Cost of public funds,
rewards and law
enforcement
Sébastien Rouillon
GREThA
ELEA, Sept. 17, 2009
Benchmark literature
According to the literature on Law
Enforcement, it is socially worthwhile to
satisfy the following rules:
Rule 1. The fine should be maximum (Becker,
1968).
Rule 2. Some underdeterrence should be
tolerated (Polinsky and Shavell, 1984).
Cost of public funds
In this paper, we check whether rules 1 and 2
remain valid when we assume the existence
of a positive cost of public funds.
This assumption is justified if only
distortionary schemes, such as capital, income
or good taxation, are available to the
government to raise additional public funds.
Jellal and Garoupa (2002)
Jellal and Garoupa (2002) show that a positive cost
of public funds augments the degree of
underdeterrence at an optimum.
Indeed, when the government must finance the
enforcement policy through distortionary taxation,
the cost of detection and conviction is larger.
Garoupa and Jellal (2002)
Garoupa and Jellal (2002) implicitly suppose that
neither the enforcer nor the government receive the
fines paid. Thus, the government must finance the
entire cost of enforcement.
This paper builds on the converse assumption. That
is, we assume that the government ultimately
recovers the fine revenue and, therefore, only needs
to finance the enforcement expenditures net of the
fine revenue.
The (standard) model
Risk-neutral individuals contemplate whether to
commit an act that yields benefits b to them
and harms the rest of society by h.
The policy-maker observes the harm h, not the
individual’s benefit b. However, he knows the
distribution of b among the population,
described by a general density function g(b)
and a cumulative distribution G(b).
The (standard) model
The government sets the enforcement policy, by
choosing a fine, f, and a probability of
detection and conviction, p.
The maximum feasible sanction is F.
The expenditure on detection and conviction to
achieve a probability p is given by c(p), where
c’(p) > 0 and c’’(p) > 0.
The marginal cost of public funds is l.
Assumption. c’(0) = 0 and c’() is arbitrary large.
Deterrence, Revenue and Social
Welfare
An individual will commit a harmful act if, and
only if, b  p f.
Thus, the expected revenue will be:
t = p f (1 – G(p f)) – c(p),
and the social welfare will be:
pf (b – h) g(b) db – c(p) + l t.
For all l, we will denote by f*(l) and p*(l),
the choice of f and p that maximizes the social
welfare.
Deterrence, Revenue and Social
Welfare
For all l, we will denote by f*(l) and p*(l),
the choice of f and p that maximizes the social
welfare.
We present it in the following three slides,
beginning with the polar cases when l = 0 and
l = , and using these to expound the general
solution (0 < l < ).
Deterring harmful activities (l = 0)
When l = 0, the social problem is to deter
harmful activities only. At an optimum, the
fine should be set as high as possible:
f*(0) = F,
and the probability of detection and
conviction p*(0) should equalize the marginal
benefit and the marginal cost of enforcement:
F (h – p*(0) F) g(p*(0) F) = c’(p*(0)).
Raising public funds (l = )
When l = , the social problem is to raise public
funds only. At an optimum, the fine should be set as
large as possible:
f*() = F,
and the probability of detection and conviction p*(0)
should equalize the marginal revenue and the
marginal cost of enforcement:
F [1 – G(p*() F) – p*() F g(p*() F)]
= c’(p*()).
Considering both objective jointly (0 <
l < )
Proposition 1. (a) The optimal fine f*(l) is the
maximal fine F. (b) There exists a threshold
level for the harm h, denoted h, such that
p*(0) <, = or > p*(), whenever h <, = or > h.
(c) The optimal probability of detection and
conviction p*(l) is monotone and varies from
p*(0) to p*() as l goes from zero to infinity.
Case where h is large (i.e., h > h)
p*(l)
p*(0)
p*()
p*(l)
l
As p*(0) > p*(), p*(l) is decreasing, for all l.
Case where h is small (i.e., h < h)
p*(l)
p*()
p*(l)
p*(0)
l
As p*(0) < p*(), p*(l) is increasing, for all l.
Over-deterrence can be optimal
Corollary 1. Let h° = p*() F > 0. If h < h°, it will
be socially worthwhile to overderdeter
harmful activities for sufficiently large l.
(Precisely, there exists l° > 0 such that p*(l) F
> h if, and only if, l > l °). Otherwise, some
underterrence is always optimal.
Case where h is large (i.e., h  h°)
b
h
p*(0) F
h°=p*() F
Under-deterrence
p*(l) F
Deterrence area
l
As p*(0) F < h and p*(l) is decreasing,
some under-deterrence is optimal, for all l.
Case where h is small (i.e., h < h°)
b
h°=p*() F
h
p*(0) F
p*(l) F
Under-deterrence
Deterrence area
l°
l
As p*(l) is increasing and converges to p*(), if
h° = p*() F > h, over-deterrence is optimal, for l > l°.
Why not rewarding good guys ?
In some areas of law enforcement (such as the
compliance with the traffic laws, the tax codes
or the environmental regulations), the
enforcer visits the individuals at random and
convicts them to pay the fine, whenever he
finds they have had the wrong behaviour.
The government could also ask him to pay a
reward r (with r  R) to those individuals that
he finds compliant.
Deterrence, Revenue and Social
Welfare
An individual will commit a harmful act if, and
only if, b  p (f + r).
Hence, the expected revenue will be:
t = p [f (1 – G(p (f + r))) – r G(p (f + r))] – c(p),
and the social welfare will be:
p(f+r) (b – h) g(b) db – c(p) + l t.
For all l, we will denote by f°(l), p°(l) and
r°(l), the choice of f and p that maximizes the
social welfare.
Deterrence, Fine Revenue and Social
Welfare
Let P(l) be the solution of the problem of
choosing the probability of detection p to
maximize l p F – (1 + l) c(p).
To interpret, notice that this objective
function coincide with the social welfare if we
assume that the individuals always engage in
the harmful activity (i.e., each time b > 0).
Optimal enforcement policy with
rewards
Proposition 2. (a) The optimal fine f°(l) is the
maximal fine F. (b) There exists l0 and l1, with
0 < l0 < l1, such that:
(i) If l < l0, then p°(l) > P(l) and the optimal
reward r°(l) is the maximal reward R;
(ii) If l0  l  l1, then p°(l) = P(l) and the
optimal reward r°(l) lies between 0 and R;
(iii) If l > l1, then p°(l) < P(l) and the optimal
reward r°(l) is the minimal reward 0.
Shape of the solution
p°(l), r°(l)
P(l)
R
p°(l)
r°(l)
l0
l1
l
When is it socially worthwhile to
reward?
Proposition 3. The threshold level l1 of the
cost of public funds below which it is optimal
to reward the individuals found in compliance
is decreasing in the maximal fine, and
increasing in the harm and the cost of
enforcement.
Thank you for your attention.
When is it socially worthwhile to
reward?
Suppose that:
c(p) = kp2/2;
b is uniformly distributed on [b0, b1].
Let B = b1 – b0.
The social welfare will be:
p(f+r) (b – h)/B db
+ l p [f – p (f + r)2/B] – (1 + l) k p2/2.
When is it socially worthwhile to
reward?
The social welfare will be:
W = p(f+r) (b – h)/B db
+ l p [f – p (f + r)2/B] – (1 + l) k p2/2.
The FOC are:
dW/dp = l f – (1 + l) k p + (f + r) A,
dW/dr = p A,
where: A = (h – p (f + r))/B – 2 l p (f + r)/B.
When is it socially worthwhile to
reward?
Consider the situation where: f = F, p  ]0, 1[, r = 0.
It is socially worthwhile to reward r > 0 if:
dW/dr = p [h – (1 + 2 l) p F]/B < 0.
Deterring harmful activities (l = 0)
c’(p)
c’(p)
h
p*(0)
1
p