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Optimal Unconditional Parole David Scoones Department of Economics - - PDF document
Optimal Unconditional Parole David Scoones Department of Economics - - PDF document
Optimal Unconditional Parole David Scoones Department of Economics University of Victoria December 2009 Motivation Each time an offender on parole release commits a crime, calls for parole reform are widespread. Truth in sentencing
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Not much work in the Economics of Crime literature has examined parole. One exception is Bernhardt, Mongrain and Roberts (2006) which considers informed release. This paper ask what might be the rationale for automatic early release I find that statutory release may indeed be optimal in some contexts.
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Preview of results When sentences are long, criminals discount the future sufficiently that the extra deterrence of prison versus parole is minimal. In addition, parolees on statutory release face a significantly higher likelihood of arrest should they commit crimes, and so are more likely to be deterred from crime than are similar individuals in the community who are not on parole. However, when statutory release periods increase, the discounted value of eventual release becomes small, at least at the beginning of parole, and parolees are more likely to return to offending. Provided that they avoid arrest, the incentive to not engage in crime rises as the end of the parole period draws near. Thus the model predicts that statutory release parolees are less likely to re-offend as their time
- n parole passes.
When sentences are short, increasing the portion of the sentence served on parole reduces the deterrence of
- punishment. In this case, society may be better of
without statutory release.
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This suggests that far from a trade off, as implicitly assumed by the arguments for "truth-in-sentencing", statutory release and long sentences might be complementary. Provided that sentences exceed a certain length, some short period of statutory release is welfare improving. With long sentences, longer periods of statutory release become optimal.
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Among what I ignore The dominant explanations for parole, and these too might be used to justify statutory release. The most common argument for statutory release is that releasing prisoners without restriction or community support greatly increases the chance they will re-offend. Another argument is that people change, and jailing the no longer criminally-minded is unduly costly. (This is the focus of Bernhardt, Mongrain and Roberts (2006)) In the present model types are fixed and there is no prospect of reform. Trying to learn these types is still potentially useful. To explain statutory release on this basis, it must be that some aspect of offenders' types is better measured
- n parole than in prison.
Finally, a little noted value of parole is how it allows local law enforcement officials (including parole
- fficers) with the modus operandi, family contacts and
likely whereabouts of a town’s criminal class.
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The Canadian Federal Parole System All persons sentenced to two years or longer enter the Federal Corrections system All parolees are monitored by regular contact with a parole officer, and possible others. Restrictions are set by the Federal Parole Board at the time of release. Day parole: release into a “community residential facility” – halfway house Full parole: allowed to live independently
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Opportunities for parole: ♦ Accelerated Parole Review. First time non-violent
- ffenders are eligible for day parole at six months
regardless of sentence length. The onus is on the parole officer to demonstrate that there is an uncontrollable risk to re-offend. If Day Parole is granted, Full Parole is automatic six months later. ♦ Full Parole eligibility is at 1/3 the sentence. Day Parole is Full Parole less 6 months. ♦ Statutory Release is at 2/3 the sentence. ♦ IF a parolee is suspended and (eventually) revoked, the process restarts at the date the warrant is executed. ♦ Section 810 Orders are sought by a local police force to impose parole-like conditions on inmates for one year at the end of their sentence. Can be renewed. ♦ Long Term Supervision Orders
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Some Federal Numbers
(Corrections and Conditional Release Statistical Overview 2008)
Cost per annum (2006-7) Men Women Prison $90,744 $166,830 Parole $23,076 Population Compared to European countries Canada has a high incarceration rate: 108 per 100K population. Germany is 91 and Denmark 63. Canada’s rate has fallen in recent years. The US is 762, and rising. Men Women Total 13,500 495 57.7% were under age 40 in 2007-08. The population is aging. (Aboriginals are greatly overrepresented: in 2007-08 17.3% of federal offenders were Aboriginal compared with 4.0% of the population)
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Parole outcomes Men Women
Day parole
Granted 69% (3K) 89% (283) Success 83.5% Revoked 13% New offence 4%
Full Parole
Granted 41% (14K) 71% (168) Success 72% Revoked 19% New offence 8%
Statutory release
Detained 266 Success 59% (3348) Revoked 30.5% (1739) New offence 10% (606 – 110 violent)
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The model ♦ A unit population of infinitely lived, potentially criminal citizens. ♦ Time is discrete. At the start of each period every agent is in one of the following: a free non-criminal, a free criminal, in jail, or on parole. ♦ Free citizens and parolees must choose between engaging in criminal activity and working in the legal
- sector. The environment is stationary.
♦ All criminals are subject to arrest and conviction, leading to a prison term of length J followed by
P periods of parole.
♦ Each agent is characterised by a type, ψ , distributed according to F , that affects his return from crime. All other parameters are assumed to be identical for every agent. ♦ The higher is an agent's ψ -coefficients the greater is his benefit from crime, c
ψ
♦
j p w > . >
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Jail time only: .
= P
Value functions Value for non-criminals
i NC i
V w V δ + =
The value of a crime is given by ] ) 1 ( [
i A i i C i
V V c V λ λ δ ψ − + + = The value of a prison term of length followed by release
J
i J J J t t A i
V j j V δ δ δ δ + − − = = ∑ =
−
1 ) 1 (
1 1
Thus
) ( 1 ) 1 )( 1 (
1 1 + −
− + − − − + =
J J j C i
j c V δ δ λ δ δ δ δλ ψ
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The marginal criminal for P=0 Characterised by the ψ -coefficient for which
C i NC i
V V =
The marginal criminal is then
) 1 ( ) ( ) ( *
1
δ δ δ λ ψ − − − + =
+
c j w c w
J
. The population is partition:
- 1. if
* ψ ψ >
i
, agent i is a criminal
- 2. if
* ψ ψ <
i
, agent i is not a criminal Notice that if there is no chance of arrest, = λ , then
w c = * ψ
.
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Comparative Statics 1.
* < ∂ ∂ j ψ
2.
* > ∂ ∂ w ψ
3.
) 1 ( ) )( ( *
1
> − − − = ∂ ∂
+
δ δ δ λ ψ c j w
J
4.
) 1 ( ln ) ( *
1
> − − − = ∂ ∂
+
δ δ λδ ψ c j w J
J
5.
) 1 ( ) (ln ) ( *
2 1 2 2
< − − − = ∂ ∂
+
δ δ λδ ψ c j w J
J
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Parole On parole the arrest rate is higher: λ γ > The population partitions into
2 + P
groups: ♦ the λ -deterred never commit crimes (non- criminals) ♦ the
−
i
γ
deterred for
1 ... 1 + = P t
. The
−
t
γ
deterred commit crimes when free and on parole up to the
- period. When
th
t
1 + = P t
the criminals are never deterred. Depending on the parameters of the model, each of these partitions can range from empty to containing the entire population.
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Value functions For non-criminals life is as before. The value of arrest for a
−
t
γ
deterred agent is , which depends on type as well as the parole period in which the agent is deterred.
A Dt
V
1
P Dt
V is the value at the first period of parole. This is
defined recursively by: If
1 + < P t
C D P D
t P t
V p V δ + =
If , in the penultimate period
P t <
C D P D P D
t P t P t
V p V p V
2
) 1 (
1
δ δ δ + + = + =
−
And so on until period , at which point
t
C D t P D P D
t t t t
V pF V
− +
+ =
1
δ
where
δ δ − − =
− +
1 1
1 t P Dt
F
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Prior to this the agent is not deterred and earns the higher payoff from crime, but faces the possibility of arrest:
) )( 1 ( ) ) 1 ( (
1
1
C D t P D A D P D A D P D
t t t t t t t t
V pF V c V V c V
− +
+ − + + = − + + =
−
δ γ δ δγ ψ γ γ δ ψ
. This continues until .
) ( ) 1 ( ) (
1 1 1
1
C D t P D t t A D D P D
t t t t t
V pF V c Q V
− + − −
+ − + + = δ γ δ δγ ψ
where
) 1 ( 1 ) 1 ( 1
1 1
γ δ γ δ − − − − =
− − t t Dt
Q
.
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The value at arrest
1
P D J A D
t t
V K V δ + =
where
δ δ − − = 1 ) 1 (
J
j K
. Hence
t t t t t
D J C D t P D t t J D J A D
Q V pF K c Q V γ δ δ γ δ ψ δ
1 1 1 1
1 ) ( ) 1 (
+ − + − − +
− + − + + =
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Longer periods of parole lead to recidivism Consider the difference between the value at the start
- f parole for any two types, tand , which can be
written:
s
( )
C D s C D t P D s s A D D D t t A D D P D P D
s t s s s t t t s t
V V pF V c Q pF V c Q V V
1 1 1 1 1 1
) 1 ( ) 1 ( ] ) 1 ( ) ( [ ) 1 ( ) (
1 1
− − − − − −
− − − + − + + − − + + = − γ γ δ γ δ δγ ψ γ δ δγ ψ
Likewise, when grows large, the value of freedom disappears from .
P
V A
Dt
Thus as P grows large, the difference between these values depends only on the difference between the values for time spent in some form of incarceration. The incentive effect of the end of parole disappears, and the only consideration in the choice to re-offend is the relative payoff bouts of crime and terms in jail. Freedom is
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The value of crime
1 1 1 1 1
) 1 )) 1 ( 1 )( 1 ( ) 1 ( )) ( 1 (
− + + + − + +
− ( − − − − − + + − − =
t P J D J D t t J D J C D
t t t t
Q pF K c Q V γ λ δ λ δ γ δ γ λδ δγ ψ λ γ δ
for .
} 1 ,.., 1 { + = P t
Thus,
λ δ λ δ λδ δγ ψ
1 1
)) 1 ( 1 (
1 1
+ + +
− − − + + =
P J D J C D
pF K c V
and
P P J D J D J C D
t P P
Q K c Q V ) 1 )) 1 ( 1 )( 1 ( )) ( 1 (
1 1 1
1 1
γ λ δ λ δ γ δ δγ ψ λ γ δ − ( − − − − + − − =
+ + + +
+ +
.
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The marginal criminal In the equilibrium, higher agent types are more likely to violate parole for longer periods of time. To see why this is so, consider an agent considering violating parole in period . The benefit is the short-term gain from crime; the cost is the greater chance of arrest.
t
P
Since the value of arrest is increases in type, this cost is lower for higher types. Further, the gain for cheating is higher for higher types. Together, these conditions imply that if type
* ψ will
violate in period , so will all other agents of type
t
P
. * ψ ψ >
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The Equilibrium Partition Intervals of
− ψ
coefficients, with the lowest interval containing the non-criminals, the next higher the always
− γ deterred, then those deterred in the second
period, and continuing to the never deterred. An example of one such equilibrium is depicted below. The easiest case to study analytically is that in which the marginal criminal is
−
1
γ
deterred, as the value functions take a particularly simple form. For this case, the type
- f the marginal criminal solves
, where the latter is found as in equation 1.
NC C D
V V =
1
The type of the marginal criminal is then
) 1 ( ] ) ( ) ( [
1
δ δ δ λδ ψ − − − − − − + =
+
c j p p w j w c w
J P J
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The type of the marginal criminal increases with an increase in either or . However, an additional period in jail increases the marginal type by more than an additional period on parole:
J P
[ ]
ln ) ( ) 1 (
1 1
> − − = ∂ ∂ − ∂ ∂ δ δ δ λδ ψ ψ
J
p j c P J
Thus parole has a deterrence effect on free citizens, but it is, not surprisingly, less than that of additional periods in prison. This is due to paroles higher payoff. Notice that this difference goes to zero as criminals become more impatient and as the number of periods in jail increases.
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The Social Costs of Crime Depends on the total measure of agents in each state. Criminal careers are Markov Chains. Distinguish states, one for each year of prison and parole, and one for freedom.
1 + + P J
Criminals in each state impose costs on society: when free, criminals commit crimes with a low chance of being caught. Despite the higher chance of arrest, some criminals on parole commit crime, and all parolees impose direct costs on taxpayers. Finally, incarceration prevents crime, but is extremely costly, particularly when prison populations rise. The cost of prisons and policing are ignored. Implicitly prison costs are assumed to be so large that total incapacitation is not a viable option. Perhaps this is because of considerations of differential deterrence.
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The transition matrix for a
−
2
γ
deterred parole violator when
2 = = P J
is Free Jail 1 Jail 2 Parole 1 Parole 2 Free 1-λ λ Jail 1 1 Jail 2 1 Parole 1 γ 1-γ Parole 2 1 For the
−
1
γ
deterred criminals it as if
= γ
, and the process on parole looks like that in prison. For the never deterred, the final row has
) 1 ( γ −
in the first column, and γ in the second. For non-criminals, all the rows contain one 1 and four zeros.
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Time shares Four interesting states for each of the
2 + P
types. Denote the time spend by agent
} , {
t
D NC i ∈
in state as .
} , , , { PD PN J F j ∈
j i
ω
Then and .
1 =
F NC
ω
= =
P NC J NC
ω ω
For the
−
t
γ
deterred
t t F D
L S
t
) 1 ( γ ω − =
t J D
L J
t
λ ω =
t t P D
L S
N t
γ ω =
t t P D
L t P
D t
) 1 ( ) 1 (
1
− + − =
−
γ λ ω
where
1 1
) 1 ( 1 (
−
− − =
t t
S γ
γ
and
) 1 ( ) 1 ( ( 1
1
t P S J S L
t t t t
− + − + + + − =
−
γ λ γ
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Increases in either or
J Preduces the time spent in
crime. However, also increases in
N t
P D
ω
P for the reason
remarked upon above: as parole terms lengthen the incentive for not committing crimes induced by a return to freedom is weakened, increasing recidivism during parole.
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The Incapacitation Effect If there were only positive measures of non-criminals and two values of , the always deterred and the never deterred, the incapacitation effect would be
t
) )]( ( ) ( [ )] ( 1 [
1 1 1
1 2 2 P D J D J D I
F F F M
P
ω ω ψ ψ ω ψ + − + − =
+
, where is the critical value for type and the critical value for type .
1
ψ
1
D
2
ψ
1 + P
D
The measure of active criminals would be
F D P D F D C
F F F M
P P 1 1 1
)] ( ) ( [ ) )]( ( 1 [
1 2 2
ω ψ ψ ω ω ψ − + + − =
+ +
.
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A Numerical example “Social Welfare”
2 1 .. 1
) (∑
+ =
− =
P t P D C
t
k HM W ω
where is the harm from each instance of crime.
< H
Assume that types are distributed uniformly on , and
] 1 , [
w c p j δ λ γ H k 0.5 1 0.4 0.2 0.95 0.05 0.2
- 10 10
Then the relationship between the length of the jail term and the optimal level of parole is: J P* 1 4 2 11 3 12
When , the social cost minimizing level of parole term is also zero: time on parole is insufficient deterrent for crime, despite the higher arrest rate, and spending money on monitoring is inefficient.
= J
As the term in jail increases statutory release becomes
- ptimal. In fact, as jail terms become harsher, it makes sense
to use parole as a low cost way to extend incapacitation.
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Infinite Parole? Infinite parole is not optimal for two reasons: the increasing marginal cost of parole and the fact that increasing parole leads to greater recidivism. To get a sense of the latter effect, consider the type distribution for for various lengths of parole.
2 = J
When , for example types are distributed by
3 = P
Type t ψ- min ψ-max NC 0.0 0.55 D1 0.56 0.57 D2 0.58 0.61 D3 0.62 0.67 D4 0.68 1
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For , the type distribution is
8 = P
Type t ψ- min ψ-max NC 0.0 0.57 D1
- D2
- D3
0.57 0.58 D4 0.59 0.6 D5 0.61 0.63 D6 0.64 0.67 D7 0.68 0.74 D8 0.75 0.86 D9 0.87 1 Thus with eight periods of parole there are no types who are always deterred or even deterred in the second period.
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When
11 * = = P P
, recidivism has increased further. Type t ψ- min ψ-max NC 0.0 0.58 D1
- D2
- D3
- D4
- D5
0.58 0.59 D6 0.60 0.61 D7 0.62 0.63 D8 0.65 0.66 D9 0.70 0.71 D10 0.77 0.78 D11 0.79 0.93 D12 .94 1 How much time is spent violating parole? When , around 6% of all available time is used by parole violators.
3 = P
For , this share rises to nearly 15%.
8 = P
When
11 = P
, over 19% of all time is spent in parole violations.
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