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The Enforcement of Speeding:Should Fines be Higher forRepeated Offences?Eef Delhaye a ba Center for Economic Studies , Katholieke UniversiteitLeuven , Leuven, Belgiumb Transport and Mobility Leuven , Leuven, BelgiumPublished online: 31 May 2008.

To cite this article: Eef Delhaye (2007) The Enforcement of Speeding: Should Fines beHigher for Repeated Offences?, Transportation Planning and Technology, 30:4, 355-375

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ARTICLE

The Enforcement of Speeding:Should Fines be Higher for

Repeated Offences?

EEF DELHAYE

Center for Economic Studies, Katholieke Universiteit Leuven, Leuven, BelgiumTransport and Mobility Leuven, Leuven, Belgium

(Received 17 November 2006; Revised 6 May 2007; In final form 20 May 2007)

ABSTRACT When the fine structures for speeding offences are observed, it is oftenfound that fines depend on speeders’ offence history. In this paper we devise twofine structures: a uniform fine, and a fine which depends on offence history. Ifdrivers differ in their expected accident costs, the literature prescribes that thefine for bad drivers should be higher than for good drivers. However,governments do not know the type of driver. We develop a model where thenumber of previous convictions gives information on the type of driver. We findthat the optimal fine structure depends on the probability of detection, and onthe strength of the relationship between the type of driver and having a record.We illustrate this by means of a numerical example and show that, for reasonablevalues for the probability of detection, a uniform fine is preferred.

KEY WORDS: Drivers; speeding; regulation; enforcement; fines; repeat offences;traffic safety

Introduction

Since speed plays an important role in most accidents (c.f., Aarts & VanSchagen, 2006), speed limits are an important instrument for govern-ments to improve road safety. However, the imposition of speed limits

Correspondence Address: Eef Delhaye, Center for Economic Studies, Katholieke Universiteit

Leuven, Naamsestraat 69, 3000 Leuven, Belgium. Email: eef.delhaye@econ.kuleuven.be

ISSN 0308-1060 print: ISSN 1029-0354 online # 2007 Taylor & Francis

DOI: 10.1080/03081060701461758

Transportation Planning and Technology, August 2007

Vol. 30, No. 4, pp. 355�375

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alone is not sufficient; they have to be supplemented with enforcement(c.f., Becker, 1968; Stigler, 1970; Polinsky & Shavell, 2000). Enforce-ment, typically, consists of two elements: the probability of detectionand the magnitude of the fine. If the goal is to maximize social welfare,the probability of detection and the fine should, according to Polinskyand Shavell (2000), be such that

fine�expected damage due to speeding

probability of detection(1)

The faster you drive, the higher the expected damage and hence, for agiven probability of detection, the higher the fine should be. Thiscoincides with reality, since in all European countries the fine increaseswith the level of speeding and with the number of previous convictions(European Commission, 2004).

The goal of this paper is to find a rationale for this observation. Atfirst glance, it seems that the analysis of optimal fines for repeatedoffences would not differ from the analysis of a single offence. If thefine is set optimally with respect to the first offence, and the harmcaused by the second offence is the same, there is no apparent reason toset the fine differently for a second offence (Polinsky & Rubinfeld,1991). There are, nonetheless, three reasons why it might be desirableto condition fines on offence history.

First, the use of offence history may provide an additional incentivenot to violate the law when detection not only leads to an immediatesanction, but also increases the sanction for future violations. Land-sberger and Meilijson (1982) were first to analyze how prior offencesshould affect the expected fine. They focused on the probability ofdetection rather than the level of punishment, and showed that given afixed enforcement budget, a higher level of deterrence can be achievedby targeting potential violators based on past compliance rather thanby treating everyone equally. This is feasible for environmentalviolations and tax evasion, but in traffic it is difficult to control oneparticular party more than another. When the inspection frequencycannot be differentiated, it might be a logical idea, which is alsoobserved in reality, to make fines higher for repeated transgressions.However, the literature on this is ambiguous. Harrington (1988)examined increasing fines for environmental violations, but did notminimize the control costs for a given total pollution reduction. Firmswith identical pollution cost functions end up polluting at differentlevels. If one takes these costs into account, Harford and Harrington(1991) have argued that a static solution, where all firms are treatedalike, will often be superior to a state-dependent solution. Emons(2003), on the other hand, states that given that people’s wealth is

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fixed, the optimal fine scheme is decreasing rather than increasing withthe number of past violations. Emons (2002) finds that, allowing foraccidentally committing the violation, the optimal scheme is decreasingif the benefit of the harm is small, and increasing if the benefit is large.He assumes that people choose whether they will always commit theviolation or always try to comply. Hence, they cannot change theirbehavior depending on the fines they face. Polinsky and Shavell (1998)found that it is optimal to reward good behavior. The optimal sanctionfor a first time offence equals the sanction for a repeated offender in thesecond period, while the sanction is lowered in the second period if theoffender does not have a record. We cannot apply this model tospeeding since there is no record of drivers passing speed cameras whocomply with the speed limits.

Second, the offence history may provide information on thecharacteristics of individuals and the need to deter them. Polinskyand Rubinfeld (1991) explained increasing fines by assuming thatpeople receive an acceptable as well as an illicit gain from criminalactivity. In a traffic situation, the acceptable gain of speeding could bethe gain in time; unacceptable could be the thrill that joy ridersexperience of driving too fast.

Third, the traditional Becker (1968) result states that, with costlydetection and costless fines, the fine should be set as high as possible.However, there are limits on the magnitude of the sanctions imposed;for example, the maximum amount that people can pay or themaximum amount that is politically and/or socially acceptable. Ifenforcement is imperfect, the Becker result leads to higher fines forrepeat offenders if the upper limit is determined by the political and/orsocial acceptability and if people accept higher fines for repeatoffenders. This means that the expected fine increases, which in turnincreases compliance at no additional cost.

In this contribution, we explore the second reason and state that apositive relationship between previous convictions and the probability ofbeing involved in an accident may rationalize increasing fines. Driversdiffer among others in their skills and risk-taking behavior. This impliesthat drivers differ in two aspects: their propensity to have an accident andtheir ability to comply with regulations. In other words, for the same levelof speed, the probability of being involved in an accident is higher for a‘bad’ driver than for a ‘good’ driver. Moreover, even if a bad driverdecides that they want to comply, there is a probability that they willspeed ‘by accident’. This means that bad drivers have, for the same speed,higher expected accident costs, and given the structure of the optimal fine(Eq. 1), should be fined more severely.

However, governments do not know who the bad drivers are, butprevious accidents and speeding violations may act as a ‘signal’ for

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being a bad driver. The literature on the relationship between previousconvictions and the probability of being involved in an accidenttypically finds a positive relationship. Gebers (1990) stated that thenumber of previous traffic convictions (speeding, not stopping, noseatbelt) is one of the best single predictors of accident risk. Boyer et al .(1991) found that the number of accidents is an increasing function ofthe number of previous offences. Stradling et al . (2000) argued that thekind of drivers recently caught for speeding are 59% more likely tohave also been recently involved in a car accident. Dagneault et al .(2002) focused on the relationship between previous convictions andthe risk of subsequent accidents for drivers older than 65 years. Theyalso found that convictions can predict the probability of an accident,but that prior accidents are a better predictor than prior convictions.Gebers and Peck (2003) again showed that increased accident involve-ment is associated (among others) with increased prior traffic citationfrequency and increased prior accident frequency. They state thattraffic conviction frequency reflects risk-taking, social non-conformityand exposure.

The remainder of the paper is organized as follows: we first developour model and calculate the private and socially optimal speeds. Next,we calculate the speed limit. Then, we devise two fine structures,uniform and differentiated, and determine which structure performsbetter. We illustrate our model for interurban roads.

Model

We start with some notation. Next, we derive the socially and privatelyoptimal level of speed and the optimal speed limit. Subsequently, wefocus on the enforcement of this speed limit. As is common in theliterature, we assume that the speed limit and fines are chosenindependently. We derive the expressions for the uniform fine andthe differentiated fines. Finally, we look at the influence of these fineson the chosen speed and calculate the welfare losses in order tocompare the two systems.

Notation

For the individual driver, the private cost of a trip C(x) depends on thespeed x. C(x) consists of the resource cost, the fuel cost and the timecost. We assume that this cost function is convex, Cxx ]0. If speedincreases, the private cost first decreases and then increases. Indeed ifspeed rises, the time needed to complete a certain trip decreases, and,hence, the time costs decrease. This may also be interpreted morebroadly. People may simply value fast driving positively, not for the

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time gain, but for the thrill and the excitement of it. On the other hand,the fuel costs increase if speed increases. For low to intermediatespeeds, the gain in time dominates; for high speeds, the fuel costsdominate.

We consider unilateral accidents, that is, accidents in which only oneparty, the injurer, influences the probability of the accidents and theother party, the victim, bears all the losses. Think for example of anaccident between a car driver and a cyclist. Of course, in reality thecyclist also influences the probability of an accident and the driver canalso have losses, but note that the qualitative results will not change ifwe include private accident losses into the private costs C(x). In theremainder of the text, we use ‘car driver’ for the injurer and ‘cyclist’ forthe victim.

We distinguish two types of driver, ‘good’ and ‘bad’, who differ intheir ability to comply with regulations and in their expected accidentcosts. The probability of an accident p(x , ui) depends on the level ofspeed x , and on the individual propensity to have an accident ui . Peopleare either good (ui �g) or bad (ui �b) drivers. For a given level ofspeed, the probability of being involved in an accident is higher for baddrivers than for good drivers, p(x; b)�p(x; g) � x: For a given typeui , the probability of an accident is increasing in the level of speed,px(x , ui)�0 (pxx(x ,ui)]0). If an accident happens, the victim incursharm h . Note that we can make the harm dependent on speed and notthe probability, or make both dependent, but this does not influence theanalysis. Drivers also differ in their ability to comply with regulations.A good driver who wants to comply will comply. We assume that baddrivers who want to comply can still speed unintentionally. If baddrivers unintentionally speed, they drive at speed xa

b�xo�o; o�0;where xo is the intended speed. The probability of speeding by accidentis denoted by q �[0,1]. We assume that all drivers think they are gooddrivers. However, this is not a very strong assumption; in generaldrivers overestimate their abilities. Svensson (1981) has shown that80% of drivers think that they are above average drivers. Hence, baddrivers make decisions as if they are good drivers. For example, they donot take into account that they can speed by accident. There are g gooddrivers and (1�g) bad drivers. g is given exogenously with 0BgB1;we normalize the population to one.

The government only knows this distribution, but not the individualdriver types. We further assume that drivers are risk neutral.

Private and Social Optimum

When the government does not intervene, the car driver only takes theirprivate costs into account and increases their speed until

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minx

C(x)[C?(x)�0 (2)

Their private optimal level of speed xprivate is determined by the pointwhere their marginal cost of increasing speed by 1 km/hour no longerprovides a net benefit. The social optimum, on the other hand, takesinto account the expected accident costs and is determined by

minx

C(x)�p(x;ui)h[C?(x)��px(x;ui)h (3)

Note, however, that unlike Rietveld et al . (1998), we do not take intoaccount environmental costs in determining the socially optimal speed.

The socially optimal level of speed x�ui

is determined by the pointwhere the marginal cost of lowering the speed equals the marginalsocial utility of lowering the speed, which equals the decrease inexpected social accident costs. Without government intervention, cardrivers do not take the full costs of driving into account, and drive toofast (see Figure 1). On the horizontal axis we represent the level ofspeed, and on the vertical axis, the costs in euro. The upward slopingcurve represents the marginal private cost reductions of driving fasterC ?(x). The downward sloping lines are the negative of the marginalaccident cost for the good and the bad driver, �px(x , u)h . The privateoptimal level of speed is given by the intersection of the marginal costwith the horizontal axis. The socially optimal speed levels are given bythe intersections of the marginal cost with the marginal accident costs.It is clear from this figure that x�

bBx�gBxprivate: The government can

bring the private optimal speed closer to the socially optimal level bythe use of liability rules, infrastructure, vehicle regulation or speedlimits (Delhaye, 2006). In this paper we focus on the use of a speedlimit.

Figure 1. Private and social optimal level of speed

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Speed Limit

The government influences the drivers’ choice of speed by setting aspeed limit. Due to the differences in accident propensity, it would beoptimal to set a different speed limit for each type. However, theregulator lacks the information to do this, and, for this reason, sets auniform standard. This is also what we observe in the real world.

The speed limit is denoted by s . The regulator minimizes the expectedsocial costs, taking into account the distribution of drivers’ types andthe probability q with which bad drivers speed unintentionally.

minx

C(x)�gp(x; g)h�(1�g)(1�q)p(x;b)h�(1�g)qp(x�o; b)h

[C?(x)��gpx(x; g)h�(1�g)(1�q)px(x; b)h�(1�g)qpx(x�o;b)h(4)

This gives s+ the optimal uniform speed limit with x�b5s�5x�

g: s* islower than if bad drivers would be able to comply (q�0). In Figure 1,s+ is represented by the dotted line. The uniform speed limit means thatbad drivers drive faster than optimal, while good drivers drive too slow.The grey areas in Figure 1 represent the welfare losses under perfectcompliance compared to the social optimum. For the proportion q ofbad drivers that do not comply, there is an additional welfare loss equalto the arched trapezium.

However, if there is no enforcement and given that s+Bxprivate , nodriver has an incentive to comply, and they all drive at their privateoptimal speed. Therefore, we need to discuss enforcement.

Enforcement

The government uses a fine 8(x) and a probability of detection d toenforce the speed limit. In this paper, we assume that the probability ofdetection is given and fixed. Even though the probability of detectiondoes not depend on the level of speed, the fine does.

We consider two cases. In the first case, the government sets auniform fine. In the second case, the information imbedded in theoffence history is used. This makes the fine dependent on the offencehistory.

Uniform fine. A uniform fine only depends on the level of speed and noton the drivers’ compliance record. The fine is equal to zero if people donot speed, and larger than zero if people speed. This is

8(x) with8(x)�0 for x5s�

8(x)�0 for x�s�

�(5)

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The government determines the optimal fine by setting the private costof driving at a chosen speed x�s equal to the expected social cost ofdriving at speed x�s .

C(x)�d8(x)�C(x)�E[p(x;ui)h][d8(x)�gp(x; g)h�(1�g)p(x;b)h

[8�(x)�gp(x; g)h � (1 � g)p(x;b)h

d

(6)

The fine thus equals the expected social cost of speeding, corrected forthe probability of detection. Given this fine, the driver can choosewhether to speed or not. They will not speed if the cost of speeding,taking into account the expected fine, is larger than the cost of drivingat the speed limit. Hence, if they do not speed, they will drive at thespeed limit and 8(x�0). They will not drive slower than the speed limitbecause s*Bxprivate . If they speed, the problem for the driver becomes,using Eq. (6)

minx

C(x)�d8(x)[C?(x)��d8�?(x) (7)

Inserting Eq. (6) in Eq. (7) and comparing with Eq. (4), we see that thisfine makes the driver want to drive at the speed limit s+. However,q percent of bad drivers will speed by accident and drive at speed xa

b�s+�o; o�0: Given that all drivers think they are good drivers, theywill not take this into account when choosing their level of speed.

We show this in Figure 2. The ‘thick line’ gives the negative firstderivative of the expected fine. People choose the speed where the firstderivative of the private costs, which is the marginal benefit of speed,equals the marginal cost of speed, which is the expected fine. Thishappens at s+. Hence, good drivers drive slower than socially optimal,and bad drivers drive faster. q percent of the bad drivers will drive at

Figure 2. Uniform fine

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speed xab�s+: The social welfare loss for a good driver (WLg) equals

the black triangle in Figure 2. They have to drive at a speed where themarginal benefit of increasing ones’ speed � the private cost � is higherthan the marginal social costs � the accident costs. The grey trianglerepresents the social welfare loss for a bad driver who complies (WLb).They drive at a speed where the marginal benefit of speed is lower thanthe marginal social costs. The welfare loss for a bad driver who fails tocomply equals the grey triangle, (WLb), plus the hatched trapezium,/(WLa

b): Total social welfare loss of a uniform fine (WLuf) then equals

WLuf �gWLg�(1�g)(1�q)WLb�(1�g)q(WLb�WLab)

�gWLg�(1�g)WLb�(1�g)qWLab

(8)

Fine depends on offence history. The government does not know who thegood and bad drivers are. However, it does know that there is a positiverelationship between the number of previous convictions and theprobability of an accident. Therefore, the drivers are divided into twogroups: a group with no record, and a group with a record. A drivergets a record if they caused an accident and/or if they are caughtspeeding. Note that the model does not change if we assume that onlyaccidents or only speeding is recorded. However, since accidents are arare event, accidents on their own will probably not provide enoughinformation to distinguish between both groups. Similarly, not allspeeding is recorded, and the occurrence of accidents may giveadditional information about the drivers’ type if the probability ofhaving an accident for bad drivers is distinguishably larger than forgood drivers. If not, including accidents on the record only createsnoise.

Both groups will consist of good and bad drivers. This is animportant difference with the uniform case. Denote Pg

nr as theproportion of good drivers without a record, Pg

r the proportion ofgood drivers with a record, Pb

nr the proportion of bad drivers without arecord and Pb

r the proportion of bad drivers with a record. Note thatPg

nr�Pgr �1 and Pb

nr�Pbr �1: We calculate these proportions in

equilibrium using Markov chains later in this paper.The government then sets a fine 8(x, k), which depends firstly on the

level of speed x and secondly on the history k of the driver.

8(x; k) with8(x;k)�0 for x5s8(x;k)�0 for x�s

�(9)

where k equals 0 if the driver has no criminal record, and equals 1 if thedriver has a criminal record.

We assume that the government set fines in the following way. If thedriver has no record, the regulator assumes that they are a good driver

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and equates the private costs with the social costs for a good driver.This means that

C(x)�d8(x;0)�C(x)�p(x; g)h

[8o(x;0)�p(x; g)h

d

(10)

If the driver has a record, the government assumes that they are a baddriver and the fine equals

C(x)�d8(x;1)�C(x)�p(x; b)h

[8o(x;1)�p(x;b)h

d(11)

We do not assume that the regulator sets the fines socially optimal.Comparing Eqs. (11), (10) and (6) yields immediately that 8o(x;1)]8�(x)]8o(x; 0): How will this structure influence the speed choice ofthe drivers, and, hence, the welfare losses? Drivers again choosewhether to speed or not. If they choose not to speed, they drive atthe speed limit since s+Bxprivate and they will not pay a fine. Rememberthat bad drivers can speed unintentionally with probability q . If they dospeed, they pay a fine, which depends on their criminal record. Thereare four cases we need to consider: good drivers with and without arecord, and bad drivers with and without a record. We first discuss,with the help of Figure 3, the case for drivers without a record and thenturn, using Figure 4, to the case where drivers have a record. Bothfigures have the same structure as Figure 1. For x5s+, 8(x , k) equalszero and coincides with the horizontal axis. For x�s+, 8(x , 0) is givenby the wide line in Figure 3, and coincides with the marginal accidentcost for the good drivers. 8(x , 1) coincides with the marginal accidentcost for the bad drivers and is represented by the wide line in Figure 4.

Good drivers with no record. When good drivers speed their problemis represented by

minx

C(x)�d8(x; 0)[C?(x)��d8x(x;0) (using Eq: (9))

[C?(x)��px(x; g)h(12)

Note that Eq. (12) is the same as Eq. (3), and that good drivers with norecord speed (x�

g�s�) and choose the socially optimal level of speed.Hence, there are no welfare losses for this group.

Bad drivers with no record. Bad drivers with no record face the sameproblem as in Eq. (12), and, hence, choose speed x�

g�x�b: The welfare

loss (WLnrb ) of this equals the gray triangle. However, a proportion q of

them will drive faster unintentionally. The additional welfare loss(WLnra

b ) of this subgroup is represented by the black trapezium.

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Good drivers with a record. When good drivers with a record speed, theyminimize the following problem

minx

C(x)�d8(x;1)[C?(x)��d8x(x;1) (using Eq: (10))

[C?(x)��px(x;b)h(13)

The solution to this problem is speed x�b at which point the expected

fine is zero. Hence, they will drive faster than x�b; and choose to drive at

the maximum speed limit. The welfare loss for this group (WLrg) equals

the black triangle in Figure 4.

Bad drivers with a record. Bad drivers with a previous record also faceproblem Eq. (13); hence they try to comply. A proportion (1�q) of baddrivers drive at the maximum speed level and their welfare losses (WLr

b)are denoted by the grey triangle. The other q percent of bad driversspeed unintentionally and their welfare losses equal the grey triangle(WLr

b); and the small hatched trapezium (WLrab ):

Figure 3. Differentiated fine for drivers without a record

Figure 4. Differentiated fine for drivers with a record

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Total welfare losses for a differentiated fine (WLdf) then equal

WLdf �gPnrg 0�Pnr

b (1�g)(WLnrb �qWLnra

b )�gPrgWLr

g

�(1�g)Prb(WLr

b�qWLrab ) (14)

Comparison of welfare losses. Which structure of fines should theregulator choose? They should compare the welfare losses under auniform fine, given by Eq. (8), with the losses under a differentiatedfine, given by Eq. (14). The differentiated fine is better if the welfarelosses are lower than under a uniform fine, this is, if

0�(1�g)Pnrb (WLnr

b �qWLnrab )�gPr

g WLrg|ffl{zffl}

�WLg

�(1�g)Prb

�WLr

b�qWLrab|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}

�WLb�qWLab

�

BgWLg�(1�g)(WLb�qWLab)

[(1�g)Pnrb (WLnr

b �qWLnrab )Bg(1�Pr

g)WLg�(1�g)(1�Prb)(WLb�qWLa

b)[(1�g)Pnr

b (WLnrb �WLb�q(WLnra

b �WLab))BgPnr

g WLg

(15)

We cannot say that one structure always dominates the other. In orderto be able to compare the welfare losses, we have to calculate Pr

uiand

Pnrui: If past violations are a good predictor of the drivers’ type, Pnr

b verylow and Pnr

g very high, it is more likely that Eq. (15) will hold. We cancalculate Pr

uiand Pnr

uiin equilibrium if we know the movements from

the drivers in and out the two groups. We argue that good drivers withno records will speed and move to the ‘record group’ for two reasons:they are caught with probability d or they cause an accident withprobability p(x�

g; g): Good drivers with a record will comply and movewith an exogenous probability u back to the ‘no record group’ if theydo not have an accident. This reflects the fact that, if the driver is notcaught and did not cause an accident, after a period of time their recordis cleared. Bad drivers with no record will also speed and have the sameprobability d of being caught and transferred to the ‘record group’. Theprobability that they have an accident, p(x�

g;b)�p(x�g; g) is higher.

Hence, the probability that a ‘bad driver without a record’ receives arecord is higher than the probability for a ‘good driver without arecord’. If bad drivers have a record, they will try to comply. However,with probability q they will speed unintentionally, and with probabilityd they are caught. Moreover, they will also stay in the ‘record group’ ifthey have an accident. Hence, their probability of moving to the ‘norecord group’ is lowered to u �qd �p(s*,b). We summarize thesemovements in the transition matrices represented in Table 1.

The long run equilibrium, or steady state, probabilities Pnrg ; P

rg; P

nrb

and Prb may be found by solving the following sets of linear equations

(Winston, 1994):

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Pnrg �Pr

g�1

Pnrg � (1�d�p(x�; g))Pnr

g �(u�p(s�; g))Prg

(16)

and

Pnrb �Pr

b�1Pnr

b � (1�d�p(x�; b))Pnrb �(u�qd�p(s�; b))Pr

b

(17)

Using this information, we can calculate the steady state equilibriumand find that

Pnrg �

u � p(s�; g)

d� u � p(x�g; g) � p(s�; g)

; Prg�

d� p(x�g; g)

d� u � p(x�g; g) � p(s�; g)

Pnrb �

u � qd� p(s�; b)

d� qd� u � p(x�g; b) � p(s�; b)

; Prb�

d� p(x�g; b)

d� qd� u � p(x�g; b) � p(s�; b)

(18)

In general Pnrg �Pnr

b and PrgBPr

b; so proportional to the population, itis most likely that there are more good drivers than bad drivers in the‘no record group’, and that there are more bad than good drivers in the‘record group’.

The best structure is the one with the lowest welfare losses. Hence,we prefer a uniform fine if WLu BWLdf and vice versa. At first sight, itis still impossible to see which structure will perform the best. Itdepends mainly on the level of g , Pnr

ui; Pr

ui: However, for one case,

although unrealistic, the situation is clear-cut. If the probability ofdetection equals the probability of speeding unintentionally equals theprobability of having your record cleared equals one, d�q�u�1;then Pnr

g �Prg�1=2 and Pnr

b �0; Prb�1: This is, the bad group

coincides perfectly with the record group, and the good drivers areevenly distributed into the two groups. In this case, the differentiatedfine outperforms the uniform fine for any g . The welfare losses for thebad drivers are the same under both fine systems, but under thedifferentiated fine, half of the good drivers will drive at their sociallyoptimal speed.

Table 1. Transition matrices

Good drivers Bad driversa

No record Record Record Record

No record /1�d�p(x+g; g) /d�p(x�

g; g) /1�d�p(x�g; b) /d�p(x�

g; b)Record /u�p(s+; g) /1�u�p(s+; g) /u�qd�p(s+; b) /1�u�dq�p(s+; b)

aAlthough it is more correct to replace p (x , b ) in the formulae by (1�q ) p (x , b )�qp (x�o , b ), we choose not to do this for ease of notation.

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In reality, d , q and u will not take such extreme values. Whichstructure performs best, switches for a certain values for d , q and u .This is shown in the illustration discussed in the next paragraph.

Numerical Example

We apply the model to interurban roads since accidents between carsand cyclists are most likely on this type of road. The current speed limitin Belgium on interurban roads is either 70 or 90 km/hour. We firstcalculate the private and socially optimal level of speed and the optimaluniform speed limit. We then compute the fines and compare them withthe current fine structure. We end this section by calculating the welfarelosses to determine which fine structure performs best. The private costfor a driver is assumed to equal the sum of the resource costs, the timecosts and the fuel costs. The resource cost consists of the purchase cost,the insurance cost, maintenance, etc. We assume that this cost isindependent of speed and equals 0.2355 t/km (t1�US$1.35) (DeBorger & Proost, 1997). The fuel cost depends on the fuel type, theprice, and the consumption. All elements needed to calculate the fuelcosts are shown in Table 2.

The time cost equals the value of time divided by the level of speed.For the value of time, we make a weighted average of the value of timeof commuters, business and others. Based on Gunn et al . (1999) andHuber and Toint (2002), we obtain a value of time of 6.3917 t/hour.The sum of the resource cost, the weighted fuel cost, and the time costmean that we can express the private costs as

C(x)�0:30599�6:3917

x�0:3919 �10�3x�0:536 �10�5x2 (19)

The private optimum for drivers then equals 98 km/hour (min C(x)).In order to derive the expected accident cost (p(x , ui)h), we first

consider the present accident risk (accrisk (a)), given a current speed of80 km/hour, and following Elvik et al . (2000), correct this for changesin speed. We then multiply this risk with the value for the harm done

Table 2. Fuel costs

Fuel type Fuel price (t/l) Consumption (l/km) % share

Diesel 0.811 0.13778�0.00242x�0.000016x2 40.6Gasoline 1.068 0.0396�0.00064x 59.4

Source : Ministerie van Economische Zaken (2001), MEET Project (1998), IEA(2002), Ministerie van Verkeer en Infrastructuur (2000).

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(h(a)) with a the accident type. We focus on accidents with slightlyinjured, heavy injured and deaths. Hence, we do not take into accountaccidents with only material damage. The expected accident costs for agood driver can then be expressed as

p(x; g)h�X

a�acctype

accrisk(a)��

x

current speed

�m(a)

h(a) (20)

with m (slight injury)�2, m (heavy injured)�3, m (fatal)�4.We assume that the accident risk for bad drivers is 1.59 times

the accident costs for capable drivers (Stradling et al ., 2000), this is,p(x , b) h�1.59 p(x , g)h . Table 3 gives the Current accident risk andthe cost of an accident.

Taking these accident costs into account yields the socially optimalspeeds of x�

g�71 and x�b�64 km/hour.

Given this information, and assuming that 20% of the population arebad drivers and o�5 km/hour, we first calculate the socially optimaluniform speed limit and find that s+�68.8 km/hour. Note that this isclose to the current 70 km/hour speed limit. Note also that if no onespeeds by accident, q�0, the optimal speed limit increases and equals69.2 km/hour. The reason is that the gain in private costs due tounintentionally speeding is lower than the increased expected accidentcost. Next, we calculate the uniform and the differentiated fines,assuming a probability of detection of 0.9% per trip and an averagetrip of 13 km. In order to compare these with the speed fine structure inBelgium, we average these fines over the same classes as the currentstructure. Table 4 shows the results.

If we assume a probability of detection of 0.9% (there is no dataavailable for Belgium on what the actual probability of detection couldbe), the calculated fines for small offences equal the existing ones. Forlarger offences, the current fines increase more steeply in the level ofviolation than the calculated fines.

In order to compare these two fine systems, we need to calculate thewelfare losses. We begin by removing the assumption on theprobability of detection d , we assume that 80% of the drivers are

Table 3. Accident risk and accident cost

Accident type Cost accident (t) Accident risk

Light 26.273 7.92*10�7

Serious 965.131 1.27*10�7

Fatal 2.197.540 0.25*10�7

Source : Own calculations based on De Brabander (2005), NIS (2005).

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good drivers (g�0.8), that the probability that bad drivers speedunintentionally equal 40% (q�0.4) and that the probability to returnto the ‘no record group’ equals 30% (u�0.3). The last figure means,for example, that you move to the ‘no record group’ after three years ifyou were not caught or did not have an accident during these threeyears. Given this information, we calculate Pr

ui; Pnr

uiand the difference

in social welfare, DWFL(‘welfare losses uniform fine’ less ‘welfarelosses differentiated fine’). The result is given in Figure 5. Notice thatthe kink is due to the fact we imposed the condition thatPr

ui;Pnr

ui� [0; 1]:

The optimal structure switches for a certain probability of detectiond . We see that for dB0.417 the uniform fine performs better than thedifferentiated fine. If d�0.417 the differentiated fine performs better.It is hard to know the real probability of detection in Belgium, but it ismost likely lower than 41.7% per trip (the percentage obtained fromautomatic speed control sources). In this case, we should prefer auniform fine. The reason for this is that the signal is not sufficientlystrong. For example, if we set q�1, where all bad drivers speedunintentionally, the differentiated fine is preferred as soon as d�0.179.Moreover, if q�1 and the expected accident cost of bad drivers is fourtimes the cost of good drivers, the differentiated fine should be chosenonce d�0.12. Thus, if the signalling is better, the differentiated fineperforms better. When the probability of an accident is the same forboth types, and, hence, the only difference is the unintentionallyspeeding (q�0.4), the differentiated fine is chosen when d�0.44. Also,when q�0, when there is no unintentional speeding and the onlysignalling is the higher probability of an accident for the bad drivers, auniform fine is always preferred. We show this in Figure 6.

Table 4. Comparison with the existing structure

Speeding(km/hour)

Results

Present structureAverage immediate

collection (t)

Uniform fine (t) Differentiatedfine (t)

Fine no record Fine if record

B10 50 51 44 7810�40 128 89 79 136�40 Court 256 226 381

Source : Wegwijs in het Belgisch Verkeersreglement, KB 30 September 2005, owncalculations.

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As the probability of detection d increases, the differentiated fineperforms better. The reason for this is that, if the probability ofdetection increases, the proportion of bad drivers with a record alsorises, and this rise is larger than the rise in good drivers with a record.

If the probability of clearing the record rises to 80% (u�0.8),the uniform fine always outperforms the differentiated fine. If theprobability of clearing the record decreases to 10% (u�0.1), thedifferentiated fine is preferred once the probability of detection is largerthan 7%. Hence, for a given q , the probability of detection underwhich the differentiated fine performs better decreases in the prob-ability of clearing the record.

If we assume that there are more good drivers (g�0.95), we find thatthe differentiated fine performs better if the probability of detection isgreater than 0.38. The reason for this is that for a large part of thepopulation, the welfare losses will be zero. If g rises, the right hand sideof Eq. (15) becomes larger, while the left hand side becomes smaller,making it more likely that the condition is fulfilled. If there are less gooddrivers (gB2%), we find that a uniform fine is preferred. For the twocorner solutions (g�0/1) the uniform fine structure is the only solution.

The exercise is not very sensitive for different values of o . If o�1 km/hour, we prefer uniform fines as long as dB0.42. If o�10 km/hour theswitching probability of detection equals 0.40. Thus, the probability ofdetection for which differentiated fines become socially a better optionincreases with o . However, given the interpretation of unintentionallyspeeding, o cannot grow very large.

Figure 5. Difference in welfare losses if g�0.8, q�0.4, u�0.3, o�5

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This numerical example shows that, for reasonable probabilities ofdetection, the uniform fine should be preferred. The reason for this isthat, in general, the correlation between driver type and having a recordor not is not very strong. Having a record is not a good signal for drivertype. Only when bad drivers drive really badly, that is, they have a highprobability of unintentionally speeding and their accident costs aremuch higher, the differentiated fines are to be preferred at relativelylow probabilities of detection.

Conclusion

When fine structures for speeding offences are analyzed, two char-acteristics are often found. First, the level of the fine increases accordingto the severity of the violation. Second, fines increase with offencehistory. The first result is commonly found in the literature. For thesecond result, there is much more controversy. Increasing fines in linewith offence history are often found in the real world, but remain atheoretical puzzle.

We focused on the structure of the fines and on repeated offences.We did not look for the optimal structure, but merely compared twosystems: a uniform fine and a fine dependent on offence history. Ourrationale for having offence-dependent fines was as follows. Peoplediffer in their ability to follow the rules and in their propensity to causean accident. That is, there are good and bad drivers, and bad drivers

Figure 6. Difference in welfare losses if g�0.8, q�0, u�0.3, o�5

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can speed unintentionally even if they want to comply. Moreover, theexpected accident cost for bad drivers is higher than for good drivers.Standard theory then prescribes that bad drivers should be fined moreseverely than good drivers. Moreover, increasing fines may be preferredover uniform fines because they allow good drivers to drive faster thanthe speed limit, which is set too low for them. However, thegovernment does not know who is a good and who is a bad driver.The literature shows that there is a relationship between the probabilityof being involved in an accident and the number of previous offences.Hence, we claim that a record of offences may act as a signal for thetype of driver.

A uniform fine means that good drivers are fined too harshly and baddrivers not enough. However, the differentiated fine system also doesnot work perfectly because there is no perfect correlation betweendriver type and the record group. There are bad drivers in the ‘norecord group’ and good drivers in the ‘record group’. The choicebetween these two systems depends on how good the relationship isbetween type of driver and the record of the driver.

We devised a numerical illustration which examined two aspects.First, we calculated the optimal values for the speeding fines andcompared these with the existing fines in Belgium. We found that thecurrent fine structure increases faster than our calculated fines. We alsofound that for the current fines to be optimal, the probability ofdetection should be around 0.9% per trip. Further, we also studied thecritical values for the probability of detection, which determine thechoice between the two fine structures. The analysis showed that forreasonable values for the probability of detection a uniform fine shouldbe preferred.

There are two important extensions which could be made to themodel. First, it would be interesting to calculate the socially optimalincreasing fine structure. However, this cannot be done within thisframework as it stands. Second, it would be realistic for drivers to learntheir driver type. Bad drivers with a record want to comply, hence ifthey are caught speeding, they should conclude that they are baddrivers. Intuitively, we can say that by incorporating this would makebad drivers drive slower under both fine structures, and, hence, wouldnot qualitatively change the comparison between the two finestructures.

Acknowledgements

The author would like to acknowledge the financial support of the Belgian FederalScience Policy Research Program � Indicators for Sustainable Development � ContractCP/01/38 (Economic Analysis of Traffic Safety: Theory and Applications), S. Rousseau

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for useful comments, C. Billiet for advice on the legal system, and two referees for

helpful suggestions.

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