Cost of public funds, rewards and law

enforcement

Sébastien RouillonGREThA

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)

As p*(0) > p*(), p*(l) is decreasing, for all l.l

p*(0)

p*() p*(l)

p*(l)

Case where h is small (i.e., h < h)

As p*(0) < p*(), p*(l) is increasing, for all l.l

p*(0)

p*() p*(l)p*(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°)

As p*(0) F < h and p*(l) is decreasing,some under-deterrence is optimal, for all l.

l

b

p*(0) F

h°=p*() F p*(l) F

hUnder-deterrence

Deterrence area

Case where h is small (i.e., h < h°)

As p*(l) is increasing and converges to p*(), ifh° = p*() F > h, over-deterrence is optimal, for l > l°.

l

b

p*(0) F

p*(l) Fh Under-deterrence

Deterrence area

h°=p*() F

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

l

P(l)p°(l), r°(l)

p°(l)

r°(l)

R

l0 l1

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)

p

c’(p)

h

p*(0) 1