Optimal Penalties for Repeat Offenders – The Role of Offence History

1 Introduction

This paper is in the economics of crime literature tradition building upon the fundamental work of Becker (1968). Here, the decision on whether to commit a crime is modelled as a rational choice of the potential offender. It is rational in that it is based on utility maximization balancing expected benefits from crime against expected costs. There is an extensive literature on the benefits and costs of crime starting with the seminal papers by Ehrlich (1973, 1975) highlighting the importance of the expected size of punishment. ^[1]

Of course, criminals impose costs on society, particularly the damage done to their victims but also indirect costs like higher uncertainty in economic and social interactions. Consequentially, society allocates scarce resources to crime prevention. Consistent with the understanding of the potential criminals’ decisions sketched above, the decision of society on how much to spend on crime prevention is also interpreted to be a rational one balancing benefits of crime reduction and its costs. ^[2]

Obviously, the decisions of potential criminals and the decisions of the designers of criminal law are highly interdependent. Not only do potential criminals acknowledge the size and structure of penalties as an important determinant of their decisions for (against) crime. Also, the lawmakers should take the cost-benefit-analysis of potential criminals into account in the making of criminal law. Any assessment of the deterrence effect of a certain form and size of punishment (and thereby of the benefits of this punishment) must be based on a hypothesis about how potential criminals might adjust their decisions to this specific punishment.

To date, most of the contributions to the literature tradition referred to above use static economic models. A price to be paid for the analytical simplicity thereby achieved is that certain intertemporal aspects of crime and punishment cannot be satisfactorily analysed. ^[3] One of these aforementioned aspects is the intertemporal profile of punishment for repeat offenders. ^[4]

A central question in this context is: “We observe that repeat offenders are punished more severely than first time offenders. What (if any) is the economic rationale for this escalating penalties structure?”

Papers that explore the escalating (or synonymously: increasing) penalties question address two issues: First, how does a potential criminal react to alternative intertemporal punishment structures for first time and repeat offenders? Second, what is the optimal intertemporal structure of penalties (given the criminal’s cost benefit analysis explored in the first step)? The optimality concept used for the choice of the intertemporal penalties structure varies within the “escalating penalties literature”. Most authors assume that the lawmaker strives to design the penalty structure which maximizes social welfare in that the social cost of crime is minimized. ^[5] Typically the social cost of crime consists of the damage suffered by the victims and the cost of crime prevention. Some authors differentiate the cost of crime reduction distinguishing between the cost of apprehending offenders and the cost of punishment. ^[6] Others differentiate the benefits of crime reduction subtracting the utility losses suffered by the criminals from the gains of the potential victims which are spared. ^[7]

Most of the contributions to the escalating penalties literature use a finite game framework and confine the analysis to two periods and two sanction levels, one for first time and one for repeat offenders. ^[8] The advantage of this framework is analytical simplicity. The drawback is well-known in the game theoretic literature: The approach suffers from all kinds of “finite horizon paradoxes” (Rubinstein 1998, 165), the discussion of which began with the seminal contributions of Luce and Raiffa (1957) and Selten (1978), and is ongoing.

Apart from this methodological problem the results produced by the established escalating penalties literature are puzzling, in that a decreasing penalty scheme (one that punishes repeat offenders less severely than first time offenders) is optimal at least for certain parameter values in many of these finite models. ^[9] However, this kind of penalty scheme is nowhere to be observed in reality, and seems implausible.

To solve the “puzzle of escalating penalties” (Dana 2001), we analyse the impact of the time horizon on the structure of the optimal penalty schemes. In the main parts of the paper (Sections 2–5) we extend the model of the “criminal career” of an individual from the two-period case presented in the literature to an infinite game framework in order to formalize the unknown endpoint of lifetime more appropriately. Specifically, we assume that a decision maker who is alive in period t expects still to be alive in the next period with a positive probability q<1. ^[10],[11]

In Section 6 we consider a corresponding traditional two-period framework and show that in this finite model the optimal penalty scheme might be of the escalating as well as of the decreasing type as it is also the case in Emons (2007), Friehe (2009), Miceli (2013) and Polinsky and Rubinfeld (1991). This highlights the fact that the time horizon may have a crucial impact on the optimal penalty structure. ^[12]

The infinite game framework is similar to the one of Endres and Rundshagen (2012). However, opposed to the paper at hand, in Endres and Rundshagen (2012) only two sanction levels are considered, i. e., the sanction for repeat offenders is not allowed to depend on the number of previous records, which is an important feature of the paper at hand. This allows us to rule out the optimality of hybrid penalty schemes, which combine decreasing and increasing steps of penalization. Moreover, Endres and Rundshagen (2012) do not compare their outcome with a corresponding finite game, which is a central part of the paper at hand.

In both frameworks of this paper, the infinite and the finite one, the game is modelled in three stages. In the first two stages the legal system is chosen. First a regulator, who strives to minimize the sum of harm from crimes and cost of punishment chooses a target criminality level and in the second stage he chooses the corresponding optimal penalty scheme. In the third stage which comprises the infinite game described above or a traditional two-period game, respectively, potential offenders decide in each period of their life whether they commit a crime.

For the infinite game framework we derive three main results: First, any decreasing or hybrid penalty scheme can be replaced by a (weakly) escalating penalty scheme that leads to the same criminal activity and the same social penalization cost. Second, for any (exogenously) given aspired criminality level (chosen in stage one), the cost minimizing penalty scheme (equilibrium of stage two) is monotonically increasing in the number of previous police records but not strictly increasing. Third, although crime prevention is cheaper under optimal escalating penalty schemes than under uniform penalty schemes, the socially optimal level of crime may be higher than the level which would be optimal under the restriction of uniform penalties.

Moreover we show that there is a fundamental difference between the equilibrium strategies in the finite and infinite game framework. In the infinite game a decreasing sanction scheme never leads to higher welfare than a uniform penalty scheme. The driving force behind this result is the fundamental divergence between the equilibrium strategy of a potential offender under a decreasing sanction scheme in a game with ex ante known number of periods, and this strategy in an infinite game. In the finite game under a decreasing penalty scheme the decision of the potential offender in the second (or a later) period, depends on whether the offender has been convicted for previously committed crimes. However, in the infinite game a potential offender either commits the crime in each period or in no period under a decreasing penalty scheme as well as under a uniform penalty scheme. In both cases he commits the crime if and only if his benefit from committing the crime exceeds his average expected penalty. Thus, for any decreasing penalty scheme which induces the same criminality level as a given uniform penalty, the average expected penalty coincides with the uniform penalty. Hence, the critical benefit that separates offenders from non-offenders is identical under both penalty schemes, too. This equality also implies identical penalization costs. To prove this conjecture we proceed as follows: for a given penalty scheme we relate the decision of a potential offender to commit a crime to a sequence of critical benefits {bn}n=0,1,…. For criminal record level n an outsider will observe that the offender commits the crime if and only if his benefit is larger than bn. We then show that each penalization scheme that leads to the same sequence of critical benefits (as it is the case, e. g. for a decreasing and corresponding uniform penalty scheme) implies identical penalization cost. Moreover, for the infinite game we show that each set of penalty schemes that lead to the same sequence of critical benefits contains an escalating penalty scheme (including the uniform penalty scheme as border case). Hence the task of finding the overall optimal penalty scheme is reduced to the task of finding the optimal penalty scheme within the set of escalating ones (including the uniform penalty schemes as border case). We show that the structure of the unique optimal penalty scheme is of the following type. There exists at most one criminal record level n0 for which the penalty level is non-degenerate (0<sn0<smax). All the other records result in either zero penalty (sn=0 for n<n0), or the maximum penalty (sn=smax for n>n0).

The intuition behind this result may be explained as follows. Consider a uniform or escalating penalty scheme s which induces the maximum tolerable criminality level, as chosen by society. Now consider a change from this scheme to a scheme of escalating penalties sˆ which just induces the same criminality level but differs from s by a lower penalty for offenders with a low criminal record and a higher penalty for offenders with a high criminal record. The change in the penalties must be such that the increased criminal activity of offenders with low criminal record is exactly offset by the decrease in the criminal activity of offenders with high criminal record. However, even though the level of crime is unaffected by the envisaged change in the penalty structure, the aggregate cost of punishment goes down: the cost of punishment for the additional criminal activity of offenders with low criminal record is lower than the saved cost of punishment for the reduced criminal activity of offenders with high criminal record. This is so since the (new) penalty for offenders with low criminal record is lower than the (old) penalty for offenders with high criminal record and thus the aggregate penalization effort is reduced. ^[13],[14]

In Section 6 we demonstrate that in the finite game framework, as well as in the infinite one, cost might be reduced by using escalating instead of uniform penalties. However, due to the final round effect, different from the infinite scenario, they also might be reduced by using decreasing penalties instead of uniform ones. Here it depends on the parameter values whether increasing of decreasing penalties are globally optimal in this setting.

We proceed as follows: In Section 2 we present the basic model. In Section 3 we derive the equilibrium strategies of the potential offenders, that is, we first solve the infinite game of stage 3. In Sections 4 and 5 we derive the equilibrium strategies of the legal authority. In Section 6 we investigate the consequences of a change from our infinite game framework to a corresponding two-period framework. We conclude in Section 7.

2 The Basic Model – Assumptions

Suppose that some individuals, the “potential offenders”, receive non-negative, constant gross benefits, b, from crime. ^[15] We assume that in each period a set of potential offenders with benefits distributed according to a continuous distribution function F: B→[0,1], B⊂ℝ0+, is born. ^[16] Additionally we assume that the probability density is bounded away from zero and that the hazard rate ^[17] defined by h(b)=f(b)/1−F(b) is increasing. ^[18] The probability that an individual which is alive in period t still lives in period t+1 is given by q∈(0,1). ^[19] Hence in each period the total number of potential offenders is given by 1/(1−q). ^[20] Moreover, each individual is assumed to be able to commit at most one crime per period in which it is alive. ^[21]

We formalize the decision process as a three stage game. ^[22] In the first two stages the regulator chooses the maximum tolerable (“aspired”) number of crimes c per period (stage 1) and a penalty scheme (stage 2) {sn}n=0,1,... with 0≤sn≤smax∀n≥0, where the index n denotes the number of previous convictions and smax the maximal possible sanction. ^[23] We illustrate our point using three types of penalty schemes: uniform penalty schemes defined by sn=sˉ∀n≥0, decreasing penalty schemes, defined by sn+1≤sn∀n≥0 and increasing penalty schemes, defined by sn+1≥sn∀n≥0. ^[24] The Propositions however are also valid for hybrid penalty schemes for which there exist criminal histories n,m≥0, such that sm+1<smand sn+1>snhold. In the third stage potential offenders decide whether they commit a crime.

Let p∈(0,1) denote the (exogenously given) probability of apprehending offenders which is assumed to be independent of the type of offender (defined by his previous number of crimes and his benefit). ^[25] We assume that F(psmax)<1 holds, i. e., even if the regulator chooses the maximum available penalty for each offender there are some potential offenders who may not be deterred from committing a crime.

In this paper, we follow the lines of neoclassical welfare economics. There, the regulator is stylized to maximize social welfare. In our model the effect of crime on social welfare is two-fold. First, crime inflicts harm to the victim and second, punishing criminals is costly to society. ^[26] Thereby, the strive to maximize social welfare translates into a strive to minimize the sum of harm from crimes and cost of punishment. This implies that the optimal crime control policy is defined by using “the minimum amount of punishment to achieve any given level of crime control” (Kleiman 2009, 3). ^[27]

Harm from crimes is given by H(c) with H′(c)>0∀c∈(0,cmax), where cn(s) denotes the number of crimes per period committed by the group of potential offenders with n police records given the penalty scheme s, c=∑n=0∞cn(s) denotes the aggregate number of crimes committed per period and cmax denotes the aggregate number of crimes committed if there are no sanctions at all. Penalization costs are given by Pp∑n=0∞cn(s)sn with P denoting a strictly increasing function of the sum of imposed sanctions. ^[28] Thus, the objective function of the regulator may be written as

In the following three sections we analyse the game backwards and begin with the third stage. That is, we first solve the infinite game (where the potential offender plays versus nature, which determines the offender’s lifetime). Replacing the third stage by its equilibrium outcome reduces the infinite game to a finite one.

3 Third Stage: Equilibrium Number of Crimes – Individual Choice and Aggregation

4 Second Stage: Lawmaker’s Choice – The Optimal Time Profile of Sanctions

Having determined the equilibrium criminality levels, we now consider the optimization task (1) of the regulator. As already mentioned in Section 2 this task is divided into two steps. In the second stage the regulator has already chosen the aspired number of crimes cˉ per period. His remaining task is to choose a sanction scheme which achieves cˉ with minimal penalization cost. Since the penalization costs are a strictly increasing function of total sanctions (P(p∑n=0∞cn(s)sn) with P′>0), without loss of generality we may assume that P equals the identity function. Hence in the remaining parts of Section 4 we consider the equivalent optimization task

To assure that a reduction of crime (or equivalently an increase of a sanction) raises punishment cost, we need the following assumption:

If the regulator chooses the minimal (maximal) sanction sˉ=0 (sˉ=smax) the corresponding criminality level is given by cmax=1/(1−q) (cmin=1−F(psmax)/(1−q)).

In the remaining part of Section 4 we assume that for the criminality target cˉ∈(cmin,cmax) holds in order to assure that there exist different penalty schemes that enforce cˉ and thus the solution of eq. [3] is non-trivial. ^[33]

In Section 3 we have seen that a given criminality target cˉ can be induced by different penalty schemes, which either may lead to the same or to different sequences of critical benefits {bn}. Proposition 3 shows that the penalization cost under a given penalty scheme is determined by the sequence of critical benefits.

The intuition behind Proposition 3 can be explained by the following example. Consider two penalty schemes which lead to the same sequence of critical benefits {bn}=(1,1,1,5,5,5,5,7,7,8,12,…). Under both penalty schemes a potential offender with benefit b=1 is indifferent between “never committing the crime” and incurring the cost of “committing the crime until he is caught 3 times”. From this indifference condition, it follows that under both penalty scheme the penalization cost of punishing any criminal for the first 3 times are identical. Analogously it can be argued that the aggregate penalization cost for punishing offenders for the fourth to seventh time are identical and so on.

Proposition 3 can be interpreted as follows: If two penalty schemes induce the same crime decisions their implementation costs the same to the society. This directly implies the following statement:

Proposition 4 shows that any penalty scheme can be replaced by an equivalent escalating penalty scheme (in terms of critical benefits and social cost).

The assertion directly follows from the fact that under an escalating penalty scheme the critical benefits are given by bn=psn (see Section 3).

Using Proposition 4, the task of finding the globally optimal penalty scheme can be reduced to the task of determining the optimal scheme within the class of escalating penalty schemes. From Proposition 2 directly follows that the corresponding aggregate and type specific criminality levels are given by c=∑n=0∞(1−F(psn))∑t=n∞qtBn|t and cn=(1−F(psn))∑t=n∞qtBn|t. That is, in case of escalating penalty schemes, cn is a function of sn only.

It turns out (see Proposition 5) that the optimal penalty scheme has the following structure: Offenders with a high criminal record receive the harshest punishment and offenders with a low criminal record are not punished at all. The number of offences which is “necessary” to receive a positive punishment depends on the criminality target. The maximal sanction is imposed the earlier, the lower the aspired criminality level is. In case of a strict criminality target the punishment for the first detected offence is already positive. (Only) the first positive sanction may differ from smax to assure that the criminality target is exactly fulfilled. In particular it is not optimal to choose a strictly increasing penalty scheme with multiple (i. e. more than three) sanction levels.

The proceeding of the proof of Proposition 5.a is as follows: We assume that in the initial situation we have an escalating penalty scheme s with c(s)=cˉ which is not of the type described in Proposition 5.a. We define the minimum number of previous convictions a potential offender must have under s to receive a positive punishment by n0, i. e. n0:=min{n|sn>0}. Since s is not of the type given in Proposition 5.a, there must be a number of previous records n˜>n0 for which the corresponding sanction sn˜ is not maximal. We show that in this case it is possible to increase one or more sanction(s) for offenders with high criminal record and instead decrease the number of offenders with rather low criminal record in such a way that the following three conditions are fulfilled: i) The new penalty scheme is of the escalating penalties type, too, ii) the criminality level remains constant and iii) the shift of sanctions decreases penalization cost.

To clarify the driving forces that generate smallest penalization cost for the escalating penalty scheme given by Proposition 5 we conclude this section with an example, which compares the penalization cost of two escalating penalty schemes.

Since the criminality level and hence also harm from crime is identical under both penalty schemes, we only have to show that penalization cost under sˆ is lower than under s.

Therefore we consider the change of penalization cost, which is given by

with

from which follows

Hence, the increase of penalization cost for offenders with one previous record is overcompensated by the decrease of penalization cost for offenders without criminal record. The main reason behind this assertion is that the cost of punishment for the additional criminal activity of offenders without criminal record and with b∈[0,0.5) is lower (in our example it takes the value zero) than the saved cost of punishment for the reduced criminal activity of offenders with one previous sanction and b∈[1,1.61ˉ). This is so since the (new) penalty for offenders with low criminal record (sˆ0=0) is lower than the (old) penalty for offenders with high criminal record (s1=2) and thus the aggregate penalization effort is reduced. In addition to that we have the following effects. For offenders without criminal record and benefit b>0.5 penalization cost is also reduced (in our example they even vanish) due to the reduced sanction for first time offenders. For offenders with benefit b>1.61ˉ there is an additional increasing cost effect for offenders with one previous sanction. However for our equal distribution function this cost effect is just offset by the decreasing cost effect for offenders without previous sanction and with benefit b>1.61ˉ. ^[36] Adding all effects reveals that aggregate penalization cost is decreasing if the system switches from the suboptimal escalating penalty scheme to the optimal one.

5 First Stage: Lawmaker’s Choice – The Optimal Level of Crime

In the previous section we have compared penalization cost under different penalty schemes for a given criminality target c∈(cmin,cmax). Knowing the optimal penalty scheme for each criminality level, the regulator may determine the optimal level of crime after inserting the minimized value of punishment cost for each level of crime in the objective function [1]. We denote the optimal level of crime by cI∗ in order to indicate that the optimal increasing penalty scheme is used to induce the criminality target. Only in the cases cI∗∈{cmin,cmax} the penalty scheme is of the uniform type as a border case.

Economic intuition might suggest that cI∗≤cU∗, with cU∗ denoting the level of crime that would be optimal under the constraint of uniform penalties. After all the costs of reducing crime go down if society switches from the latter to the former penalisation scheme. However, Proposition 6 shows that this plausible presumption does not always hold.

The statement of Proposition 6 is rather counterintuitive. To understand this counterintuition consider a marginal increase of the advised criminality level starting from the criminality level cmin, which is induced if the maximal sanction, smax, is applied irrespective of the number of previous convictions. We compare the welfare effects in the following two scenarios: in scenario 1 the criminality level is induced by the corresponding uniform penalty and in scenario 2 the criminality level is induced by the optimal escalating penalty scheme. In both scenarios the welfare effect is composed of two components: the increase of social harm from crimes and the decrease of enforcement cost. Whereas the increase of social harm coincides in both scenarios the decrease of enforcement cost is larger in the escalating penalty scenario than in the uniform scenario. Therefore it is possible to construct an example, as is done below, where in scenario 1 marginal welfare is strictly decreasing in the crime level, and thus cmin is the equilibrium criminality level, whereas in scenario 2 marginal welfare is positive in a non-empty interval (cmin,c˜) and thus the optimal level of criminality exceeds cmin.

First note that H,H′>0∀c∈(cmin,cmax), i. e., the marginal utility of crime prevention is positive and decreasing. Moreover note that g(c)=−dPU/dc holds. ^[37] Thus, for the uniform penalty scheme we obtain dKUdc=H′+dPUdc=ε>0 from which follows the optimal criminality level cU∗=cmin=5, with corresponding penalty scheme s˜=(smax,smax,smax,…). Hence, to prove Proposition 6 we have to show that there exists an ε>0 such that within the class of escalating penalty schemes the penalty scheme of border type s˜=(smax,smax,smax,…) is not optimal. Therefore it suffices to demonstrate that there exists an ε>0 such that the penalty scheme s=(0,smax,smax,…) (with corresponding higher criminality level) leads to lower total cost than s˜. Since under penalty scheme s the corresponding criminality levels cn, n≥1 coincide with those under s˜ and only differ for potential offenders without criminal record, the aggregate criminality level under s can be determined by

Thus a change from s˜ to s increases harm from crimes by

On the other hand penalization costs under s are lower than under s˜.

Since the penalty and the criminality level coincide under both penalty schemes for potential offenders with at least one criminal record the change of penalization costs can be determined by

Hence the change of total cost is given by ΔK=ΔH+ΔP=4.95868ε−3.71901 which implies that for ε<0.75 we have cI∗>cU∗.

To clarify the driving forces in our example in a first step we assume ε=0. In this case we obtain dKU/dc=0. Hence in case of uniform penalization each enforceable level of criminality and thus also cmin is optimal. I.e., if c is increased, the increase of harm is just outweighed by the corresponding decrease of penalization cost H′(c)=g′(c)=−dPU/dc. However under the escalating penalty regime s, penalization cost is lower, from which follows

i. e., under escalating penalties cmin can’t be the optimal criminality level.

In the second step, we depart from the assumption ε=0 and discuss how the additional term εc>0 in the marginal harm function affects equilibrium criminality levels. Under uniform penalties ε>0 assures that the social cost function is strictly increasing in c, and cmin is the unique equilibrium. On the other hand if ε is small enough, the advantage of the escalating penalty regime with respect to marginal penalization cost is not overcompensated.

6 Infinite Versus Finite Horizon – The Consequences

In the following we show that if we change the infinite game framework presented above to a two-period and thus finite one, the optimal penalty scheme can be of the escalating as well as of the decreasing type. The result that a decreasing penalty scheme is optimal at least for certain parameter values has been received in many of the finite models presented in the literature. Other papers do not consider decreasing penalty schemes at all and only compare welfare under uniform and escalating penalty schemes. ^[38]

We restrict our attention to the cost minimizing implementation of a given criminality target. Hence we (only) proceed along the structure of Sections 2–4 of our main model. Obviously, what will be said for any given target also holds for the socially optimal one.

7 Summary and Outlook

We have analysed positive and normative issues of criminal law in an intertemporal framework. In contrast to the earlier literature our analysis compared an infinite game setting with a finite one. Criminals have been assumed to maximize their net benefits from crime and the lawmaker to minimize social cost. The decision variables have been the number of crimes committed, and the time profile of punishments.

For the infinite game framework it has been shown that for any given criminality level penalization costs under decreasing and uniform penalty schemes are identical. However penalization cost may be reduced by choosing an escalating penalty scheme. We have shown that the optimal penalty scheme shifts the punishment as far into the future as possible. We further have shown that the optimal criminality level under escalating penalties may exceed the optimal criminality level under uniform penalties. Moreover we have shown that in the corresponding finite game framework the optimal penalty scheme may be of the escalating as well as of the decreasing type.

Our basic infinite game approach may be generalized in several directions. One is to endogenize the apprehension probability. Following Polinsky and Shavell (1998) this can be done by adding an additional term A(p) (with A′>0) which captures the costs of apprehending offenders. If we additionally assume that neither the choice of p=1 nor the choice of p=0 is socially optimal, ^[41] it can easily be proven by contradiction that also in this framework the optimal escalating penalty scheme still dominates the uniform and decreasing ones. Once the optimal level of p is determined, finding the optimal penalty levels boils down to the problem that is solved in Sections 4 and 5 of this paper.

Examples for future extensions allow for age-dependent benefits, risk averse individuals or more general distribution functions of offenders’ benefits. Moreover, all kinds of “criminal hysteresis” (Fajnzylber, Lederman, and Loayza 2002, 1328) are particularly suggestive to be integrated into our intertemporal model. For example the criminal history of a certain offender might affect his cost of carrying out a particular criminal activity, the probability of apprehension and other determinants of the decision for crime.