Jean Pinquet
email: jean.pinquet@polytechnique.edu

Georges Dionne
email: dionne@hec.ca

Charles Vanasse

Mathieu Maurice

INCENTIVE MECHANISMS FOR SAFE DRIVING: A COMPARATIVE ANALYSIS WITH DYNAMIC DATA

Keywords: Mécanismes incitatifs, permis à points, sécurité routière Point-record mechanisms, incentive effects, road safety Classification JEL: D81, C23

Les politiques de sécurité routière utilisent souvent des mécanismes incitatifs basés sur les infractions pour améliorer le comportement des conducteurs. Ces mécanismes sont par exemple des amendes, des primes d'assurance ou des permis à points. Nous analysons l'efficacité incitative de ces mécanismes. Nous obtenons leurs propriétés théoriques par rapport au nombre de points associés aux infractions et par rapport au temps contrat. Ces propriétés sont ensuite testées empiriquement avec des données issues du système public d'assurance au Québec. Nous concluons à la présence d'aléa moral dans les données, qui traduit le fait que les conducteurs qui accumulent les points deviennent plus prudents car ils sont plus sous risque de perdre leur permis. Par ailleurs, la prime indicée sur les points introduite en 1992 a réduit de 15% la fréquence d'infractions. Nous utilisons ce résultat pour calculer des équivalents monétaires pour les infractions et les retraits de permis.

Introduction

Since the 1970s fatality rates due to road-tra¢ c accidents have decreased steadily in developed countries, although risk exposure increased concomitantly (see OECD, 2005). For example, over the last ten years, the road fatality rate decreased by forty percent in France. However, the implied social cost of road accidents remains high (Doyle, 2005). A major reason for the improvement of the situation in the OECD has been the development of incentives for safe driving. Experience rating schemes used by the insurance industry have incentive properties (see Boyer and Dionne, 1989; Abbring et al, 2003). They are supplemented by point-record driver's licenses based on tra¢ c violations. In many countries, each convicted tra¢ c o¤ense is …led with a speci…c number of demerit points. When the accumulated number of points exceeds a given threshold, the driver's license is suspended. Point removal clauses are added so that this penalty can be avoided in the long run. 1 A point-record driver's license was implemented in Quebec in 1978, together with a no-fault insurance plan for bodily injuries which replaced a tort system. [START_REF] Boyer | The Economics of Road Safety[END_REF] The road fatality rate decreased by …fty percent during the …fteen years that followed. [START_REF] Boyer | An Empirical Analysis of Moral Hazard and Experience Rating[END_REF] In Quebec, the Société de l'Assurance Automobile du Quebec (referred to as SAAQ below) is a public monopoly that provides coverage for bodily injury.

The SAAQ is also in charge of accident prevention and control, including the management of driver's licenses. Before 1992, the rating structure for bodily injury insurance was completely ‡at. Since December 1, 1992, the public authorities in Quebec have implemented an experience rating scheme based on accumulated demerit points. This mechanism was added to other incentives, i.e., …nes, the point-record driver's license in force since 1978, and the private sector insurance rating for coverages other than bodily injury. This paper analyzes the incentive properties of …nes, point-record driver's licenses and experience rating based on tra¢ c violations. Studies on incentive 1 hal-00414479, version 1 -9 Sep 2009 mechanisms for road safety have appeared in the economic literature for many years (Peltzman, 1975; Landes, 1982; Boyer and Dionne, 1987). In the presence of asymmetric information, insurers use partial insurance or experience rating to improve resource allocation. Both schemes have proved to be e¢ cient for handling moral hazard and adverse selection. Empirical tests have measured the e¤ectiveness of such mechanisms for road safety (Sloan et al, 1995; Boyer and Dionne, 1989) and the presence of residual asymmetric information problems in insurers' portfolios (Chiappori and Salanié, 2000; Dionne, Gouriéroux and Vanasse, 2001). More recently, Abbring, Chiappori and Pinquet (2003) designed a new test based on the dynamics of insurance contracts to detect the presence of moral hazard. Their model makes it possible to separate the moral hazard e¤ect on accidents from unobserved heterogeneity. They found no evidence of moral hazard in the French car insurance market. The convex structure of the French "bonus-malus" system is used to show that the optimal e¤ort level exerted by a rational policyholder increases after a claim at fault. In our study, insurance pricing is not the major incentive scheme but rather a measure used to complement …nes and the point-record driver's license. Moreover, the pricing scheme of the Quebec public automobile insurance is an increasing step function of past demerit points.

Insurance pricing may not su¢ ce as a tool for designing an optimal road safety policy because it may not create the appropriate incentives for reckless drivers (Sloan et al, 1995). Bourgeon and Picard (2007) show how point-record driver's licenses provide incentives for road safety among normal drivers (those who respond to the usual incentive schemes) when the judicial system or the insurance market fail to provide optimal incentives. Point-record driver's licenses also allow the government to incapacitate reckless drivers because …nes for tra¢ c violations are bounded above in many jurisdictions. Bounded …nes exist for di¤erent reasons: 1) many o¤enders are judgment proof and are unable to pay optimal …nes; 2) many drivers are expected to escape from paying tickets issued by the authorities when …nes are very high; 3) society thinks it is unfair that rich and reckless drivers will pay high …nes and continue to drive dangerously (Shavell, 1987a, 1987b). However, …nes do reinforce the e¤ectiveness of the point record mechanism by providing normal drivers with more incentives.

In Bourgeon and Picard's model, which uses only two levels of prevention, the optimal …ne must be set at the maximal level and must be neither progressive nor regressive. These authors also discuss the optimality of point removal mechanisms as a screening device. Public intervention can be justi…ed when there is a signi…cant di¤erence between the private and the social cost of human lives (Viscusi, 1993). Finally, drivers may be unaware of their own accident or infraction probabilities or may misunderstand some features of the incentive environment.

We present the data base in Section 2 as well as our …rst empirical results related to the introduction of the new pricing policy implemented in 1992. The point-record mechanisms (driver's license suspensions and insurance pricing) are described in Section 3 and their incentive properties are investigated in an optimal behavior model in which time and e¤ort are continuous. If incentives are caused by …nes and by the point-record driver's license, we show that the optimal e¤ort level increases globally with the number of demerit points accumulated and decreases with the seniority of non-redeemed tra¢ c violations, if any. Tra¢ c violation risk varies conversely, as it decreases with the e¤ort level.

These results are compared with the data in Section 4. The observed dynamics on the drivers result from the incentive e¤ects and from the revelation with time of hidden features in risk distributions, i.e. an unobserved heterogeneity e¤ect. Let us compare these two e¤ects at the time and event (accident or tra¢ c violation) level of the data dynamics. Drivers with more tra¢ c violations committed during a given period are riskier with respect to hidden features in risk distributions. Hence unobserved heterogeneity entails a risk reassessment after each event, and the event e¤ect of risk revelation counters the corresponding incentive e¤ect of the point-record driver's license. The revelation e¤ect on the risk level of a period without tra¢ c violations is negative. [START_REF] Bourgeon | Point-Record Driving License and Road Safety: An Economic Approach[END_REF] The time e¤ect of unobserved heterogeneity is also converse to that of the incentive e¤ect, which raises an identi…cation issue in the interpretation of the observed dynamics on the drivers. Abbring et al's (2003) test for moral hazard can be used if there are no time e¤ects in the incentives. Because the time e¤ects are important in Quebec's point-record system, we use another approach. We test for an increasing link between tra¢ c violation risk and the number of demerit points. Rejecting this assumption amounts to …nding evidence of moral hazard (ab absurdo, because of the increasing event e¤ect created by unobserved heterogeneity). In Section 4.1, the incentives created by the threat of driver's license suspension are found to increase with accumulated demerit points. These …ndings con…rm the theoretical analysis.

The insurance rating scheme introduced in 1992 reduced the frequency of tra¢ c violations by 15%. In Section 4.2, we link the incentive levels of the three point-record mechanisms as derived from the theoretical analysis of Section 3 and the observed e¢ ciency of these mechanisms. Monetary equivalents for tra¢ c violations and license suspensions are derived from this analysis. Conclusions are drawn in Section 5 and technicalities are relegated to an appendix.

Presentation of the data base and preliminary empirical results

Our data base represents roughly one percent of the SAAQ portfolio. The panel covers the period from January 1, 1983 to December 31, 1996. A …rst sample of 40,000 license holders was selected at random at the beginning of 1983. About 300 young drivers were added each following year. [START_REF] Chiappori | Testing for Asymmetric Information in Insurance Markets[END_REF] The attrition rate per year is close to 1.5%, which is very low compared with the private sector. Due to the monopolistic status of the SAAQ, leaving the company means that drivers lose their license, which explains the low attrition rate. The endogenous attrition is not very high. It was estimated from a bivariate probit model on tra¢ c o¤enses and departures from the sample. A score test for the nullity of the correlation coe¢ cient between the two equations [START_REF] Cox | Regression Models and Life Tables[END_REF] was performed with the regression components set used in Section 4. The null hypothesis was not rejected at a …ve percent signi…cance level. Hence the attrition risk adds no signi…cant information to the assessment of tra¢ c violation risk.

The personal characteristics of each driver are available on the driver's license for the current period. These characteristics are used as regression components in the empirical study. Several types of events are recorded in the data base; they are listed below with related variables in addition to the date. 1) Accidents that have led to a police report. Only those with bodily injury are compensated by the SAAQ. 2) Convicted violations of the Highway Safety Code, together with the number of demerit points used in the point-record mechanisms.

The number of demerit points is based on the severity of the tra¢ c violation.

Their distribution is given in Section 3.4. 3) Driver's license suspensions, which are spells rather than events, and 4) Premium payments, which since the 1992 reform are related to accumulated demerit points. These payments are made every two years on the policyholder's birthday.

Between 1985 and 1996, the average yearly frequencies of accidents with bodily injuries, accidents of all types (not including joint reports …led with private insurers) and tra¢ c violations are equal to 1.4%, 6.7% and 16.9% respectively. Figure 1 represents the relative frequencies derived from a one year centered moving average. [START_REF] Dionne | Testing for Evidence of Adverse Selection in the Automobile Insurance Market: A Comment[END_REF] Insert Figure 1 about here There is an overall decline in the frequency of accidents, whereas the frequency of tra¢ c violations remains more stationary. This may seem surprising, but it is explained by the evolution of the tra¢ c control environment. For instance, the number of tra¢ c control devices such as radars increased during the 1980s and 1990s. An increase in the rate of tra¢ c o¤enses recorded by devices or police o¢ cers among those committed explains this relationship. Figure 1 shows evidence of several periods where the frequency of tra¢ c violations increased along with opposite variations in the frequencies of accidents. A step-up in tra¢ c control during these periods may well explain such observations. A tra¢ c violation committed by a driver must be selected twice in order to be …led with demerit points. It must …rst be recorded by a control device or a police o¢ cer. We already mentioned that the related selection rate had previously increased. Second, the o¤ender must be convicted of the recorded tra¢ c violation. The …ling of a tra¢ c violation is somewhat discretionary. Since the 1992 reform for instance, drivers are being forced to pay more in premiums given demerit points, and we might expect policemen to be more hesitant to hand them out, and to give warnings instead. [START_REF] Dionne | The Role of Memory in Long-Term Contracting with Moral Hazard: Empirical Evidence in Automobile Insurance[END_REF] The conviction rate is less likely to vary with time than the recording rate.

In Figure 1, a downturn is also observed in the frequency of tra¢ c violations just before the date (December 1, 1992) of the reform that introduced the experience rating structure based on demerit points. The reform was announced in August 1, 1992, which may explain why the downturn starts slightly before December 1, 1992. The experience-rated premium was …rst applied at the contract anniversary following December 1, 1992 and used the complete two year history of demerit points. Hence the new incentives for safe driving took e¤ect for most of the drivers at August 1, 1992 (i.e. drivers with their next birthday after December 1, 1992). [START_REF] Doyle | Health Insurance, Treatment and Outcomes: Using Auto Accidents as Health Shocks[END_REF] On average, the annual frequency of tra¢ c violations was equal to 17.6% before the reform and 15.4% afterwards, which corresponds to a 12.5% decrease. The 1992 reform can be interpreted as a laboratory experiment to test whether experience rating based on demerit points reduces tra¢ c violations. Nonetheless, the lower rate of tra¢ c violations following the 1992 Quebec reform may be due to the changes in other factors that in ‡uence driver behavior. Identifying the in ‡uence of these factors necessitates a control group that is not a¤ected by the policy change. Unfortunately, we do not have access to such a control group because the insurer is a monopoly and bodily injury insurance is compulsory. [START_REF] Fagart | Liability Insurance under the Negligence Rule[END_REF] In Section 4.2, we shall link the average decrease in the frequency of tra¢ c violations before and after the 1992 reform to the overall e¤ectiveness of incentive schemes.

Monetary and non-monetary incentives for safe driving are based on tra¢ c violations as well as the optimal behavior models presented in Section 3. However the actual social cost of road tra¢ c is caused by accidents. To reconcile these two approaches, two results are worth mentioning. First, demerit points are good predictors of accidents. This is well documented in the literature and is con…rmed on our data in Section 4.1. Second, the global stationarity of convicted tra¢ c violation frequency observed in Figure 1 concurs with a probable decrease in the frequency of committed tra¢ c violations (see the aforementioned developments on selection rates). Lowering tra¢ c violation risk through point-record mechanisms should also lower accident risk and the related social cost.

Finally, Figure 1 shows that accidents with bodily injuries evolve in much the same way as those recorded in the SAAQ …le. We include accidents of all types in the empirical analysis in order to obtain more stable results. [START_REF] Helland | Does Three Strikes Deter? A Nonparametric Estimation[END_REF] 3 Incentive e¤ects of point-record mechanisms

Point-record mechanisms in Quebec

In this section, we describe Quebec's point-record mechanisms which are based on tra¢ c violations, both monetary (insurance premiums) and non-monetary (point-record driver's license). Comparisons are made with respect to the mechanisms used by other countries. [START_REF] Landes | Insurance, Liability, and Accidents: A Theoretical and Empirical Investigation of the E¤ect of No-fault Accidents[END_REF] We investigate the incentive properties of point-record mechanisms in Sections 3.2 to 3.4.

In many countries, driver's license suspensions are based on demerit points.

In Quebec, demerit points are assigned to convictions for tra¢ c o¤enses, and their number depends on the tra¢ c violation severity. When the accumulated number of demerit points reaches or exceeds a given threshold, the driver's license is suspended. Before January 1990 this threshold was set at twelve in Quebec and has since been increased to …fteen.

In order to mitigate the social cost of license suspensions, point removal systems exist for most real-world point-record driver's licenses. In Quebec, the demerit points related to a given driving o¤ense are removed after two years.

Hence, driver's license suspensions depend on the demerit points recorded during the last two years. Most of point removal systems used by American states follow the same approach. The average number of demerit points per convicted o¤ense is 2.4 in Quebec. It takes about six tra¢ c violations within two years to trigger a license suspension, an unlikely outcome when the annual tra¢ c violation frequency is 16.9%. However, the heterogeneity of risks is high and a point-record driver's license is also an incapacitating device for risky and reckless drivers through the license suspensions. Another point removal system consists in cancelling all the demerit points after a given period of violation-free driving.

This mechanism was recently implemented in Spain, with a two year seniority.

The French driver's license uses the two point removal systems.

The experience rating structure introduced by the SAAQ in December 1, 1992 links each premium paid every two years to the demerit points accumulated over the previous two years. The rating structure is given in Section 3.4. Once the premium is paid, the driver is reinstated with a fresh zero point record.

Thus the length of the record relevant to the derivation of optimal behavior never exceeds two years.

3.2 Basic model for a point-record driver' s license without point removal Bourgeon and Picard (2007) analyze the incentive e¤ects of point-record driver's licenses. Their model uses a binary e¤ort variable. We extend their approach with a continuous e¤ort level. Hence the e¤ectiveness of e¤ort may also be a continuous function of contract time, a desirable property for empirical validation. We show that under fairly general conditions, a rational policyholder's safe driving e¤ort will increase with the number of demerit points accumulated.

We assume that the driver's license is revoked when the driver reaches a total of N demerit points. For the sake of simplicity, each convicted tra¢ c violation is linked to one supplementary demerit point in this section. A driver with a suspended driver's license is reinstated after a period D with a fresh zeropoint record like that of a beginner. [START_REF] Manning | Health Insurance and the Demand for Medical Care: Evidence from a Randomized Experiment[END_REF] The duration D may be …xed or random in the model. In Quebec, a license suspension is of random length because drivers must pass a new exam after a given period before recovering their license. 14 A rational driver maximizes his expected lifetime utility expressed in $ and derived from:

An instantaneous driving utility, d u .

A time-dependent disutility of e¤ort, denoted as e(t). [START_REF] Peltzman | The E¤ects of Automobile Regulation[END_REF] This e¤ort level is linked to an instantaneous tra¢ c violation frequency risk, denoted as (e(t)). The hazard function (e) corresponds to a probability p(e) in static or discrete time incentive models, and is assumed to be a positive, decreasing and strictly convex function of the e¤ort level.

In this section, we suppose that there is no point removal mechanism. In that case, the lifetime expected utility (we assume an in…nite horizon) depends only on the number n of accumulated demerit points, and is denoted as u n . The Bellman equation on the expected utility leads to

u n = d u r c (u n u n+1 ) r ; (0 n < N ); (1)
where r is a subjective discount rate, and where c is de…ned as follows

c ( u) = def min e 0 e + [ (e) u] : (2)
Technical details can be found in Appendix A. u 0 and u N , the lifetime expected utility just after the suspension of the driver's license. For instance, if the private disutility of driver's license suspension is only the loss of driving utility during a period D, we have that

u N = u 0 ; = E[exp( rD)]: (3)
The utilities are then derived from the recurrence equations ( 1) and ( 3). Optimal e¤ort depends on the variation of lifetime utility u because it minimizes the function de…ned in equation ( 2). The variable u (the argument of c in equation ( 2)) which determines optimal e¤ort will be referred to later as the incentive level. In this setting, optimal e¤ort depends on the number n of accumulated demerit points but not on time, and we denote it as e n . It is shown in Appendix A.1 that e n increases with n for any given value of N . The related frequency of violations n = (e n ) thus decreases with n. The intuition behind this result is the following. A given reduction in tra¢ c violation risk is more ef-…cient as the threat of the license suspension gets closer. Hence the e¢ ciency of e¤ort increases with the number of demerit points accumulated, and we obtain an increasing link between this number and the optimal e¤ort level. A parallel can be drawn with the "three strikes and you're out" rule enforced in California to deter crime. The deterrence e¤ect increases from one to two strikes (Helland and Tabarrok, 2007). [START_REF] Shavell | The Optimal Use of Nonmonetary Sanctions as a Deterrent[END_REF] Fines represent another monetary incentive scheme applied in Quebec throughout the period investigated in this study. Let us denote f a as the average …ne for a tra¢ c violation conviction. Given that …nes and premiums are low in comparison to average wealth, risk aversion is not signi…cant. If …nes are combined with the preceding point-record driver's license, the incentive level is equal to f a + u n u n+1 . This means that the average …ne is added to the utility loss in the variable which determines optimal e¤ort. Besides, the optimal e¤ort still increases with n for a given value of the average …ne.

Point-record driver' s licenses with point removal

In Quebec, each tra¢ c violation is redeemed at the end of a two-year period.

Integrating this feature in the optimal behavior model is di¢ cult, because all the seniorities of non-redeemed driving o¤enses must be included as state variables in the dynamic programming equations. Lifetime utility is expected to increase with time for a given number of demerit points accumulated. Optimal e¤ort depends on the di¤erence between the present utility and a substitute utility (i.e., that reached after an additional tra¢ c violation). With the point removal system in force in Quebec, the substitute utility increases with time as does the present utility. Time should have more value for worse situations, which

implies that the substitute utility should increase faster than the present utility.

Optimal e¤ort should thus decrease with time. Appendix A.2 provides proof that optimal e¤ort is continuous at the time of a point removal. Optimal e¤ort is then expected to increase with each tra¢ c violation in order to compensate for the decreasing link between time and e¤ort. To summarize, we expect the e¤ort level to increase globally with the number of demerit points accumulated and to decrease with the seniority of non-redeemed tra¢ c violations, if any.

Insert Figure 2 about here In Section 4, we will use the model of Section 3.2 as a proxy for the incentive environment in Quebec.

3.4

Incentive e¤ects of premiums indexed on demerit points: The example of Quebec Table 1 presents the rating structure enforced for each driver's license on the …rst contract anniversary following December 1, 1992. The premium paid every two years after this date depends on the number of demerit points accumulated in the last two years. It does not represent the total premium for bodily injury insurance but rather the additional premium related to demerit points. This average premium is equal to $54.60, and complements a yearly driver's license fee for insurance coverage equal to $107.

Insert Table 1 about here

In this section, the incentive properties of this rating structure are analyzed separately from the point-record driver's license. An important input is the distribution of demerit points for a given driving o¤ense, which we left out in Section 3.2. Denoting f j as the proportion of tra¢ c violations with j demerit points, we have the following values

f 1 = 4:71%; f 2 = 52:32%; f 3 = 38:34%; f 4 = 2:83%; f 5 = 1:80%: (4)
Note that f 5 actually refers to o¤enses with …ve points and more. Table 1 shows that the premium is a step function of the accumulated demerit points.

Because of the local non-convexity of the premium, the incentives may not always increase with the number of demerit points accumulated. Consider, for instance, a policyholder just before her contract anniversary. The incentive level will be stronger with two accumulated demerit points than with four. With four points, it is indeed less than likely that the next tra¢ c o¤ense will trigger an increase in premium. The corresponding probability is 2:83+1:80 = 4:63%, if we assume that the distribution of the f j is independent of the accumulated demerit points. The incentives for safe driving are stronger at a two point level because the probability of climbing a step in the rating structure after a tra¢ c o¤ense is close to one. The aforementioned result is in contrast to that obtained by Abbring et al (2003) for the French "bonus-malus" scheme and its exponential structure. Hence, step pricing schemes may have poor incentive properties. They are employed because they are simple.

Let us design an optimal behavior model based on this rating structure.

Once the premium is paid, the driver is reinstated with a fresh zero point record.

Hence the optimal control model can be designed with the next contract anniversary as the horizon. Let n be the premium paid for n demerit points accumulated during a period of T = 2 years between two contract anniversaries.

As we disregard the point-record driver's license and its possible deprivation, the driving utility is no longer a parameter. However, we retain …nes in the in-centives for safe driving. We denote v n (t) as the expected disutility of premiums and …nes paid until the next contract anniversary, where t is the seniority of the last anniversary and n is the number of demerit points accumulated since that date. We have the terminal conditions v n (T ) = n ; 8n = 0; : : : ; N:

If incentives are related to …nes and insurance premiums, the incentive level is the sum of the average …ne f a and of the expected variation of v n (t)

after a tra¢ c o¤ense (see Appendix A.3). Hence optimal e¤ort is determined by

f a + v n (t), with v n (t) = P j = f j >0 f j v min(n+j;N ) (t) v n (t): Derivations
show that the average of v n (t) with respect to n does not vary much with time. The terminal values of v n are derived from equations ( 4), ( 5) and Table 1. For n = 0; 2; 4; we obtain v 0 (T ) = $2:32; v 2 (T ) = $47:65; v 4 (T ) = $3:43:

The incentives with two points accumulated are much stronger than with four points, which con…rms the analysis following equation (4). The overall average of v n (t) with respect to t and n is close to $12, a 9% increase for an $130 average …ne. In Section 4.2, this increase will be compared with the variation in tra¢ c violation frequency before and after the reform.

4 Empirical results on the incentive e¤ects of point-record mechanisms

Point-record driver' s license

In this section, we analyze the data before the 1992 reform that introduced the experience rating scheme based on demerit points. Thus the point-record driver's license interacts only with …nes. Regressions are performed from January to December 1992, the date of the reform enforcement. We try to con…rm the theoretical …ndings of Sections 3.2 and 3.3 (i.e., the e¤ort level increases globally with the number of demerit points accumulated and decreases with the seniority of non-redeemed tra¢ c violations, if any), and to …nd evidence of moral hazard in the data.

We estimate the hazard functions of convicted tra¢ c o¤enses and accidents with a proportional hazards model (Cox, 1972). We retained the following spec-

i…cation j i (t) = exp( x i (t) j + g j (adp i (t))) h S i (t) j (c i (t)): (7)
In equation ( 7), j i (t) is the hazard function of type j (j = 1 : tra¢ c violation or j = 2 : accident) for driver i at calendar time t. Regression components are denoted by the line-vector x i (t). We retained the gender, driver's license class, place of residence, age of the driver, calendar e¤ects related to years and months, and the number of past license suspensions. [START_REF] Shavell | Economic Analysis of Accident Law[END_REF] The number of demerit points accumulated in the last two years is denoted as adp i (t): Let us comment the expected shape of g 1 ; which links tra¢ c violation risk to accumulated demerit points. The revelation e¤ect of unobserved heterogeneity is increasing in adp, and is converse to the incentive e¤ect. In the absence of moral hazard, g 1 should then be increasing. We will test the null hypothesis that g 1 is weakly increasing against the alternative that it is not (i.e. g 1 strictly decreases for at least some values). Rejecting the null amounts to …nding evidence of moral hazard. [START_REF] Sloan | E¤ects of Tort Liability and Insurance on Heavy Drinking and Driving[END_REF] E¤ort is expected to decrease with time only if the number of demerit points accumulated is greater than zero. Hence we speci…ed a strati…ed proportional hazards model. [START_REF] Viscusi | The Value of Risks to Life and Health[END_REF] The baseline hazard functions h S i (t) j depend on the risk type j and on the stratum S i (t). There are two strata, depending on whether the variable adp i (t) is equal to zero or not. Lastly, contract time c i (t) is integrated into the baseline hazard function h j . The function c i is set at zero at the beginning of the whole period. Then it is reset to zero at each event that trig-gers a variation of the accumulated demerit points (i.e., tra¢ c violation or point removal). This event-driven operation should eliminate interactions between calendar and contract-time e¤ects for the stratum associated with adp > 0.

Insert Table 2 about here

Table 2 shows that b g 1 is increasing until adp reaches the seven point threshold, at which point the increasing property is no longer ful…lled, and b g 1 reaches a maximum at seven points. It is worth mentioning that the SAAQ warns the policyholders when their accumulated demerit points increase beyond a seven point threshold. 20 When the accumulated demerit points are less than or equal to seven, the unobserved heterogeneity e¤ects outweigh the incentive e¤ects because b g 1 is increasing. Controlling for other covariates, a driver with seven accumulated demerit points is exp(0:974) = 2:65 times riskier than a driver with a violation-free record. Hence unobserved heterogeneity e¤ects strongly outweigh incentives to drive carefully between zero and seven points. This result is reversed beyond seven points. For instance, a driver with seven accumulated demerit points that gets a supplementary two point o¤ense has a tra¢ c violation risk that is 19% lower. As the unobserved heterogeneity e¤ect is increasing with adp, the estimated risk reduction due to the incentive e¤ect is greater than 19%. A 20% value for the risk reduction after a tra¢ c o¤ense will be retained in the next section, in order to derive monetary equivalents for tra¢ c violations and license suspensions. Table 2 presents tests where the null assumption is g 1 (7) g 1 (adp); for adp > 7. The null is rejected for these values of adp at signi…cance levels greater than 1:3%, except for adp = 11: We also performed a global test on the weakly increasing shape of g 1 : The test statistic is a squared distance between the unconstrained estimation of g 1 given in Table 2, and the set of weakly increasing functions of adp (see Appendix A.4 for more details).

A graphical illustration is given in Figure 3. We reject the increasing shape assumption for every usual signi…cance level. As an absence of incentive e¤ects entails an increasing shape for g 1 (unobserved heterogeneity e¤ect), rejecting the null shows evidence of moral hazard.

Insert Figure 3 about here Table 2 shows less evidence of moral hazard on accident risk than on tra¢ c violation risk, as the estimation of g 2 increases until adp = 10. A possible interpretation is that we cannot separate at fault from no-fault accidents. In the literature, the incentive e¤ect is usually higher with at fault accidents. Besides, drivers nearing the license suspension threshold might also apply opportunistic strategies regarding tra¢ c violations (e.g. paying more attention to radars) without otherwise modifying their attitude towards road tra¢ c risk.

The estimated hazard functions from the tra¢ c violation equation globally decrease with time, including the one related to the stratum adp > 0. 21 This means that the continuous time e¤ect created by the revelation of unobserved heterogeneity outweighs the incentive e¤ects (see Figure 2 and the discussion that follows). This is not surprising, given that the average increase of g 1 in Table 2 (a vast majority of drivers have less than seven demerit points) must be balanced by a negative time e¤ect in order to obtain stationary risk levels on average.

Incentive e¤ects of the 1992 reform and monetary equivalents for tra¢ c violations and license suspensions

In this section, we assess the e¢ ciency of the 1992 reform. We compare its incentive level to those of …nes and of the point-record driver's license. We relate the e¢ ciency of the reform to its relative weight in the three incentive mechanisms, and we infer results on the shape of the link between the cost and e¢ ciency of e¤ort. Lastly, we derive monetary equivalents for tra¢ c violations and license suspensions, from the estimated e¢ ciency of incentives created by a supplementary tra¢ c violation.

In Section 2, we mentioned a 12.5% decrease in the average frequency of tra¢ c violations before and after the reform which introduced the experience rating structure based on demerit points. This result changes slightly if we control with the regression components used in Table 2. A regression estimated from 1985 to 1996 with the covariates of Section 4.1 and a dummy related to the period following December 1, 1992 associates the reform with a 15% decrease. 22 The results of Section 3.4 (for instance equation ( 6)) suggest that the number of demerit points accumulated since the last anniversary should in ‡uence the e¤ectiveness of the 1992 reform. However, we did not obtain signi…cant results in this direction. The drivers'limited knowledge of the environment could explain this poor result, a point developed later.

We perform an overall comparison of the three incentive schemes. We use the model without point removal of Section 3. We now apply this result to the 1992 reform. As the reform entailed a signi…cant reduction in tra¢ c violation risk regardless of the number of demerit points accumulated, we can assume that incentives are e¤ective for a representative driver. 23 This leads us to analyze the elasticity between tra¢ c violation risk and the incentive level. Suppose that we leave out the modi…cations of incentive levels due to the aggregation of incentive mechanisms. Then we can relate:

On the one hand, a 15% reduction in the frequency of tra¢ c violations since the 1992 reform.

On the other hand, a relative increase in the incentive level ranging between 9% and 10%. Indeed, the 1992 reform entails a $12 average increase in the incentive level. This increase supplements the other components of the incentives, i.e., the $130 average …ne and the utility variation for the point-record driver's license. In Table 2, the point-record driver's license o¤ers signi…cant incentives beyond a seven point threshold, a result corresponding to only a minority of drivers (1.4%). The contribution of the point-record driver's license to the incentives is low as compared with that of …nes.

This suggests that the elasticity between the optimal frequency risk and the incentive level is less than -1 in this case. This result is linked to a locally concave shape of log( ) for the representative driver. However external e¤ects could also explain the reduction in the frequency of tra¢ c violations. We cannot eliminate these e¤ects because there is no control group that is not a¤ected by the reform.

Besides, the elasticity would be modi…ed if the distribution of demerit points for a given driving o¤ense (see equation ( 4)) was wrongly perceived by the drivers.

The $12 contribution of the reform to the incentive level is low, because of the high frequency of drivers without demerit points since the last birthday (87%),

and the low incentive level of the reform for these drivers (see equation ( 6)). This violations and license suspensions. In the discussion on the elasticity between the optimal frequency risk and the incentive level, we argued that an overestimation of the frequency of severe tra¢ c violations would increase the perceived incentive e¤ect of the 1992 reform. The $12 average e¤ect of the reform on the incentive level should then be upgraded, and the monetary equivalents of tra¢ c violations and license suspensions should also be upgraded.

Conclusion

In this article, we analyze the properties of policies designed to promote safe driving. Three important incentive mechanisms for road safety are used in Quebec. When the Quebec government introduced a no-fault insurance regime in 1978, it implemented a point-record driving license to maintain incentives for road safety. Because prevention activities are not observable by the public insurer, the new mechanism uses drivers'accumulated demerit points to approximate past safety activities and to increase incentives to reduce infractions. Under moral hazard, drivers who accumulate demerit points become more careful because they are at threat of losing their license. This result was con…rmed from the data. Otherwise, we would have accepted an increasing link assumption between accumulated demerit points and tra¢ c violation risk. Fines are on average the most e¢ cient device, but they are associated to a uniform incentive level. We designed our incentive models with a representative driver, but there is of course heterogeneity in the individual parameters, such as the threshold beyond which the incentives are e¤ective. We did not have wealth variables at hand, and an interesting empirical issue would have been to cross such variables with a reform dummy in risk assessment.

The experience-rated premium based on accumulated demerit points is a monetary point-record mechanism. The empirical results exhibit a rather uniform e¤ectiveness after its enforcement in 1992, i.e. a 15% decrease in the frequency of tra¢ c violations. Its incentive e¤ects do not strictly increase with the accumulated demerit points, however, because of the steps in the rating structure. The actual incentive e¤ect of the reform looks more like that induced by an increase in the average …ne. The SAAQ modi…ed its rating policy in 2008, with a premium increase from the …rst demerit point.

Regulations are not self-enforcing. To successfully achieve a reform, the regulator must implement enforcement activities and bear the costs thereof. In our application to road safety, the Police is the most important enforcer of the di¤erent incentive schemes. Police o¢ cers have a limited capacity to modify the penalties for a given infraction but may a¤ect the detection probabilities based on the aggregate convictions cycles, the driver's accumulated demerit points or other tasks. When analyzing our results, we must consider the e¤ect of these potential behaviors. For example, is it possible that when accumulated demerit points are greater in a given …le, police o¢ cers have an incentive to issue them more or less often because of risk of the driver's losing his license.

We did not have precise information to control for these factors. It is well documented that the Police collaborates very well with the Quebec public insurer and Government. There is consequently a low probability that aggregate safety scores would have a¤ected the Police behavior without the agreement of the Government, which is assumed to be fully committed to its road safety policies. Regarding the e¤ect of accumulated demerit points, we do not believe there is a general policy in the Police force on the behavior to adopt based on drivers' past experience although some individuals may modify their behavior in one direction or the other. This e¤ect is implicitly considered as a purely random e¤ect in our analysis and should not have a¤ected our results on the point record incentives.

In this study we have not examined the long-term evolution of accidents in detail, because we did not have access to the control variables of interest. In recent years, many road safety initiatives have had an impact on accidents but did not necessarily have any e¤ect on violations. These initiatives include measures such as occasional campaigns to prevent fatal accidents, increased police patrols to reduce speeding and designated driver campaigns to prevent drinkers from getting behind the wheel. The decline in deaths and serious injuries can also be explained by vehicular improvements and the wearing of seat belts. All such measures are complementary to those studied in this article.

Notes

1 These clauses and their incentive properties are detailed in Section 3. [START_REF] Boyer | The Economics of Road Safety[END_REF] The North American continent preceded Europe in the design of such systems. Pointrecord driver's licenses were introduced in 1947 in the USA. Germany, France, and Spain implemented these mechanisms in 1974, 1992 and 2005, respectively. [START_REF] Boyer | An Empirical Analysis of Moral Hazard and Experience Rating[END_REF] We do not have information on factors di¤erent from the incentive schemes analyzed in this article that might have explained this decrease. This study is performed in a no-fault environment. For a recent comparison of strict liability and a negligence rule for risk-incentives trade-o¤, see Fagart and Fluet (2009). [START_REF] Bourgeon | Point-Record Driving License and Road Safety: An Economic Approach[END_REF] Increases in premium after an event and "no claims discount" clauses observed in non-life insurance can be explained by the revelation e¤ect of unobserved heterogeneity. [START_REF] Chiappori | Testing for Asymmetric Information in Insurance Markets[END_REF] Selecting at random one percent of the new license holders every year would evidently have been a preferable sampling procedure. One thousand new license holders would then have been selected every year, as the entry rate in the SAAQ portfolio is close to 2.5%. [START_REF] Cox | Regression Models and Life Tables[END_REF] Binary variables related to tra¢ c o¤enses and attrition were created on a monthly basis, and explained with the covariates used in Section 4.1. The score test statistic is equal to 0.34. Hence we do not reject the nullity of the correlation coe¢ cient at the usual signi…cance levels.

with a contract anniversary falling between the announcement of the reform and its enforcement are not incited by the experience-rated premium before this anniversary. Incentives exist otherwise (for these drivers after their birthday, and for all the other drivers since the announcement of the reform). A referee suggested using this natural experiment in order to disentangle the incentive e¤ects of the reform from calendar e¤ects. We did not obtain significant results. Four months is a short period, and on average only one driver out of twelve was not incited by the rating scheme during the period. [START_REF] Fagart | Liability Insurance under the Negligence Rule[END_REF] See Manning et al (1987) for the use of a control group in the assessment of a cost sharing modi…cation in the health insurance market. [START_REF] Helland | Does Three Strikes Deter? A Nonparametric Estimation[END_REF] Important variables in the regressions such as the number of accumulated demerit points have low frequencies for the highest values. It is hard to make an accurate estimation if the frequency of events is low, as is the case for accidents with bodily injury. in the same stratum and at risk at the same date. However, the same coe¢ cients for the covariates are used across all strata. 20 Drivers are not informed when o¤enses are redeemed. 21 Results are available from the authors upon request. 22 We retained the covariates used in Table 2 except for dummies related to calendar e¤ects and to the number of past license suspension spells. The estimated parameter for the reform dummy is equal to -0.163, and the related standard deviation is equal to 0.008. Hence the reform e¤ect is conclusive at the usual tests signi…cance levels.

23 From Section 3.2, a su¢ cient condition to have this result is that the average …ne is higher than the threshold 1= 0 (0) beyond which the incentives are e¤ective. 24 These values correspond to an e¤ort level that increases with the number of demerit points accumulated. Notes: The premium is a step function of accumulated demerit points. The incentives for safe driving depend on the accumulated demerit points. For example, we document in the text that the incentive level is stronger with two accumulated demerit points than with four because the probability of climbing a step in the rating structure after a traffic violation is higher. Notes: (*) is the reference level. Controlling for other variables, a driver with seven accumulated demerit points who gets a supplementary two point offense has a traffic violation risk reduced by 1exp (0.764 -0.974) = 19%. As the unobserved heterogeneity effect is increasing with adp, the estimated risk reduction due to the incentive effect is greater than 19%. The "p-values" refer to tests for the null assumption: g₁ (7) ≤ g₁(adp) (with adp > 7) vs. the alternative, with one-sided rejection regions. For instance, the null assumption related to adp = 8 is rejected at the level α if α is greater than the "p-value" 1.3%, and accepted otherwise. The results suggest that g₁ is not increasing beyond adp = 7, which shows evidence of moral hazard. A global test for the increasing shape of g₁ is presented in Figure 3. Additional regression variables are: gender, driver's license class (9 levels), place of residence (16 levels), age of the driver (5 slopes), number of past suspension spells (3 levels), as well as calendar effects related to years (8 levels), and months (12 levels). Number of observations: 3,492,868 duration-event indicator pairs, derived from 41,290 driver's licenses. The durations are bounded above by one month. Calendar time

Relative frequency

Accidents with bodily injury Accidents Traffic violations December 1992

Notes: Figure 1 represents relative frequencies of traffic violations and accidents derived from a one-year centered moving average. The reform introducing the insurance experience rating structure based on demerit points took effect on December 1, 1992.

( ) Notes: This figure presents a graphical test where the null is the weakly increasing shape of g₁ as a function of adp, the number of accumulated demerit points. The test statistic is a squared distance between the unconstrained estimation of g₁ (thick line) given in Table 2, and the set of weakly increasing functions of adp. The metric is that of the Wald asymptotic test, and the weakly increasing function which is closest to g₁ is represented by the dotted line. The statistic is equal to 43.18 and its asymptotic distribution is less than that of a χ²(13) in the sense of first order stochastic dominance. This leads us to reject the increasing shape assumption for every significance level greater than 42×10⁻⁶. As an absence of incentive effects entails an increasing shape for g₁ (unobserved heterogeneity effect), rejecting the null shows evidence of moral hazard. We evidently assume that d u > c (f a); i.e. e u max > 0. If the two incentives are mixed, we have u n e u max and we deduce from (13) the properties of utilities and of optimal e¤ort levels as functions of n that we obtained in the …rst place.

λ
In addition, we have

e n > 0 ; 8n; , f a + u n u n+1 > u ; 8n:
This condition is ful…lled if

f a > u = 1= 0 (0);
in which case the incentives are e¤ective at every level.

A.2 Incentive e¤ects of point-record driver' s licenses: Model with point removal

The Bellman equation on a holistic incentive model can be written as follows

d u ru(S) + d dt [u(S t )] t=0 + = c (f a + u(S) E [u(T R(S))]): (14)
The state variables S are the seniorities of each non-redeemed tra¢ c o¤ense (if any), the related demerit points and the seniority of the last contract anniversary if the premium is included in the incentives. The related lifetime utility is u(S).

The state S t is reached from S with an eventless history (no tra¢ c o¤ense, point Let us prove the continuity of optimal e¤ort after a point removal in a system where each tra¢ c violation is redeemed beyond a given seniority threshold, equal to T . We suppose that each tra¢ c violation is associated with one demerit point, and that incentives are related to …nes and to the point-record driver's license. The state variables are then the seniorities of each non-redeemed tra¢ c o¤ense, if any. Let us denote these variables as S = (t 1 ; : : : ; t n ); 0 t 1 < : : : < t n < T:

The corresponding optimal e¤ort is denoted as e(S). The states reached without tra¢ c o¤ense before the next point removal are then S t = (t 1 + t; : : : ; t n + t); 0 t < T t n :

We denote the state reached from S after an additional tra¢ c o¤ense (if n < N ) as (0; t 1 ; : : : ; t n ) = T R(S): As the lifetime utility is continuous after a point removal, we have the following result:

n 1 : lim t!(T tn)
u(S t ) = u(S R ); S R = (t 1 + T t n ; : : : ; t n 1 + T t n ):

The state S R is reached from S if there is no tra¢ c o¤ense before the …rst point removal. Then it is easily seen that

lim t!(T tn) u [T R(S t )] = u T R(S R ) = u(0; t 1 + T t n ; : : : ; t n 1 + T t n ):
This means that the left continuity at T t n of the map t ! u(S t ) also holds for the map t ! u [T R(S t )] ; which is associated with the states reached after an additional tra¢ c o¤ense. The reason is that removal of past o¤enses occurs regardless of the future individual history.

From the three last equations, we obtain lim t!(T tn) e(S t ) = e(S R ) replaced by g 1 , but we use here the usual notations for asymptotic tests. The level adp = 13 also includes the 14 point total. As adp = 0 is the default level in the regression, the increasing shape of g 1 is related to the null assumption g 1 (0) = 0 ) is rejected from our data at every usual level, as W = 1933:12.

The increasing shape of g 1 is the null to be tested for with inequalities, and is expressed as follows H 0 : f i ( ) 0 8i = 1; : : : ; 13 , f ( ) 2 O + ;

where O + is the positive orthant of R [START_REF] Manning | Health Insurance and the Demand for Medical Care: Evidence from a Randomized Experiment[END_REF] . In order to test for H 0 against the alternative, a natural statistic derived from the Wald test uses the distance between the unconstrained estimation of positivity constraints and the positive orthant, i.e. The metric M was already used for the Wald statistic. As 0 belongs to the positive orthant, we have W + W . The derivation of the projection

P O + (f ( b ))
of f ( b ) on the positive orthant O + can be seen numerically as a quadratic programming problem. As this projection belongs to the frontier of the positive orthant, the issue is to …nd the set of coordinates that are strictly positive.

The asymptotic properties of the statistic W + under the null are described by Wolak (1991). The statistic is not asymptotically distribution free under the null, as is the case in the equality setting. The limit distribution is a mixture of 2 (i) distributions (0 i 13), where 2 (0) is the unit mass (Dirac) at zero.

If f ( ) belongs to the interior of O + , the limit distribution of W + is obviously equal to 2 (0), due to the pointwise convergence of f ( b ) towards f ( ). Otherwise the weights of the mixture depend on the position of f ( ) at the frontier of the positive orthant. As a 2 (i+1) variable is obtained as the sum of a 2 (i) variable and of a nonnegative variable, the 2 (i) distributions increase with i in the sense of …rst order stochastic dominance. Then all the limit distributions of W + will be …rst order-dominated by a 2 (13) distribution, owing to their de…nition as mixtures. This result is also simply obtained from the inequality W + W .

On our data, the projection P O + (f ( b )) is found equal to the projection of f ( b ) = J b on the subspace of R 13 de…ned by fz 2 R 13 = z 7 = : : : = z 13 = 0g.

We have: The statistic W + is a squared distance between the unconstrained estimation of g 1 and the set of weakly increasing functions of adp. If F is the distribution function of a 2 (13) variable, we have that F (43:18) = 0:999958: If the null assumption: f ( ) 2 O + is tested for with rejection regions of the type [W + > a], it will be rejected at any signi…cance level greater than 1 0:999958 = 42 10 6 . To see this, it is enough to apply the …rst order stochastic dominance property given before. Then the increasing shape of g 1 is rejected at every usual signi…cance level, which shows evidence of moral hazard.

P O + (f ( b )) = f ( b 0 ) ) b 0 = J 1 h P O + (J b )

A.5 Elasticity between the optimal frequency risk and the incentive level

We relate the shape of and the elasticity between optimal frequency risk and

1 .

 1 From equation (2), incentives are e¤ective if we have u > 1= 0 (0). In this equation, u is the lifetime utility loss between the current state and the one reached after an additional tra¢ c o¤ense. Once quanti…ed, u is the monetary equivalent of this tra¢ c violation. Values of this type are derived in Section 4.2. The function c minimizes the sum e + [ (e) u] ; which is the disutility ‡ow of both e¤ort (short-term component) and the expected lifetime utility loss (long-term component). All the u n are lower than u max = d u =r, the private lifetime driving utility without the point-record driver's license. Equation (1) means that c (u n u n+1 )=r is the minimal private utility cost of the point-record mechanism for a driver with n demerit points. The cycle of lifetime utilities is closed with a link between

Figure 2

 2 Figure 2 illustrates the changes in the optimal tra¢ c violation risk during a period that includes two violations and their point removal. The continuous time e¤ect of incentives balances the downward jump in risk after each violation. The incentive e¤ects at the time and event level are the opposite of those created by the revelation of unobserved heterogeneity. The incentive properties of the point removal system in force in Quebec are similar to those of a mechanism without point removal as regards the number of demerit points accumulated.

  level depends largely on the probability of moving up a step in the premium schedule after an additional tra¢ c violation, which must be associated with four demerit points or more. If the perceived frequency of corresponding tra¢ c violations was greater than the actual one (i.e., f 4 + f 5 = 2:83 + 1:80 = 4:63%), the variation of the incentive level induced by the reform would increase. In that case the elasticity would be closer to zero.Lastly, let us assess monetary equivalents for a tra¢ c violation and a license suspension. The monetary equivalent of a tra¢ c violation is the component of the incentive level related to the point-record driver's license. It is equal to the loss of lifetime utility after a tra¢ c violation, which depends on the number of tra¢ c violations accumulated. A value can be derived from the e¤ectiveness of e¤ort estimated in Table2and from the aforementioned link between e¢ ciency of e¤ort and the incentive level. An additional tra¢ c violation beyond seven accumulated demerit points entails a reduction of tra¢ c violation frequency of close to 20%. Although these drivers cannot be seen as representative, we will apply the elasticity derived from the preceding developments. If the 9% increase in the incentive level induced by the 1992 reform entails a 15% reduction in the frequency of tra¢ c violations, a 20% decrease in tra¢ c violation frequency is associated with a 12% increase in the incentive level. The implied loss of lifetime utility depends on the tra¢ c violation frequency risk but mostly on the discount rate r (see Appendix A.6). With = 0:15, the monetary equivalent of an additional tra¢ c violation for these drivers belongs to the interval [$120; $195] if r = 3%, and to [$41:1; $55:7] if r = 6%. Besides, the growth rate of this monetary equivalent with respect to the number n of non-redeemed tra¢ c violations falls between r= n and r= n+1 ; where n is the optimal traf-…c violation frequency related to n. Monetary costs for license suspensions are then obtained by adding the costs of tra¢ c violations until the crossing of the demerit point threshold. Starting from a zero-point record and assuming that six tra¢ c violations are needed to trigger a license suspension, the monetary cost of a license suspension is bounded by $700 and $1178 if r = 3%; and if 0 = 0:17; 1 = 0:17; 2 = 0:16; 3 = 0:15; 4 = 0:12; 5 = 0:09. 24 A misperception of the environment could modify the monetary equivalents of tra¢ c

FIGURE 1 .

 1 FIGURE 1.-RELATIVE FREQUENCIES (IN PERCENTAGE) FOR TRAFFIC VIOLATIONS AND ACCIDENTS

FIGURE 3 .

 3 FIGURE 3.-GLOBAL TEST FOR THE INCREASING SHAPE OF THE TRAFFIC VIOLATION RISK AS A FUNCTION OF ACCUMULATED DEMERIT POINTS

  removal or contract anniversary) of duration t. The parameters d u and f a are the driving utility ‡ow and the average …ne, and E [u(T R(S))] is the lifetime utility averaged with transition probabilities on the state(s) reached from S after a tra¢ c o¤ense. Continuity equations on utility at the time of a point removal or of a contract anniversary (in the latter case, the increase in lifetime utility is equal to the disutility of the premium) and the equation linking the utility of a beginner and the utility just after a license suspension de…ne the solution together with equation (14).

1 ;

 1 amounts to …nding evidence of moral hazard. The null assumption is obviously expressed as a set of inequalities of the type f i ( ) 0 (i = 1; : : : ; 13); with f ( ) = vec i test for the nullity of f ( ) derived from b ; the maximum likelihood estimation (m.l.e.) of -estimated here together with nuisance parameters -is the Wald test. The related statistic writes as followsW = jjf ( b )jj 2 M = d 2 M (f ( b ); 0); M = b V h f ( b ) i 1 = h J f ( b ) b V ( b ) t J f ( b ) i where J f ( b ) is the Jacobian of f (J f ( b ) = J as f is linear),and where b V ( b ) is derived from the asymptotic e¢ ciency of the m.l.e. Under the null, the Wald statistic follows a 2 (13) distribution asymptotically. The null (i.e. f ( ) = 0 , = 0 as f is one-to-one; as J is invertible, we have W = jj b jj 2 [ b V ( b )] 1

W + = d 2 M

 2 (f ( b ); O + ) = min z2O + jjf ( b ) zjj 2 M = jjf ( b ) z 0 jj 2 M ; z 0 = P O + (f ( b )):

2 M = jj b b 0 jj 2 [ b V

 22b 0:311 0:500 0:522 0:794 0:854 0:873 0:974 ::::::: 0:571 ; t b 0 = 0:312 0:500 0:523 0:795 0:855 0:870 0:870 ::::::: 0:870 ;The missing components of b can be found in Table2, and we have b 0 6 = : : : = b 0 13 . Hence b 0 is the vector of parameters with a weakly increasing shape that is the 7 hal-00414479, version 1 -9 Sep 2009 closest to our estimation with respect to the metric t JM J = [ b V ( b )] 1 . Both vectors are represented in Figure 3 (with replaced by g 1 ). From our data, the statistic W + is equal to W + = jjf ( b ) f ( b 0 )jj ( b )] 1 = 43:18;

  the incentive level. Let us perform a local expansion around a value u 0 of the incentive level, in a situation where the incentives are e¤ective (i.e. u 0 > u = 1= 0 (0)). If we write e 0 = e opt ( u 0 ); e 0 + de = e opt ( u 0 + d u); the equations 1 + 0 (e 0 ) u 0 = 0; 1 + h 0 (e 0 + de) u 0 + d u i = 0 lead to de = 0 (e 0 )

TABLE 1 .

 1 -SAAQ INSURANCE PREMIUMS FOR BODILY INJURY AS A FUNCTION OF ACCUMULATED DEMERIT POINTS SINCE THE LAST CONTRACT ANNIVERSARY

	Accumulated demerit points	Premium for the next	Frequency
	(last two years)	two years (Canadian $)	(%)
	0,1,2,3	50	93.7
	4,5,6,7	100	4.9
	8,9,10,11	174	1.1
	12,13,14	286	0.2
	15 and more	398	0.1

TABLE 2 .

 2 -ESTIMATION OF THE HAZARD FUNCTION FOR TRAFFIC VIOLATION AND ACCIDENT FREQUENCY RISKS

					P-value (H0:
			Frequency	Traffic violation	( ) ( ) 1 g 7 g 1 ≤ adp	,	Accident
	Variable	Level	(%)	risk (g 1 )	adp > 7)		risk (g 2 )
	adp:	0 point (*)	76.60	0			0
	Number of	1 point	0.39	0.311			0.243
	accumulated demerit points	2 points	9.36	0.500			0.328
	(last two years)	3 points	6.23	0.522			0.449
		4 points	1.92	0.794			0.496
		5 points	2.09	0.854			0.595
		6 points	1.25	0.873			0.647
		7 points	0.72	0.974			0.601
		8 points	0.55	0.876	0.0130		0.736
		9 points	0.43	0.764	< 0.0001		0.797
		10 points	0.32	0.786	0.0002		0.842
		11 points	0.06	0.906	0.2328		0.522
		12 points	0.04	0.378	< 0.0001		0.810
		13-14 points	0.04	0.571	0.0004		0.222

  represented by the thick line. In this example, the first traffic violation occurs at t 1 , the second at t 1 +1. The two demerit points are removed at t 1 +2 and t 1 +3. The optimal effort level is denoted as ( ) e t . It is determined by all the seniorities of non-redeemed traffic violations, if any. The minimum effort level is denoted as e min and may be greater than zero, depending on the individual characteristics of the driver. After each traffic violation, there is an upward jump in the optimal effort level and an implied drop in traffic violation risk if incentives are effective. The continuous time effect of incentives counters the event-driven effect (i.e. ( )

		FIGURE 2.-TIME EVOLUTION OF TRAFFIC VIOLATION RISK RELATED TO OPTIMAL EFFORT.
						EXAMPLE WITH TWO TRAFFIC OFFENSES
	e	min
			1 year	t 1	t 1 +1	t 1 +2	t 1 +3	t (time)
	Notes: The function	t	→ λ	( ) ( ) e t is e t
	decreases with t and	( ) ( ) λ e t increases). Before t 1 , the effort level is minimal. It increases after the
	first traffic violation. The drop in traffic violation risk is the opposite of the unobserved heterogeneity
	effect. The effort is maximum after the second offense, and then we have ( e t 2 e t 1 at the ) ( ) + = + 1 1
	time of the first point removal (one non-redeemed traffic violation with a one year seniority in both
	cases). We also have ( e t 3 e at the time of the second removal. ) + = 1 m i n

We begin in 1985 in order to match the regressions that follow, because a two-year history is needed to obtain the accumulated demerit points. Data are …rst averaged over one year, to account for strong seasonal e¤ects. A centered moving average derived on …ve fortnights is then performed twice in order to reduce the volatility of the series.

The SAAQ had the information on the drivers'tra¢ c violation history since 1978. Drivers

We focus on license suspensions in the comparisons. However …nancial penalties are also associated with demerit points in some American states.

This reinstatement can be seen as a removal of demerit points. In the present paper, we consider a point removal mechanism to be a cancellation of demerit points applied before the suspension of the driver's license.

The failure rate at the …rst exam after the license suspension is 25%.

Safe driving e¤ort can also reduce the expected disutility of accidents. If e ! (e) is the implied decrease in the disutility ‡ow, replacing e by e (e) in the model includes the in ‡uence of safe driving e¤ort on accident disutility.

We thank a referee for suggesting this connection. A "strike" is a conviction for a serious felony, the equivalent of a demerit point. Three strikes entail a prison sentence of 25 years to life, which in our context would be the license suspension. Comparing the time to rearrest for criminals with one or two strikes in California, the authors conclude that those with two strikes are less risky than those convicted twice, with one strike only. They thus observe an increase in the deterrence e¤ect at the second strike.

Comprehensive regressions based on two-year periods can be found in Dionne, Maurice, Pinquet, and Vanasse (2001).

We thank a referee for suggesting this formulation of the test.

Strati…cation in a proportional hazards model means that Cox likelihoods (of a multinomial logit type) are derived for each stratum and then multiplied together. In other words, an individual with an observed event is assumed to have competed only with other individuals

(e 0 ) u 0 d u + o (d u) ;

A Appendix

Hence the function c is de…ned on the real line as the optimal e¤ort. From the last equation, we obtain

and c is linear in the neighborhood of 0, which corresponds to no e¤ort. The function c is strictly increasing because is strictly positive. If u 0, we have that:

Hence c is an increasing homeomorphism on the real line.

The function c is concave. From the envelope theorem, we have

Hence c is concave from the assumptions on and from the properties of e opt .

We give a proof of the increasing property of the optimal e¤ort level as a function of accumulated demerit points. From equation (1), we obtain

The sequence (u n ) 0 n N is decreasing because we have u max u n . Plugging this result into equation (11) implies that the sequence (u n u n+1 ) 0 n<N is increasing. The optimal e¤ort level is denoted as e n , and expressed as

for 0 n < N , where e opt is de…ned from (8). As e opt is an increasing function, the optimal e¤ort is an increasing function of the number of demerit points for any given value of the license suspension threshold.

Let us specify the condition under which incentives are e¤ective. From (11) and ( 9), we obtain

If …nes are included in the incentives, u n+1 is replaced by u n+1 f a in equation (1), which leads to the recurrence equation (see Figure 4)

The …xed point of g is the lifetime driving utility if …nes were the only incentive scheme, i.e.

and the continuity property of the optimal e¤ort level. Since we expect a global increasing link between optimal e¤ort and the accumulated demerit points, the time-e¤ect should be decreasing globally in order to ful…ll this continuity property.

A.3 Incentive e¤ects of the experience rating system

Let us derive the Bellman equation on the expected disutility function given in (15), including an average …ne of f a j for a j demerit point tra¢ c violation. The optimal disutility function is obtained from the program

which leads to

with f a = P j = f j >0 f j f a j the average …ne. Then we obtain the Bellman equation

The argument of c ; f a + v n (t); is the incentive level.

A.4 Test for the increasing shape of a vector of parameters, and for evidence of moral hazard

Let be the vector of values reached by the function g 1 ; where g 1 is de…ned in equation (7). We have adp = g 1 (adp); Then the conclusions given in Section 4.2 are easily obtained.

A.6 Monetary equivalents of tra¢ c violations

Let us suppose that the increase in the argument of c is close to 12% after a tra¢ c violation. This is the value retained in Section 4.2 for a driver with seven demerit points accumulated, which corresponds to n = 3 tra¢ c violations on average. As the argument of c in the model without point removal and with …nes is equal to f a + u n u n+1 (see equation 13) we have that

We shall compare the utility losses u n+1 u n+2 and u n u n+1 from the recurrence equation on lifetime utility, and obtain a monetary equivalent of an additional tra¢ c violation from a derivation of the utility loss u n u n+1 . We have

(see Figure 4). From the equality c 0 ( u) = (e opt ( u)) (see equation ( 10)), we obtain

The parameter n is the frequency risk corresponding to the optimal e¤ort exerted with n tra¢ c violations accumulated. As n decrease with n, we have that

From equations ( 17) and ( 18