailed distribution can reflect criminal benefits.

A second possibility is that individuals may be unable to completely control their behavior and may commit a crime inadvertently.^[9] If there is a probability that individuals commit a harmful act even when they do not want to (and, ex post, it is not possible or socially desirable to discriminate intentional and unintentional acts), then the frequency of violation will stop declining as sanctions increase even before full deterrence is reached. This amounts to a declining hazard rate.

Finally, some individuals may not engage in a rational cost-benefit comparison and thus be unresponsive to incentives provided by sanctions. This implies that some individuals remain undeterred at all levels of punishment. This kind of “irrational” behavior by some individuals may be realistic in some cases of criminal behavior (e.g. drug addicts), although it deviates from the standard economic framework. In these instances, an interior solution to imprisonment in the general enforcement model is possible.