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Deterrence: Increased enforcement versus harsher penalties
Economics Letters 117 (2012) 561–562
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Deterrence: Increased enforcement versus harsher penalties Derek Pyne ∗ Department of Economics, Thompson Rivers University, Kamloops, British Columbia V2C 5N3, Canada
article
info
Article history: Received 27 May 2012 Accepted 19 July 2012 Available online 1 August 2012 JEL classification: K4 Keywords: Deterrence Crime Recidivism
abstract Empirical studies have found that increasing the probability of punishment has a greater effect on crime than the severity of punishment. This note explains this as the result of criminals having imperfect information on their criminal ability. As they commit crimes, they update their estimates of their ability, based on their success rate. Increased penalties deter crime in the period they are applied but offer criminals no information on their criminal ability. Crime is also deterred during a period of increased enforcement. In addition, increased enforcement leads some criminals to decrease their estimates of their ability, leading to reduced recidivism. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Several empirical studies have found that increasing the probability of punishment through enforcement has a greater effect on crime reduction than increasing the severity of punishment (for a survey of the literature, see Eide, 2000). One explanation for this is that criminals are risk lovers (Becker, 1968). As this is contrary to standard economic assumptions used in other contexts, others have offered alternative explanations. One approach is to relax the standard assumptions of the expected utility model (Neilson and Winter, 1997). Another approach is to entirely reject expected utility theory. An example of this approach is Al-Nowaihi and Dhami’s (2010) use of composite cumulative prospect theory. This note offers an alternative explanation that is consistent with both the strong form of expected utility theory and nonrisk loving behavior. Criminals are assumed to possess differing abilities, where ability is measured in terms of the probability they will not be apprehended when they commit a crime. They possess imperfect knowledge of this ability. As they commit crimes, they update their estimates of their own abilities. A period specific increase in penalties will deter some criminals from committing crime in the same period. However, it does not affect the updating of criminal abilities for those who do commit crimes, whether or not they are apprehended. A period specific increase in apprehension rates also deters crime in the same period. However, unlike a period specific increase in penalties, it will also cause apprehended criminals to reduce their estimates of their own
criminal ability. Therefore, crime may be reduced in both the period of the greater enforcement and in later periods. 2. Model The criminal is assumed to be risk neutral. However, as long as risk aversion is not too great, the results would still hold. Let the benefits from a crime be given by B. Dk represents the punishment if convicted for a kth offense. In period j, a potential criminal h perceives the probability of apprehension to be phj . It is assumed that all agents caught are also convicted. Agent h’s expected utility from committing a crime in period j when facing penalty Dk if caught is given by U = B − phj Dk . The agent’s perception of the probability of being caught given by phj (i) = a(i)phT + ehj .
Tel.: +1 250 852 7289; fax: +1 250 828 5051. E-mail address: [email protected].
0165-1765/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2012.07.029
is
(2)
The value of phT (0 ≤ a(i)phT ≤ 1) is determined by the agent’s own criminal ability. It is randomly distributed over the population. The state invests i resources in enforcement which affects the probability of being caught through a(i), where ∂ a(i)/∂ i > 0. In period 1 the error term is randomly distributed over the population such that E (e1 ) = 0. For any agent h, the actual value of ehj is restricted to ensure pj is between 0 and 1:
− a(i)phT ≤ ehj ≤ 1 − a(i)phT . ∗
(1) (phj )
(3)
In subsequent periods, agents who have committed a crime in the previous period, update their perception of their criminal
562
D. Pyne / Economics Letters 117 (2012) 561–562
ability, depending on whether they were caught in the previous period. If convicted in the previous period, ehj ≤ ehj+1 .
(4)
If not convicted, ehj ≥ ehj+1 .
(5)
Both of the above weak inequalities hold as equalities when inequality restriction (3) is already binding. If an individual continues to commit crimes, over time, his error term converges to zero: lim ehj = 0.
(6)
j→∞
The justice system cannot directly observe either the actual or the criminal’s perceived values of phj . Nonetheless it does know the distribution they are drawn from. It also knows that for a given level of enforcement, an agent’s decision to commit crime will depend on the penalty imposed. Let pM j (Dj , i) represent the marginal criminal’s perceived probability of conviction in period j. Lemma 1 will show that agents who are neither deterred nor apprehended in any period j, will also not be deterred in any subsequent period before being apprehended. Lemma 1. Consider an individual who is not deterred in some period j when facing penalty Dk if convicted. If he is not convicted in period j, he will not be deterred in period j + 1. In addition, if he is not convicted until period k + g, he will not be deterred in any period from k + 1 to k + g. Proof. The theorem will be proved by induction. As the individual is not deterred in period j, it must be the case that U = B − phj Dk ≥ 0. Given inequality (5), ehj ≥ ehj+1 . Thus, it must also be the case that U = B − phj+1 Dk ≥ 0. By the same argument, if the individual is also not convicted in period j + 1, he will not be deterred in period j + 2, and so on, until period j + g. Let two possible enforcement levels in period j be given by iH j and
L iLj , where iH j > ij . If enforcement increases the apprehension rate:
1
a iH pT d(F ) >
pM (iH ) K
1
a iL pT d(F )
pM (iL ) K
(7)
g where pM K (i ) is the marginal criminal’s perception of his abilities when the enforcement is ig where g = H , L and F is the distribution of actual criminal abilities over the population. Lemma 1 will be used to prove Theorem 1. Theorem 1 establishes that if an increase in enforcement increases the conviction rate in any given period it will not only deter crime in that period, but also in future periods.
Theorem 1. Assume inequality (4) holds as a strict inequality for additional individuals deterred after a period specific increase in enforcement. An increase in enforcement that increases the apprehension rate may provide increased deterrence in both the current period and in future periods.
Proof. For the marginal criminal in period j when facing the lower L level of enforcement: U = B − pM j (Dj , ij )Dk = 0. With greater enforcement, the same marginal individual would receive negative utility from committing crime and be deterred. The later behavior of those deterred in period j is unaffected as they have learnt nothing about their criminal ability. However, some of those undeterred are convicted and reduce their estimate of their criminal ability. Assuming inequality (4) holds as a strict inequality, for these individuals ehj < ehj+1 and phj < phj+1 . As the apprehension rate has increased, the greater enforcement has increased the number of these agents. The remaining undeterred agents are not caught. Given inequality (5), they may increase their estimates of their criminal abilities, ehj ≥ ehj+1 and phj ≥ phj+1 . However, Lemma 1 established that they would have remained undeterred anyway. Thus, the increased apprehension rate may increase future deterrence. However, Theorem 2 establishes that a period specific increase in penalties has no effect on future crime. Theorem 2. A period specific increase in penalties reduces crime in the current period but not in future periods. Proof. Let two possible penalties in period j for a kth conviction be L H L given by DH k and Dk , where Dk > Dk . For the marginal criminal in L period j when facing the lower penalties: U = B−pM j (Dj , ij )Dk = 0. With higher penalties, the same marginal individual would receive negative utility from committing crime and be deterred. Thus, H M L pM j (Dk , ij ) < pj (Dk , ij ). Regardless of whether or not an individual is convicted, higher penalties contain no additional information that allows the criminal to update his estimate of his own criminal H M L abilities. Thus, pM j+1 (Dj+1 , ij+1 , Dj ) = pj+1 (Dj+1 , ij+1 , Dj ). 3. Conclusion The different effects of increasing penalties are similar to the legal concepts of general deterrence and specific deterrence. General deterrence involves deterring crime in general, while specific deterrence involves deterring a specific individual from committing future crimes. This note has found that both increased enforcement and increased penalties provide general deterrence in the period they are imposed. However, only increased enforcement deters additional apprehended criminals from recommitting crimes in later periods. This note has shown that empirical evidence showing increasing apprehension rates reduces crime more than increased penalties may be consistent with expected utility theory even if criminals are not risk lovers. References Al-Nowaihi, Ali, Dhami, Sanjit, 2010. The behavioral economics of crime and punishment. Discussion Papers in Economics. University of Leicester. Becker, Gary S., 1968. Crime and punishment: an economic approach. Journal of Political Economy 76 (2), 169–217. Eide, Erling, 2000. Economics of criminal behavior. In: Bouckaert, Boudewijn, Geest, Gerrit De (Eds.), Encyclopedia of Law and Economics. Edward Elgar, Cheltenham. Neilson, William S., Winter, Harold, 1997. On criminals’ risk attitudes. Economics Letters 55 (1), 97–102.
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Deterrence: Increased enforcement versus harsher penalties Derek Pyne ∗ Department of Economics, Thompson Rivers University, Kamloops, British Columbia V2C 5N3, Canada
article
info
Article history: Received 27 May 2012 Accepted 19 July 2012 Available online 1 August 2012 JEL classification: K4 Keywords: Deterrence Crime Recidivism
abstract Empirical studies have found that increasing the probability of punishment has a greater effect on crime than the severity of punishment. This note explains this as the result of criminals having imperfect information on their criminal ability. As they commit crimes, they update their estimates of their ability, based on their success rate. Increased penalties deter crime in the period they are applied but offer criminals no information on their criminal ability. Crime is also deterred during a period of increased enforcement. In addition, increased enforcement leads some criminals to decrease their estimates of their ability, leading to reduced recidivism. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Several empirical studies have found that increasing the probability of punishment through enforcement has a greater effect on crime reduction than increasing the severity of punishment (for a survey of the literature, see Eide, 2000). One explanation for this is that criminals are risk lovers (Becker, 1968). As this is contrary to standard economic assumptions used in other contexts, others have offered alternative explanations. One approach is to relax the standard assumptions of the expected utility model (Neilson and Winter, 1997). Another approach is to entirely reject expected utility theory. An example of this approach is Al-Nowaihi and Dhami’s (2010) use of composite cumulative prospect theory. This note offers an alternative explanation that is consistent with both the strong form of expected utility theory and nonrisk loving behavior. Criminals are assumed to possess differing abilities, where ability is measured in terms of the probability they will not be apprehended when they commit a crime. They possess imperfect knowledge of this ability. As they commit crimes, they update their estimates of their own abilities. A period specific increase in penalties will deter some criminals from committing crime in the same period. However, it does not affect the updating of criminal abilities for those who do commit crimes, whether or not they are apprehended. A period specific increase in apprehension rates also deters crime in the same period. However, unlike a period specific increase in penalties, it will also cause apprehended criminals to reduce their estimates of their own
criminal ability. Therefore, crime may be reduced in both the period of the greater enforcement and in later periods. 2. Model The criminal is assumed to be risk neutral. However, as long as risk aversion is not too great, the results would still hold. Let the benefits from a crime be given by B. Dk represents the punishment if convicted for a kth offense. In period j, a potential criminal h perceives the probability of apprehension to be phj . It is assumed that all agents caught are also convicted. Agent h’s expected utility from committing a crime in period j when facing penalty Dk if caught is given by U = B − phj Dk . The agent’s perception of the probability of being caught given by phj (i) = a(i)phT + ehj .
Tel.: +1 250 852 7289; fax: +1 250 828 5051. E-mail address: [email protected].
0165-1765/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2012.07.029
is
(2)
The value of phT (0 ≤ a(i)phT ≤ 1) is determined by the agent’s own criminal ability. It is randomly distributed over the population. The state invests i resources in enforcement which affects the probability of being caught through a(i), where ∂ a(i)/∂ i > 0. In period 1 the error term is randomly distributed over the population such that E (e1 ) = 0. For any agent h, the actual value of ehj is restricted to ensure pj is between 0 and 1:
− a(i)phT ≤ ehj ≤ 1 − a(i)phT . ∗
(1) (phj )
(3)
In subsequent periods, agents who have committed a crime in the previous period, update their perception of their criminal
562
D. Pyne / Economics Letters 117 (2012) 561–562
ability, depending on whether they were caught in the previous period. If convicted in the previous period, ehj ≤ ehj+1 .
(4)
If not convicted, ehj ≥ ehj+1 .
(5)
Both of the above weak inequalities hold as equalities when inequality restriction (3) is already binding. If an individual continues to commit crimes, over time, his error term converges to zero: lim ehj = 0.
(6)
j→∞
The justice system cannot directly observe either the actual or the criminal’s perceived values of phj . Nonetheless it does know the distribution they are drawn from. It also knows that for a given level of enforcement, an agent’s decision to commit crime will depend on the penalty imposed. Let pM j (Dj , i) represent the marginal criminal’s perceived probability of conviction in period j. Lemma 1 will show that agents who are neither deterred nor apprehended in any period j, will also not be deterred in any subsequent period before being apprehended. Lemma 1. Consider an individual who is not deterred in some period j when facing penalty Dk if convicted. If he is not convicted in period j, he will not be deterred in period j + 1. In addition, if he is not convicted until period k + g, he will not be deterred in any period from k + 1 to k + g. Proof. The theorem will be proved by induction. As the individual is not deterred in period j, it must be the case that U = B − phj Dk ≥ 0. Given inequality (5), ehj ≥ ehj+1 . Thus, it must also be the case that U = B − phj+1 Dk ≥ 0. By the same argument, if the individual is also not convicted in period j + 1, he will not be deterred in period j + 2, and so on, until period j + g. Let two possible enforcement levels in period j be given by iH j and
L iLj , where iH j > ij . If enforcement increases the apprehension rate:
1
a iH pT d(F ) >
pM (iH ) K
1
a iL pT d(F )
pM (iL ) K
(7)
g where pM K (i ) is the marginal criminal’s perception of his abilities when the enforcement is ig where g = H , L and F is the distribution of actual criminal abilities over the population. Lemma 1 will be used to prove Theorem 1. Theorem 1 establishes that if an increase in enforcement increases the conviction rate in any given period it will not only deter crime in that period, but also in future periods.
Theorem 1. Assume inequality (4) holds as a strict inequality for additional individuals deterred after a period specific increase in enforcement. An increase in enforcement that increases the apprehension rate may provide increased deterrence in both the current period and in future periods.
Proof. For the marginal criminal in period j when facing the lower L level of enforcement: U = B − pM j (Dj , ij )Dk = 0. With greater enforcement, the same marginal individual would receive negative utility from committing crime and be deterred. The later behavior of those deterred in period j is unaffected as they have learnt nothing about their criminal ability. However, some of those undeterred are convicted and reduce their estimate of their criminal ability. Assuming inequality (4) holds as a strict inequality, for these individuals ehj < ehj+1 and phj < phj+1 . As the apprehension rate has increased, the greater enforcement has increased the number of these agents. The remaining undeterred agents are not caught. Given inequality (5), they may increase their estimates of their criminal abilities, ehj ≥ ehj+1 and phj ≥ phj+1 . However, Lemma 1 established that they would have remained undeterred anyway. Thus, the increased apprehension rate may increase future deterrence. However, Theorem 2 establishes that a period specific increase in penalties has no effect on future crime. Theorem 2. A period specific increase in penalties reduces crime in the current period but not in future periods. Proof. Let two possible penalties in period j for a kth conviction be L H L given by DH k and Dk , where Dk > Dk . For the marginal criminal in L period j when facing the lower penalties: U = B−pM j (Dj , ij )Dk = 0. With higher penalties, the same marginal individual would receive negative utility from committing crime and be deterred. Thus, H M L pM j (Dk , ij ) < pj (Dk , ij ). Regardless of whether or not an individual is convicted, higher penalties contain no additional information that allows the criminal to update his estimate of his own criminal H M L abilities. Thus, pM j+1 (Dj+1 , ij+1 , Dj ) = pj+1 (Dj+1 , ij+1 , Dj ). 3. Conclusion The different effects of increasing penalties are similar to the legal concepts of general deterrence and specific deterrence. General deterrence involves deterring crime in general, while specific deterrence involves deterring a specific individual from committing future crimes. This note has found that both increased enforcement and increased penalties provide general deterrence in the period they are imposed. However, only increased enforcement deters additional apprehended criminals from recommitting crimes in later periods. This note has shown that empirical evidence showing increasing apprehension rates reduces crime more than increased penalties may be consistent with expected utility theory even if criminals are not risk lovers. References Al-Nowaihi, Ali, Dhami, Sanjit, 2010. The behavioral economics of crime and punishment. Discussion Papers in Economics. University of Leicester. Becker, Gary S., 1968. Crime and punishment: an economic approach. Journal of Political Economy 76 (2), 169–217. Eide, Erling, 2000. Economics of criminal behavior. In: Bouckaert, Boudewijn, Geest, Gerrit De (Eds.), Encyclopedia of Law and Economics. Edward Elgar, Cheltenham. Neilson, William S., Winter, Harold, 1997. On criminals’ risk attitudes. Economics Letters 55 (1), 97–102.