people more severely. But under a legal system of any reasonable degree of reliability, the probability of convicting an innocent offender twice is relatively low. Suppose first-time
offenders and repeat offenders are punished uniformly under the original regime. To maintain the same deterrence effect as in the original regime, the state can reduce the penalty
on first-time offenders a bit and correspondingly increase the penalty on repeat offenders. By doing so, we assign a larger smaller cost coefficient to the term less more likely to contain
mistakes. The type-II error costs are, therefore, reduced with the net deterrence effect unchanged. It will be shown in later sections that this intuition is indeed correct.
Our argument is, in fact, a formal characterization of some recent informal arguments in the literature of law and economics. Posner 1992 took possibly erroneous convictions as
one of the reasons to punish repeat offenders more severely. He mentions that p. 233 “ . . . the fact the defendant has committed previous crimes makes us more confident that he
really is guilty of the crime with which he currently is charged; the risk of error if a heavy sentence is imposed is therefore less.” Cooter Ulen 1997 also point out that convicting
an innocent person is more costly than failing to convict a guilty person, and the criminal law strikes the balance between costs associated with type-I and type-II errors in favor of the
defendant. In the next section, we shall formalize these arguments and provide an uncon- ditional answer to the question of whether we should penalize repeat offenders more
severely.
The model and the main result will be described, respectively, in sections 2 and 3. Section 4 presents conclusions and extensions.
2. The model
2.1. Individual decision making Our analytical framework adopts the two-period model proposed by Polinsky Rubinfeld
1991 and Polinsky Shavell 1996. Each individual in the society lives two periods, and the discount rate is zero no time preferences. During each period, one offense at most can
be committed. All individuals are assumed to be risk neutral. If they choose to commit the crime, they obtain a benefit b. The objective of each individual is to maximize his expected
payoff, which is the sum of the payoffs in both periods.
Let Fb be the distribution of benefit b among individuals. If we consider the crime in question to be theft, then offenders with larger b can be interpreted as more “efficient” thiefs
who can steal more things than other thiefs with smaller b. The legal system can be characterized by p, q, s, s
r
, where p is the probability of convicting guilty offenders, and q, p is the probability of convicting innocent offenders. These probabilities are indepen-
dent of one’s record and are the same in both periods.
3
s is the sanction for first-time offenders and is the same in both periods, and s
r
is the sanction for repeat offenders. Those who commit crimes in period 1 but are not convicted have no record and are treated as
3
For the case when the conviction probability is subject to state control, see the discussion in section 4. 131
C.Y.C. Chu et al. International Review of Law and Economics 20 2000 127–140
first-time offenders. We allow the penalty on repeat offenders to be higher than first-time offenders: s
r
s. If s
r
5 s, the penalty structure is uniform and repeat offenders face the same sanctions as first-time offenders. Given p, q, s, s
r
, individuals choose whether to commit crimes or not in both periods.
In period 2, if one has an offense record and decides to commit a crime, his expected payoff is b 2 ps
r
. If one has an offense record and decides to be law-abiding, his expected payoff is 2qs
r
. Thus, one who has a record will commit a crime in period 2 if b 2 ps
r
. 2qs
r
; otherwise he will be law-abiding. As such, in period 2 the expected payoff for one who has an offense record, denoted p
2r
, is p
2r
5 maxb 2 ps
r
, 2 qs
r
. Similarly, the expected payoff for one who has no offense record, denoted p
2f
, is p
2f
5 maxb 2 ps, 2 qs.
In period 1, a rational individual will take the expected payoff in period 2 into account in making his decision. The expected payoff of committing a crime in period 1, denoted p
1f
, is p
1f
5 b 2 ps 1 pp
2r
1 ~1 2 pp
2f
. 1
The expected payoff of being law abiding in period 1, denoted p
1n
, is p
1n
5 2qs 1 qp
2r
1 ~1 2 qp
2f
. 2
2.2. Individual Optimal Decisions Depending on the size of b, individual decision patterns are separated into three types.
Because the analysis involved is straightforward algebra, we present the result below and leave the details to Appendix A.
T
HEOREM
1: 1 Individuals with b , p 2 qs [ b
2
will be completely deterred and will not commit crimes in either period. 2 Individuals with b . p 2 qs
r
[ b
1
can never be deterred and will commit crimes in both periods. 3 Individuals with b [ [p 2 qs, p 2 qs
r
] will commit an offense in period 1 if and only if
b . ~ p 2 q~1 2 ps 1 qs
r
1 2 p 1 q ; b
, and they will commit an offense in period 2 if and only if they are not convicted in the first
period. Fig. 1 illustrates the conclusion in Theorem 1.
2.3. The Composition of Social Costs Our objective is to minimize total social costs. In view of the remark in section 1.2, we
consider crimes without social gains. The harm from committing a crime including the
132 C.Y.C. Chu et al. International Review of Law and Economics 20 2000 127–140
direct harm and the indirect costs suffered by society is denoted by h, which is assumed to be a constant.
The total social cost, C, is composed of two parts: C 5 H 1 kE,
4
where H is the direct harm from crimes, kE is the cost of erroneously convicting innocent offenders, and k
characterizes the weight the society puts on erroneous conviction.
5
The gains of the offender are not counted in the calculation of the social cost either because the crime in question is
not socially acceptable or because the gains of offenders from crimes such as theft or other zero-sum crimes offset with the losses of victims. In view of Fig. 1, we see that the two types
of social costs in period 1 are as follows:
H
1
5 1 2 F~b
h, kE
1
5 kF~b
qs. 3
Those with b . b with probability 1 2 Fb
will commit crimes that cause direct harm. Others with proportion Fb
will not commit a crime, but q proportion of them will be erroneously convicted.
The social costs in period 2 are: H
2
5 F~b
2 F~b
2
~1 2 qh 1 F~b
1
2 F~b~1 2 ph 1 1 2 F~b
1
h 4
kE
2
5 kF~b
2
q~1 2 qs 1 ~qs
r
1 kF~b
2 F~b
2
q~qs
r
1 kFb
1
2 F~b p~qs
r
. 5
Eqs. 4 and 5 will be interpreted below. Those with b , b
2
will not commit a crime in period 2, but q proportion of them will be erroneously convicted. Among those who are
erroneously convicted in the second period, part of them with proportion q were also erroneously convicted in period 1 and have a criminal record; these people will receive the
penalty s
r
. Among those who are erroneously convicted in the second period, 1 2 q of them do not have any criminal record, and they will receive the penalty s. This explains the first
term of kE
2
Those with b
2
, b , b will not commit a crime in period 1. However, q proportion of
them were erroneously convicted in period 1 and have criminal records. These people will not commit a crime in period 2, but they still face probability q of being erroneously
convicted again and punished by s
r
. This explains the second term of kE
2
. The other 1 2 q will commit crimes in period 2 that cause direct harm to society, characterized by the first
term of H
2
. Those with b
, b , b
1
will commit crimes in period 1. Part of them p were convicted in period 1 and have a criminal record. These people will not commit a crime in period 2,
but they still face probability q of being erroneously convicted and punished by s
r
. This is the third term of kE
2
. The others 1 2 p will commit crimes in period 2 that cause direct harm to society, constituting the second term of H
2
.
4
The punishment involved can be thought of as fines, and we assume that there is no social costs associated with it.
5
If the sanctions refer to imprisonment, then k is the cost of per unit length of wrongful imprisonment. See also the discussion in section 4.
133 C.Y.C. Chu et al. International Review of Law and Economics 20 2000 127–140
Finally, those with b . b
1
will commit crimes in both periods. The harm they caused in period 2 is the third term of H
2
. We now complete the interpretation of Eqs. 4 and 5. 2.4. Social Cost Minimization
With zero interperiod discount, the total social costs are the sum of H and kE in both periods. Straightforward algebra gives us the following formula:
H 5 H
1
1 H
2
5 2h 2 F~b
2
~1 2 qh 2 F~b ~1 2 p 1 qh 2 F~b
1
ph kE 5 kE
1
1 kE
2
5 kF~b
2
~1 2 qqs 1 kF~b qs 1 ~q 2 pqs
r
1 kF~b
1
pqs
r
. The objective of the society is to minimize
L 5 H 1 kE 6
by choosing s and s
r
. An alternative interpretation of Eq. 6 is as follows. The solution to the minimization
problem in Eq. 6 will be the same as the solution to the following constrained minimization problem.
min H s.t. E E
where E is the upper bound of erroneous conviction that the society can endure. The above
problem says that we minimize the social harm subject to a tolerable type-II error, and the k in Eq. 6 in fact can be interpreted as the Lagrange multiplier, representing the shadow
price of type-II errors.
3. The rationale for differential penalties
