The economics of road safety

Transpn. Res.-B Vol. 21B, No. 5, pp. 413--431. 1987 Printed in Great Britain.

THE

0191-2615/87 $3.00+.00 O 1987PergamonJournalsLtd.

ECONOMICS

OF ROAD

SAFETYt

M A R C E L BOYER a n d GEORGES DIONNE Dept. Sciences Economiques, Universit6 de Montreal, C.P. 6128, Succ. " A " , Montreal, Canada H3C 3J7 (Received 15 January 1986; in revised form 10 June 1986) A b s t r a c t - - I n this paper we present a theoretical framework for the analysis of road safety in different contexts characterized by the following factors: the presence or the absence of extemalities~ moral hazard, taxes (subsidies), government regulation, liability rules, liability insurance and multi-period insurance contracts. The main results are the following: (i) A Pareto optimal solution for insurance coverage and road safety is characterized by full insurance and a level of prevention which takes into account externalities between drivers. (ii) Without asymmetrical information, such a Pareto optimal solution can be obtained with or without a fault system for negligence if an adequate rating system is set up in order to induce individuals to take into account externalities. Taxes and subsidies can also be efficient. Government regulation of road safety is another way to reduce inefficiencies due to externalities. However, these interventions will not generally lead to a socially optimal level of road safety under asymmetrical information. (iii) Under asymmetrical information, two mechanisms are examined in some detail: fault for negligence and multi-period insurance contracts. It is shown that one-period liability insurance contracts (assuming that the legal systen can observe the individual's level of road safety activities when accidents occur) or multi-period no-fault insurance contracts (assuming an infinite horizon with no discounting) based in part on the individual's past driving record can give individually rational self-protection activity levels which are socially efficient in presence of both moral hazard and externalities. Under less stringent assumptions, these contracts can give second-best solutions.

INTRODUCTION

The economic analysis of road safety poses a particularly difficult challenge both from the theoretical and empirical standpoints. Road safety is a good which can be considered to be private under certain aspects and public under others. Furthermore, it is subject to externalities. In fact, a driver travelling at a reasonable speed and constantly looking out for road hazards, will clearly profit personally from the increased level of safety created. However, since safe driving reduces the probability of accident of other drivers, this behavior creates benefits for other agents. But the safe driver will not receive payment for those benefits he generates. Moreover, highway safety is a good which has all the features of a public good subject to congestion: the level of highway safety for each individual driver depends upon both the individual's own behavior and the overall level of safe driving behavior of all agents. Finally, while safety may be desirable for everyone, the individual driver is not necessarily motivated to contribute to the production of this safety since actions on the part of individuals don't have a visible impact on the total production of road safety. This situation is in fact a particular form of the free rider problem. Unless there is an adequate institutional environment, resource allocation will be inefficient. In this study we will limit our definition of road safety to a private good with externalities. A resource allocation, resulting from a given group of institutions and mechanisms of a particular society, can be considered efficient if there exists no other allocation which would appear to be preferable or indifferent for all agents to the allocation considered, and strictly preferable for at least one agent. This is essentially the Pareto criterion which will be retained here since this efficiency criterion, despite its obvious drawbacks, is the most widely used criterion in economic theory today. It represents a minimal criterion in that the majority of agents can agree with it. However, it is a partial criteria since it doesn't allow the different optimal or efficient situations to be compared to each other. We can therefore qualify as efficient, a mechanism or institution which is likely to lead to a Pareto efficient allocation in a given situation. In terms of road safety, problems arising from external effects and from the difficulty to observe directly self-protection activities (Laffont, 1976) and the agent's risk (Hoy, 1982, 1984), require that special nonstandard mechanisms be instituted in order to ensure maximum efficiency taking into account individual rationality. tFinancial support was provided by the R6gie de l'assurance automobile du Qu6bec. 413

414

MARCELBOYERand GEOROESDIONNE

Imperfect information can take on many forms in the area of road safety and automobile insurance: the difficulty to evaluate the social value of road safety, moral hazard, and finally adverse selection. In this paper we shall limit our study to the problems of moral hazard and external effects. The mechanisms which could reduce inefficiencies due to externalities and moral hazard can be classified into three categories as follows: f'trstly, traditional mechanisms such as internalization by fusion of decision-making parties, taxes and subsidies and the creation of property rights giving rise to transactions between producers and victims of external effects; secondly, direct government intervention in the form of the various regulations governing the highway safety code and legal mechanisms with respect to responsibility for negligence taking into account a level of safety which is socially determined by using a more or less complex procedure; and finally, mechanisms such as insurance rating, generally nonlinear, based on observable variables correlated with the nonobservable variable which represents the individual's self-protection activities. This latter group of measures includes rate setting based on the individual's past experience such as the accumulated number of demerit points, the number of licenses revocations or suspensions and the driver's accident record. In this paper, we will lust emphasize the inefficiencies resulting from external effects and moral hazard.'t" Taxes and subsidies will then be discussed, followed by government regulation, responsibility for negligence, and finally, experience rating. Particular attention is given to a rating system based upon the driver's past driving experience. INDIVIDUALLY RATIONAL CHOICE AND SOCIALLY EFFICIENT CHOICE FOR ROAD SAFETY ACTIVITIES UNDER A PLAN WITHOUT CHOICE OF INSURANCE AND WITHOUT GOVERNMENT INTERVENTION The aim of this section is to derive the characteristics common to every efficient situation. These conditions must be satisfied in order for a situation to be qualified as efficient and will therefore represent means to analyze the efficiency properties of various institutional frameworks. In this way, we will be able to compare private versus public insurance, fixed versus variable ratings, uniform versus nonlinear ratings, and analyze regulatory measures and the issue of responsibility. The procedure we will use is the following. We will start by formalizing our previous discussion regarding road safety as well as emphasizing the social aspect of the activities which produce road safety. Then we will describe the choices of a rational agent with respect to road safety activities and determine the socially efficient choices in order to derive the differences if any between individual choices and socially efficient choices in various institutional contexts. An agent can influence road safety levels in two ways: by producing type x activities which reduce the probability of accident, and by producing type y activities which reduce the loss resulting from an accident. For purposes of simplifying this presentation, let us assume that there is only one type of accident and that the driver faces the following random situation: p(x) is the probability of accident, a function o f x which is the level of self-protection activities; l(y) is the eventual cost of the accident, a function of y which is the level of self-insurance activities. Therefore, the expected utility function is as follows: p ( x ) U ( S - l(y)) + (1 - p ( x ) ) U ( S ) - C(x, y),

where S represents initial wealth and C is a cost function in terms of utility. Moreover, the probability that a driver would be implicated in an accident depends not only on his own x activities, but also on the level of self-protection activities of other drivers. Since p is now a function of the activities of other drivers, these externalities have to be considered in the implementation of an efficient allocation of risks. Generally, the individual agent will take into consideration external benefits derived from the self-protection activities on the part of other drivers but will not consider the external benefits that his own behavior generates for the other agents. In order to simplify the presentation, we shall limit ourselves tOther aspects of incomplete informationare discussed in Boyer and MacKaay(1981).

The economics of road ~tfety

415

to type x activities and therefore the amount of loss as a result of an accident will be given by l(y) =- I. Let us assume that the economy consists of (N + 1) individuals. For purposes of simplicity, we will assume the level of exogenous insurance to be null. However, all forthcoming results are applicable to any exogenous level of partial insurance. The rational choice of the level of self-protection activities of individual # 1, x*, is given by the solution of the following problem?:

Max Zl = ~

p(x,, xi)Ul(Sl - 12) +

1 -

i=2

p(xl, xi)

Ul(Sl)

--

Cl(Xl),

(1)

i=2

where U~(.) is an increasing and concave utility function U'~(.) > 0, U~'(') < 0 o f Individual # 1 ; S~ is initial wealth; Ct(x~) is an increasing and convex cost function Ci(xO > O, C'((xO >- 0 with C~(0) = C~(0) = 0; p(x~, x~) is a function giving the probability that individual #1 will be implicated in an accident with individual i (i = 2 . . . . . N + 1) given self-protection activities x~ and xi and is assumed to be identical for all individuals and symmetrical, p(xl, xi) = p(xi, x0, and to satisfy pl(xl, xi) -

Op(x,, x,) Oxl

<0,

ap(x~, x,)

p2(xl, xi) = -

Ox~

<0,

Plt(Xl, X3 > O, Pz2(Xi, Xi) > 0 and p12(x~, xi) = 0;

and finally l, is the cost of an accident for individual # 1 . The first-order condition which the individually rational x* must satisfy can be written as follows: N+I

pl(xl, xi)[Ul(Sl - ll) -

UI(S0] = C~(xO.

(2)

i=2

Let us assume that all individuals are identical. We shall restrict ourselves to Nash symmetrical equilibria which correspond to Diamond's uniform equilibria (1974). In this context, all individuals choose the same level x* of self-protection activities. The first-order condition (2) can be written as Np,(x, x ) l U ( S - l) -

U(S)] = C'(x).

(3)

Let x* satisfy (3). At a Nash symmetrical equilibrium x* = x*, individuals do not consider the external effects of their choice of self-protection activities. This equilibrium is, therefore, inefficient. If individual # 1 were to take into account the external effects created by his choice of x, condition (2) would be written as follows: N+I

N+I

p,(x,, x,)IU,(S, - 1~) - U , ( S 0 ] i=2

+ ~

p2(x,, x,lIU,(S, - l,) - U,(S,)] = C~(x,).

(2A)

i=2

In a uniform equilibrium, with identical individuals, first-order condition (2A) will become 2Np,(x, x ) [ U ( S - t) -

U(S)] - C'(x) = O.

(4)

Let x ÷ satisfy eqn (4). The difference between expressions (3) and (4) can be explained as follows: in eqn (3), the individual only takes into account the marginal effect of his self-protection activities on his own utility level while in eqn (4), the individual also takes into account the marginal effect of his self-protection activities on the utility levels of the other individuals in the economy. tThe separability of function Z~ is quite specific but often used (Holmstrom, 1979; Diorme, 1982; Arnott and Stiglitz, 1983).

416

MARCELBOYERand GEORGESDIONNE By substituting x*, satisfying (3), in eqn (4), we obtain 2Npl(x*,

x*)[U(S

-

1) -

U(S)] -

C'(x*).

Using (3), this equation becomes Npl(x*,

x*)IU(S

-

1) -

U(S)] > 0.

(5)

Equation (5) is positive which implies that the difference (x ÷ - x*) is also positive. Hence, the equilibrium level of prevention activities x* is inferior to the Pareto superior level x ÷ , the efficient level without choice of insurance. THE DETERMINATION OF AN OPTIMUM BY THE SUBSIDIZATION (TAXATION) OF PREVENTION ACTIVITIES WITHOUT CHOICE OF INSURANCE In the preceding section, it was shown that the level of self-protection activities chosen by a given individual is lower than the efficient level because the individual neglects to take into account the positive external effects of his activities on other drivers' well being. In order to reach an optimal level under the constraint of individual rationality, one must create mechanisms which would motivate individuals to take into account these external effects. In the economic literature, there are four major methods to solve the problem of external effects: internalization by the fusion of deciding parties, taxes and subsidies, the creation of a market for external effects and direct regulation by the state. Internalization mostly concerns businesses and cannot be applied to the problem of road safety. Moreover, the creation of a market where external effects would be exchanged is not feasible because of the obvious difficulty in defining ownership rights on public roads and because of the enormous transaction costs these rights would entail. We will therefore limit our discussion to taxes and subsidies and regulation as means to reduce the problem of external effects. Let us first consider taxes. Theoretically, taxes can be imposed on care activities which are lower than the efficient level T ( x + - x ) , where x + is the efficient level and where x is the level chosen by the particular individual; the resulting tax could be a nonlinear function of (x + - x) with T(0) = 0. Apart from imposing taxes, road safety could be encouraged by subsidizing self-protection activities so that drivers increase their level of prevention to the extent that this level approaches or reaches socially efficient levels. Furthermore, the exact amount of subsidy which would motivate a given driver to choose an efficient level of selfprotection can also be calculated [see Boyer and Dionne (1984) for a detailed analysis of subsidies]. Whether one imposes taxes or grants subsidies with respect to self-protection activities, the latter must be observable. However, it is usually either very difficult or extremely costly to observe the self-protection activities of a particular individual. But indirect observations can be also used. For example, a form of tax can be found in declarations of responsibility for negligence. Consider 2, the due care level which determines the level of responsibility in the event of an accident; if an individual's x were inferior to ~, this individual would, in the event of an accident, be found negligent and depending upon the degree of negligence, would be liable to a fine, a jail sentence or some other penalty, that is a tax T(~ - x) would be imposed. An optimal solution could be achieved; conditions which would lead to this outcome in a system with responsibility for negligence will be presented later on. Another example is a nonlinear tax on activities related to accidents such as car mileage or type of car purchased (Arnott and Stiglitz, 1983). Despite the theoretical results on the use of taxes and subsidies to induce an efficient level of care, these measures are not generally used as elements of a global policy of intervention again because those self-protection activities are not directly or easily observable. Rather, alternative corrective measures are considered and some of them will be analyzed in the following sections of this paper. The use of taxes or subsidies along with these measures will also be discussed. In conclusion, it would seem that traditional measures to reduce external effects such as

The

economicsof road safety

417

internalization, taxes, subsidies and the creation of markets by granting ownership rights, cannot generally be applied to road safety. Hence, other types of solutions must be considered, if external effects are indeed found to be a major factor in road safety. The following section will present an analysis of direct government intervention in this area consisting of measures stemming from various regulations contained in Highway Codes, in particular, the compulsory seatbelt laws and speed limit restrictions on public roads. THE DETERMINATION OF AN OPTIMUM BY GOVERNMENT REGULATION WITHOUT CHOICE OF INSURANCE

Theoretical framework and evaluation As was described before, the level of road safety corresponding to an individual equilibrium (x*), is lower than the optimal level for society (x +) when insurance coverage is both exogenous and partial in nature. This difference is due to the fact that individuals do not take into account external effects when they choose a particular level of road safety.t Government intervention would be a way to remedy the situation. In fact, through its direct actions and laws, a government could modify road safety levels to the point where they reach socially optimal levels. However, one must keep in mind that government interventions would not necessarily lead to a socially optimal level. An in-depth study of the advantages and costs of each type of intervention is essential in order to evaluate the net impact of government interventions. In the following pages, the major characteristics of government interventions in the area of road safety will be presented. Even when the government takes on the role of insurer, this does nothing to remedy the problem of asymmetrical information between insurer and insured. As the insurer, the government encounters the very same difficulties as private insurance companies when it comes to direct preventive actions taken by individual drivers. Government, therefore, cannot directly encourage motorists to take care. As with private insurers, indirect measures must be taken [See Boyer and Dionne (1985a)]. Furthermore, it is for this very same reason that government cannot directly subsidize the driver's preventive activities in order to reach optimal levels. In fact, the difference between x + and x* could theoretically be eliminated by a government subsidy. However, as was indicated in the previous section, variable x is not directly observable. Hence, this type of government intervention is not feasible. Government can however intervene directly on p and I. The most common forms of intervention are regulations governing compulsory seatbelts, speed limits on public roads, impaired driving as well as other violations of the Highway Code, regulations governing the manufacturing and safety of automobiles, driving tests for the granting and renewal of permits, and advertising. Thanks to these measures, government directly reduces the probability of an accident:~ by forcing or at least encouraging motorists to produce an average level of road safety which could be, hopefully, higher than the level corresponding to the individual equilibrium. Formally stated, government intervention can be taken into account by defining the probability of accident for an individual motorist i as p~(x, a), where x is the vector of self-protection activities and a represents the vector of government interventions. Therefore, intervention al reduces the probability of accident if P~a, is negative. However, this type of intervention can generate costs, in the form of either additional taxes or direct higher costs and/or of purchase and maintenance of automobiles, and/or higher utility costs (compulsory seatbelts for example). This implies that C is also a function of a. A variation in intervention dal would be put into action if the benefits created are higher than its costs and if it produces more welfare than other competing tThe emphasis here is placed upon the presence of externalities as a justification for government intervention. Other reasonscouldjustify this action as well, such as the idea that individualsunderestimatethe probabilityof accident or because drivers are not sufficiently informed about the expected outcome of a given self-protection activity. See Oi (1973) for a detailed analysis of the reasons which could justify governmentintervention in markets for goods related to safety. These reasons are usually classifiedunder the followingcategories:externalities, lack of information, and misperceptionof risk. ~:In the formal discussion, we shall limit ourselves to interventionson p even though certain examples, such as the use of seatbelts could be represented by interventions on 1.

418

MARCEL BOYER and GEOROES D1or~qE

measures (das, u+l i=1

s =

2, 3 . . . .

). In other words, project da~ is undertaken if

(OEU~t )~i ~ da, >- O, \ 0al / u+, OEUi i~=j hi - dal = k Oal 1

i=1

hi k Oa~ / da, >-- O

fors = 2,3 .....

(6)

where hi is the relative weight of the individual or group i in the evaluation of competing measures. Assuming that there is no reaction from individual i in his choice of activity x, (6) can be written as N+I

hi{p,m[Ui(S, - l,) -

Ui(Sill - C~,} da,-> 0,

i=l N+I

Z

~ki{Pia,[Ui(Si -- li) -- Ui(Si)l - C~,} da t

i=l N+I

>-- ~ , ki{p~,IVi(Si - li) -

Vi(Si)] - C~,} da~ for s = 2, 3 . . . .

(6A)

i=l

M a j o r problems in the application o f the formula Equation (6) brings to mind the basic principle for justifying government intervention: the government's objective in all of its possible measures is not to reduce to the minimum the probability of accident but, rather, to maximize the welfare of society. Therefore, it would seem logical to calculate the optimal probability of accident for a given society rather than the minimum probability of accident. At the extreme, if the objective were to minimize the probability of accident, the most effective government measure would be to prohibit the use of automobiles all together. This somewhat extreme example demonstrates the principle referred to previously. It is therefore necessary to use eqn (6), or its monetary equivalent, in order to justify government intervention (in other words, the fact that government intervention reduces p does not alone justify this intervention). The first problem related to the use of eqn (6) concerns the calculation of individual weights hl. However, since this problem is not limited to road safety, it will not be discussed in detail here. A second problem arises when measuring the efficiency of da, interventions in reducing the probability of an accident.t An analysis of this efficiency must focus on both technical and behavioral factors. Indeed, measuring the effect of wearing seatbelts on the probability of death in the event of an accident is dependent upon the fields of engineering and medicine. Moreover, the engineer can determine the technical effect of a particular type of brake or tire. In terms of advertising, experts in human behavior or communication can be useful in analyzing the public's perception of this intervention. On a behavioral level, several authors, notably Peltzman (1975), have proposed that government intervention can affect the individual's choice of selfprotection activities.l: This implies that x* is also a function of a. For example, the compulsory wearing of seatbelts da~ can lead to faster or more dangerous driving since the driver is now protected by a seatbelt. In this case, x*tas is negative and p(x*(a,), a,) ~. p(x*(O), 0). Therefore, it is difficult to predict all the effects of a particular intervention on the probability of an accident in general or on the probability of a specific type of accident. An individual can also be affected by government intervention which creates higher costs and additional benefits to the driver. For example, the control of speed limits increases driving

tSee Arnould and Grabowski (1981) for a detailed analysis of the use of seatbelts and also, Peltzman (1975), Oi (1973), and Crandall and Graham (1984). $ See Boyer and Dionne (1983b) for a more detailed analysis of this phenomenon and also Nelson (1976), MacAvoy (1976), Joksch (1976), Robertson (1977). The latter four authors are of the opinion that the counter effects a ~ not very significant from an empirical standpoint.

The economics of road safety

419

time but reduces energy consumption and compulsory seatbelt laws force individuals to remember to buclde up. If, in the above example, the individual believes that the cost of additional driving time outweights the advantages of saving fuel, this driver may decide not to modify his behavior behind the wheel. In fact, it will not be modified if the expected net gain is less than zero, that is, if the driver prefers to pay a fine rather than respect the new regulation. This argument justifies the importance of calculating (1) the optimal probability of catching a driver who disregards the law, and (2) the optimal fine leavied against the driver who gets caught.t On the other hand, if the individual accepts to conform to the new regulations concerning speed limits, this reduces the probability of accident. But then the individual may decide to stop wearing the seatbelt because driving at slower speeds is safer. In terms of compulsory seatbelt regulation, the individual may judge that it is more profitable not to wear a seathelt and therefore run the risk of paying a fine rather than being subjected to the inconvenience (in terms of utility) of wearing a seatbelt everytime the car is driven. These examples suggest that we know the value of the loss (I) which can represent either property damage or bodily injury. Since property damage is not difficult to evaluate, we shall limit our analysis to bodily injuries. These include both injuries and fatalities. In either case, it is difficult to measure private and social monetary loss (i.e. the value of human life and/or more or less permanent injuries), since there is no market for these goods. However, the insurance market (especially life insurance) can give us an approximation of the private values and indirect indicators found in the economics of human resources can be used for public values. One must also take into account the indirect benefits of government intervention, such as the reduction in the cost of both public and private medical and hospital insurahce and the monetary cost of providing adequate services for victims of automobile accidents. And finally, nonmonetary losses must also be considered in the evaluation of disutility with respect to car accidents. It is possible to take these factors into account in a formal model by using state dependent utility functions [see Shavell (1978), Cook and Graham (1977), Dionne (1982), Debez and Dr~ze (1982), Dr~ze (1983)]. However, the evaluation of these types of benefits and costs, in order to justify government intervention, requires a great deal of information concerning the individual driver. Boyer and Dionne (1984) presented an example of this application. As an alternative to direct intervention, the government could motivate drivers to voluntarily produce safety through market mechanisms. For example, drivers can be given an incentive to use automatic seatbelts by offering a reduction in insurance premiums to those drivers who wear them.~: This type of intervention has the advantage of leaving the decision up to the individual and, if control mechanisms aren't too costly to implement, may produce greater increases in welfare than coercive measures would produce. Two other possible solutions will be examined: (a) an insurance system with fault for damage to others if the individual is guilty of negligence, (b) premium rates based on demerit points accumulated over several years, the number of license suspensions/revocations and the number of past car accidents involving the particular individual. In the first case, the driver is encouraged to produce a higher level of road safety than that which is produced when externalities are not taken into account since the driver would now bear the additional cost of damages to others when found guilty of negligence. In the second case, indirect observation of self-protection activities are used by making insurance premium rates dependent upon past driving experience (i.e. road accidents and recorded infractions to the Highway Code). This is based on the assumption that a high rate of past accidents and a high number of accumulated demerit points and license revocations and suspensions reflect a low level of self-protection activity.

;See Polinski and Shavcll (1979) for the calculation of the optimal probability and of the optimal fine. See Boycr and Dionne (1983a, 1983b) for an analysis of the impact on the risk of equivalent variations of the probability and the amount of loss. :~It would be possible to control the nonuse of seatbelts by connecting activation of the seatbelt mechanism to the ignition. Therefore, in order to drive a car without the use of a seatbelt, an individual would have to disconnect the system. Such action would then be considered illegal if, in order to obtain a lower premium, the driver of the car declares to the private or public insurer that the vehicle is equipped with automatic seathelts and that they are in working order. TRB21:5-F

420

MARCEL BOYF.Rand GEORGES D1ON'NE THE DETERMINATION RESPONSIBILITY

O F AN O P T I M U M

FOR NEGLIGENCE

WITHOUT

BY A S Y S T E M CHOICE

OF

OF INSURANCE

In order to encourage individuals to move from x* to x + , one possible method would be to implement a system of legal responsibility for negligence which takes into account the driving behavior of individuals at the time of the accident. In this section, we will analyze the effects of the introduction of fault on individual choice of self-protection activities. We will verify, among other things, whether responsibility for negligence is sufficient to motivate individuals to produce the Pareto superior level of road safety x ÷ when the level of responsibility is properly determined. Returning to the first model, let us assume that the government introduces rules of responsibility. In the economic literature, [see Brown (1973), Diamond (1974), Green (1976), Landes (1982), Lipnowski and Shilony (1983), Shavell (1980, 1982,-1984)], two models of responsibility come to mind: strict responsibility and responsibility for negligence. In the former case, the individual is responsible for damage to others without regard to degree of fault. This type of responsibility is used in the analysis of victim-aggressor type accidents such as those between vehicles and pedestrians. In the latter case, an individual must pay for damages to others only if his prevention activities at the time of the accident were deemed to be below minimum standards recognized by society (due care) and if the behavior of others involved in the accident was at a level above minimum standards. The hypotheses behind the model in Section 2 imply that the former type of responsibility is not pertinent here. In fact, contrary to Shavell's model (1980, 1982),t we assume that all individuals suffer economic loss at the time of the accident and have an equal level of monetary loss (1). Let us assume that, for a given society, the minimum level of care in order not to be responsible for an accident is exogenously determined at the level ~ (due care). One way to formalize the behavior of an individual in this given situation would be to suppose that he maximizes his utility function over x, assuming that other individuals have chosen to produce a level which is at least equal to 2. Furthermore, let us assume that the legal system is free and that it is perfect: preventive activities can be observed without cost and errors in judgement are eliminated.~; The behavior of individual #1 can be characterized by the maximization of the expected utility function ZI with respect to xj, assuming that the other individuals have a level of prevention activity equal to 2. § If xt is less than 2, the individual is responsible for his loss l and the loss I to the other. However, if xl is greater or equal to 2, individual # 1 is only responsible for his own loss l. Moreover, the probability of an accident between individuals 1 and i is still written as p(xj, To). The expected utility function for individual #1 is as follows (subscript 1 is omitted when there is no possibility of confusion):

Z

fNp(x, ~)U(S - 2l) + (1 - Np(x, ~))U(S) - C(x) "l LNp(x, x-)U(S l) + (1 - Np(x, ~))U(S) - C(x)

ifx < i f x -> ~.

In order to establish the level x** which maximizes Z, the maximum level for each domain must be found, then the higher maximum will be retained. This would lead to the first-order condition for x:

Np~(x,~)[U(S- 21)NpI(x ,

x)[U(S

-

l) -

U(S)] - C ' ( x ) U(S)]

-

C'(x)

X~ = 0,

Jr ~2 ~-- 0 ,

for x < forx

(7)

~---.~,

t W e arc only interested here in accidents involving two motor vehicles. Accidents which implicate pedestrians require a different formalization of the problem. Shavell studied models involving only one victim and one aggressor, when the victim alone experiences financial loss and where the aggressor alone or both aggressor and victim could have prevention activities. In these types of models, both forms of responsibility may apply with or without insurance. Results obtained by Shavell arc different from those presented here. :~It must also be stated that self-protection activities arc limited to those recognized by the legal system for the declaration of fault on the part of the individual. Let's take for example the repair of defective brakes (although it would be difficult to establish at the time of the accident whether or not the brakes were indeed defective before the accident actually occurred), driving on the correct side of the road, respecting speed limits, etc. On the other hand, activities such as the use of seatbelts and driving only in good weather arc not considered. §Following Diamond (1974). However, Diamond did not consider insurance and asymmetrical information.

The economics of road safety

421

where h~ and h2 and non-negative Lagrange multipliers associated with the conditions x < ~" and x -> ~, respectively. ¢ Let

Rm, = Np~(x, ~)[U(S - l) - U(S)] > 0, Rm2 = Npl(x, ~)[U(S - 21) - U(S)] > 0, Rm3 = 2Np~(x, ~)[U(S - l) - U(S)] > 0, and

~ Rm2 Rm4 = I Rml

ifx<.~ i f x -> ~.

Therefore, Rm3 -- 2Rm6 Rm2 > Rml since S > S - l > S - 2/; furthermore, Rm3 < Rm2 if U"(.) < 0 and Rm3 = Rm2 if U"(.) = 0. Assuming that ~ is the value o f x which satisfies the following

Np,(x, x-)[U(S - 2t) - U(S)] - C'(x) = 0, and recalling that Pit > 0 and PI2

=

0, we can graphically illustrate the relationship between

Rmt, Rm2, Rrn3, Rm4 (represented by the dotted line), x*, x ÷, ~, 2, as well as the individual's choice x**. The level of x** is, according to (7), given by the intersection of Rrn4 and C'(x). As can be observed from Fig. 1 x** = ~ when x* < ~ < 2. Moreover, if ~ < x*, then x** = x*; if ~ >-- ~, then x** = ~. Therefore, if ~ > x*, it is in the individual's best interest to increase the level of self-protection activities in order to avoid paying 21 at the time of an accident. However, if the legal minimum level of activity determined by society ~ is less than x*, that is, the level chosen by the individual without responsibility for negligence, the individual is better off maintaining level x* in order to protect himself against loss l. Finally, if ~ > ~, the minimum legal level becomes too costly (i.e. reduces the expected utility), and so the individual chooses .~. Therefore, if ~ = x ÷, which represents the Pareto superior solution without choice of insurance, this level will then be chosen by everyone and would therefore represent a uniform equilibrium by Diamond (1974). However, any legally determined level between x* and ~ also represents a uniform equilibrium. Moreover, the uniform equilibrium ?Since the set x < ~ is open, there may not exist a maximum in this interval. Despite the fact that this problem arises here, we will not discuss it explicitly since its solution is quite obvious.

X

Rm

/ I

• x*

g

~ Fig. 1.

x

422

MARCELBOYERand GEOROESDIOUNE

will be x* if.~ < x* and will be .~ if ~ > .~. Consequently, responsibility for negligence allows for an efficient level of prevention if the minimum legal level .~ is adequately set at x + as defined by eqn (4). In order to obtain these results, several assumptions were used: (H1) all individuals are identical, (H2) the legal system is without cost and does not commit any errors (i.e. perfect information), (H3) prevention activities are restricted to those which are officially recognized by the legal system, (H4) ~ is chosen exogenously, and (H5) no choice of insurance is introduced here. If individuals are different (H1), for example their cost functions C(x) of prevention activities are different, then the efficient level of prevention for a given individual will depend on C(x). Hence, the level of efficiency no longer remains constant. Diamond (1974) has demonstrated that it is then impossible to produce efficient levels of prevention by introducing a system of responsibility for negligence defined by a single legal level ~. In fact, if individuals have different cost functions, the level of.~ could be easily reached or even surpassed by certain individuals, but not by others. Furthermore, those who surpass level ~ will not necessarily benefit from such a high level of prevention since other individuals are below the regulated level and would therefore certainly be at fault in the event of an accident. On the other hand, it can be shown that for many individuals who are at a level between the extreme cases mentioned previously, it is in their best interest to increase their level of prevention once ~ is established, but, it is not certain whether or not the average level of prevention activities will increase as a result. This would depend upon the level of ~ chosen by a given society. In order to produce efficient choices, the legal level of prevention would have to vary according to the particular groups as defined by the cost functions C(x). The legal system can commit errors (H2) bringing about additional uncertainty which would cause an ambiguous net effect on x**: on the one hand, costs of nonprevention decrease since the individual may have to pay l instead of 2/even if he is really negligent and, on the other hand, the benefits derived from prevention activities decrease as well because he may have to pay 21 even if he is not negligent. The costs and delays of the legal system reduce the net benefits, positive or negative, of insurance plans with fault for negligence. One of the arguments put forth to justify no-fault insurance plans is the resulting lower costs of litigation and shorter delays for compensating victims (Gauvin, 1974; Landes, 1982; Witt and Urrutia, 1983; Outreville, 1984). If these costs and delays are high, as several studies seem to indicate, a society, which seeks to set up an insurance program with responsibility for negligence, must, among other things, choose between the benefits of a potential reduction in the probability of accidents and the anticipated increase in transaction costs and delays in compensation which could cause inequities. Since rules of responsibility cannot take into account all prevention activities (H3), this greatly reduces the effect of this type of intervention in order to achieve an efficient level of resource allocation. In other words, by only partially controlling preventive activities, these rules of responsibility do not allow individuals to internalize all externalities created by preventive activities. Finally, endogenous choice of a legal level of responsibility (H4) would imply the development of a theoretical model of the evolution of law, an area of study which is obviously beyond the scope of this article. We will therefore omit this case here and consider to be exogenous. Assumption 5 is the most important of all. In fact, in cases of uncertainty and in the presence of risk-averse agents, it is necessary to take into account the choice of insurance when dealing with an efficient allocation of resources. This is precisely what we will be discussing in the following sections of this article. THE DETERMINATION OF AN OPTIMUM BY A SYSTEM OF RESPONSIBILITY FOR NEGLIGENCE WITH CHOICE OF INSURANCE The Pareto superior level of prevention x ÷ satisfying (4) is defined for a given level of insurance coverage (nil, in this case). However, in order to study optimal resource allocation under uncertainty, choice of insurance as well as preventive activities must be considered simultaneously. Moreover, insurance policies are generally defined in relation to rules of

The economicsof road safety

423

responsibility. In our discussion, we will compare the following two insurance plans: (1) nofault insurance where the insurer compensates in full or in part the client's losses whether or not the latter was at fault, but does not reimburse for payments made by his client for damages to others; (2) liability insurance where the insurer reimburses his client, in full or in part, for damages to others when his client is at fault. When no one is at fault, each insurer compensates his client. When only one client is at fault, his insurer reimburses losses for both. In other words, we shall assume that a portion of the contract is in the form of property insurance. An alternative to this approach, which would yield exactly the same analytical results, would be to assume that a risk-averse individual buys two types of insurance: on the one hand, liability insurance for property damage to others and, on the other hand, property insurance for personal losses when neither party is declared at fault or when only one party is at fault. If economic agents are risk-neutral (U"(') = 0), the optimal levels of liability insurance and of property insurance are nil. In this case, a system of no-fault will imply levels of preventive activity which, under uniform equilibrium, will be given by - N p l ( x *~, x*")U'l = C'(x*").

(8)

A system with responsibility for negligence on the other hand, would produce a level of preventive activity which is a function of.~. If .~ equals x *" as defined by eqn (8), then ~ will be a uniform equilibrium. As Fig. 1 indicates, there will be discontinuity at $ for the marginal benefit associated with x; this marginal benefit will be written as follows: -Np~(x, ~ ) U ' 2 l

i f x < ~,

-Npl(x, ~)U'l

i f x -- ~.

It can be shown, as it was previously, that ~ will be a uniform equilibrium with fault if x *~ -< ~ -< $", where ~" is defined by

-2,Vp,(~, P)~,'t = c'(?~).

(9)

Therefore, liability for negligence produces higher levels of preventive activities of a riskneutral agent than the x*" level produced under a no-fault system. Liability will produce a Pareto optimal solution if the legal level chosen .~ is equal to x +', defined by - 2Npl(x +~, x+")U'l = C'(x+.).

However, if x can be observed by the government in the same way as the legal system observes x under the above assumptions, individuals could be motivated to produce level x ÷~ under a no-fault system through taxes or subsidies, as we have already discussed in a previous section. Let us now consider the case of risk-averse agents. In order to obtain a Pareto optimal solution, one must consider the choice of insurance with or without moral hazard. By definition, moral hazard exists when the insurer cannot observe perfectly the preventive activities of the insured [see Shavell (1979) and Dionne (1981, 1982)]. We shall consider, in the following sections, the cases with and without asymmetrical information. Insurance choice with no-fault f o r negligence and without moral hazard

Under a no-fault system, the behavior of a particular agent can be characterized by the maximization over x and q of the expected utility function Z given by Z = Np(x, x , ) U ( S - H - l + q) + (1 - Np(x, x~))U(S - l-I) - C(x),

(10)

where q is property insurance coverage, H is the actuarial insurance premium and xi is the common level of prevention activities chosen by all other individuals i. Assuming that the insurer is in equilibrium (no profit), then II = Np(x, xi)q.

(IOA)

424

MARCF.LBORER and GEORGESDIGNNE

This produces as first-order conditions with A -----S - Il lowing:

l + q and B = S - I] the fol-

Oz

- - = (1 - Np(x, x,))Np(x, x3[U'(A) - U'(B)] = 0, Oq

(11)

Oz

- - = Np(x, x,)[U(A) - U(B)] - Np~(x, xi)EU'q = C'(x). Ox

(12)

According to eqn (11), U'(A) is equal to U'(B) and q* is therefore equal to I. Then, U(A) equals U(B) and EU' equals U'(B). Equation (12) can now be rewritten as follows: - N p , ( x , xi)U'(S - 11)/ = C'(x).

(12A)

Let us consider the uniform equilibrium (x**, x**), where q* equals l and x** satisfies eqn (12B) -Np~(x, x**)U'(S - H)/ = C'(x).

(12B)

This equilibrium is not efficient since it does not take into account external effects produced by each agent. In fact, under full insurance, efficient choices of x, say x +* satisfy - 2Npl(x, x+*)U'(S - H)/ = C'(x).

(13)

By substituting x**, solution of (12B) into eqn (13), we obtain by using (12B) -Npj(x**, x**)U'(S - I - I * * ) / > 0, and then, x +* > x**. Therefore, under no-fault insurance with perfect information (i.e. without moral hazard), the government could intervene and, by using subsidies or taxes, could convince insurers to take into account external effects when setting premiums [Landes (1982) studies the possibility of private insurers cooperating in order to achieve similar results without state intervention]. In fact, by imposing a premium equal to 2Np(x, x+*)q to all individuals, insurers would be promoting a Pareto optimal uniform equilibrium where individuals choose q = l from (11) and x = x +* from (13). However, at this price level, insurers will experience positive profits and individuals may decide not to purchase insurance. In order to avoid this problem, a full compulsory insurance plan with a premium equal to 2Np(x, x+*)l for every individual driver could be implemented. Insurance companies could allow consumers to recover excess profits created by high premiums, i.e. Np(x ÷*, x+*), or the surplus could be transferred to the State for lump-sum redistribution. Insurance choice with responsibility for negligence and without moral hazard Let us now consider the case of insurance with responsibility for negligence. The problem for the agent lies now in the choice of x and q. We already know that for x < ~(x >- .2), an individual will experience a loss of 2l(1), again assuming that the other individuals have a level of prevention activity equal to 2. Since the insurance premium is assumed to be actuarial, the agent will select full insurance q = 2l if x < 2, or full insurance q = l if x >- 2. We can therefore turn our attention to the choice of x so that the premium to be paid will be Np(x, Yc)l if x -> ~ and Np(x, .2)2l if x < .2. Therefore, [ U(S - Np(x, .2)21) - C(x) Z = [U(S

Np(x, YOl) - C(x)

i f x < 2, i f x >- 2,

and the first-order conditions will be, respectively, with D = S Np(x, x-)l

Np(x, .2)2l, B ---- S -

425

The economics of road safety -U'(D)Np~(x,

2)2/ - C ' ( x ) - h~

over x < 2,

-U'(B)Np~(x,

2)l -

over x -> 2,

C ' ( x ) + h2

where ht and h2 are non-negative Lagrange multipliers. The choice of a particular individual will be that which leads to the highest value of Z. Since D is less than B, a similar analysis to Fig. 1 can be performed and would yield the following result: under responsibility for negligence with perfect information, risk-averse individuals will select full insurance and a level of prevention activities x*** = 2 if x** -< 2 -< Y~, where ~ is the solution of - U ' ( D ) N p ~ ( x , x-')21 = C ' ( x ) , x * * * = x * * if 2 < x** and x*** = Yc* if 2 > .~. Moreover, the resulting equilibrium will be Pareto optimal if 2 = x +* as defined by (13) since this equilibrium allows for full insurance and the internalization of externalities. Hence, we can conclude this section by claiming that with perfect observation of preventive activities, public authorities can promote both the production of preventive behavior and the choice of insurance coverage at Pareto optimal levels by either of the following two measures: (1) the application of appropriate insurance rates under a no-fault system or, (2) the establishment of a minimum adequate legal level of prevention under a system with responsibility for negligence.

Insurance choice with moral hazard

Boyer and Dionne (1985a) analyzed insurance plans with or without responsibility and with asymmetrical information. They have shown that under a no-fault system, only a secondbest solution can be found if the insurance contract is of one single period in length. The result corresponding to the case without externalities is similar to those of Holmstrom (1979), Shavell (1979) and Dionne (1982). The analysis will not be presented in detail here. Let us write :¢ and ~ < I for the second-best solution without externalities. Now, if externalities are introduced, the second-best optimal amount of insurance becomes ~ < q < l and ~ is the corresponding level of care. This second best solution cannot be improved by public insurance since, as an insurer, the government encounters the very same difficulties as private insurers in monitoring road safety activities [see ShaveU (1982) for a detailed discussion of this issue]. Boyer and Dionne (1985a) have also shown that a system of responsibility for negligence with externalities and asymmetrical information, produces a first-best solution with full insurance if 2 is appropriately related to x +*, the Pareto optimal level of prevention. Formally, the insured has a choice between two contracts: (1) a full insurance contract if x -> 2 when the accident occurs, and (2) a partial insurance contract q < 21 if x < 2. It is still assumed here that the legal system is without cost and does not commit any error even if the private insurer cannot monitor x without cost. Therefore, the choice is between U(S -

Np(x)l) -

C(x)

if x - > 2

(14)

and Np(x(q))U(S

-

qr(q)q -

21 + q) + (1 - N p ( x ( q ) ) ) U ( S

-

~r(q)q) -

i f x < 2,

C(x(q))

(15)

where rt(q) = N p ( x ( q ) ) . Since q < 21 in (15) and since U is concave, then (15) is less than U(Np(x(q))(S

-

,rt(q)q -

21 + q) + (1 - N p ( x ( q ) ) ) ( S

-

~(q)q))

-

C(x(q))

which can by rewritten using av(q) = N p ( x ( q ) ) as U(S -

Np(x(q))2l)

-

C(x(q)).

(16)

One can easily show that (14) is superior to (16) and therefore a first-best solution is obtained if 2 is appropriately related to x +~.

426

MARC~.LBOY~.~ and GEOR~ DIOr~NE

In order to adequately compare these two systems, one has to consider the costs of the legal system and compare, in terms of welfare, responsibility for negligence, that is a system with administrative costs and legal delays but without moral hazard, to no-fault insurance, a system without administrative costs but with moral hazard. This type of comparison will not be conducted here since it is beyond the scope of the present study. However, the next section will demonstrate that it is possible to improve the no-fault system by the use of multi-period contracts. The previous discussion can be modified in order to analyze the behavior of risk-neutral individuals under imperfect information. Under fault for negligence, since the agents don't purchase actuarial insurance, externalities will be internalized if ~ equals x +". Under no-fault, government must implement a nonlinear rating system for certain activities related to accidents in order to motivate individuals to take care. For example, nonlinear tax rate could be set based on car mileage or type of car purchased if this information can be fully obtained (Arnott and Stiglitz (1983) discuss in detail the use of taxation measures with moral hazard and financial externalities).

THE DETERMINATION OF AN OPTIMUM WITH I N S U R A N C E RATES BASED ON PAST E X P E R I E N C E

We know from the preceding sections that under insurance and in the presence of asymmetrical information between insurer and insured, the latter is not motivated to produce an efficient level of prevention. In order to reduce inefficiency due to moral hazard, multi-period contracts under a bonus-malus system (Radner, 1981, Rubinstein-Yaari, 1983, Henriet-Rocbet, 1984) were proposed. It can be demonstrated that there are a number of periods T, such that for every T greater or equal to T, multi-period contacts dominate one-period contracts. We will now proceed to describe the multi-period contract by assuming that there is only one insurer (public or private) in the economy. The fact that there is only one insurer reduces the costs of gathering pertinent data and renders the threat of a rate increase more credible since the insured cannot go elsewhere for another insurer [for a discussion of multi-period contracts with many insurers, see Dionne (1983) and Kunreather and Pauly (1985)]. In the previous section, it was shown that, without asymmetrical information about selfprotection activities, efficient resource allocation consists of maximizing, over a given period [Townsend (1982) demonstrated that, under perfect information, multi-period contracts are useless], the consumer's expected utility, assuming that the insurer is in equilibrium. Equations (11), (12) and (12B) have established that uniform equilibrium without consideration of externalities corresponds to (x**, q*) where q* = l and x** satisfies

- N p d x , x**)U'(S - H)/ = C'(x).

(12B)

Under asymmetrical information, since the insurer cannot observe x directly, this information cannot be used when setting insurance rates. Assuming that $ >- 0 and q <-- l are optimal solutions under moral hazard and that Z~($, q) and Z2(Y, q) are levels of welfare of the insurer and the typical insured, respectively, then Z~(Y, ~) < Zt(x **, q*) and Z2(.~, q) < Z2(x **, q*). Indirect measures of prevention activities over time can be used as a means to reduce the gaps between the various levels of welfare. This consists of negotiating a multi-period contract in which contracting parties can refer to accumulated information during the negotiation process. In this situation, the insurer can observe over time the number of accidents, demerit points, license revocations/suspensions of the insured, etc., so as to obtain as much information as possible. In the above-mentioned literature on multi-period contracts, only past loss records are considered. In addition to readjusting insurance premiums according to the risks presented by the various agents, these types of insurance contracts create incentives for agents to take care. In fact, since the insured is aware ex-ante that his future insurance premium is dependent upon

The economics of road safety

427

his present driving record, it is in the driver's best interest to take care in order to reduce future insurance costs. This type of insurance contract will now be formalized.t At the start of the first period of the contract, the insurer offers the insured a long term contract which, at the beginning of each period, takes into account accumulated information concerning say past accidents, demerit points and revocations/suspensions. Based on this data, the insurer announces a premium at the start of each period, and the insured must decide whether or not to continue to purchase the insurance and what level of self-protection activities to choose. Coverage doesn't change over time and partial coverage is not considered here since we want to isolate the effects of time and since we want to implement a first best insurance contract. Under this system, each agent is aware of the possible actions of the other as well as of the information available and can anticipate future information. For example, the insured is well aware of the fact that if he is not cautious during period t, the probability of accumulating demerit points, of being implicated in accidents or even of having one's license revoked or suspended during this period will be greater. Furthermore, the insurance premium will be likely to increase for period t + 1. At the end of each period t, the insurer knows: (1) if the individual was insured (0'~ = 1) or not (0'~ = 0); (2) if the individual was involved in an accident or not (for purposes of simplicity, we shall assume for the moment that it is possible to have a maximum of one accident during a given period and that all accidents generate the same loss l) that is if (0'z = l) or (0~ = 0); (3) if the individual has accumulated demerit points (0~ = 0, 1 . . . . ) during the given period; (4) if the individual has accumulated license revocations/ suspensions (0~ = 0, 1 . . .) during the given period. Let Oq be the insurer's information at the end of period t. It is composed of four vectors which contain information about period t as well as all preceding periods as follows: 0'~ = (Oc~, 07 . . . . .

09,

o'z = (e~, o,~. . . . .

e~),

o ~ = (0 l, 02 . . . . .

09,

o~ = (or~, o,~ . . . . .

0~).

The insured obviously is aware of all the information contained in O[ and, therefore, it is not pertinent to repeat this information. Let O[ represent the insured's information vector at the end of period t. The insured is also aware of the method used to calculate premia and knows that this method will not change over time. The insurer's actions are limited to offering insurance premia based on accumulated information. The insured, on his part, in addition to deciding whether or not to purchase insurance for each given period, must also decide what level of prevention activities to choose, knowing that this level will influence 0c, 0d, and 0y. An important issue here concerns the various strategies employed by the parties. More specifically, which strategy the insurer will use in order to motivate the insured to choose x **. Let's first examine how Rubinstein and Yaari (1983) and Radner (1981) studied this question by assuming that the insurer can only observe past accidents. By definition, a strategy is a function which transforms information into action. Let f = ( f l , f2 . . . . ) and g = (g~, g2 . . . . ), for t = 1, 2 . . . . . represent the sets of strategies of the insurer and insured, respectively, f ' + t is a function of O~ and belongs to the set of insurance premiums, g'÷~ is a function of O~ and has two components, (g~+J, g~+l), where g'/~ E {0, 1} with 1 for insurance and 0 otherwise, and g'x+ ~ C [0, X], the set of prevention activities levels. Vectors @~(f, g) and 0 [ ( f , g) are, ex-ante, random variables and represent the agents' tThe model presented here is adapted from Rubinstein and Yaari (1983), and from Radner (1981, 1985). Henriet and Rochet (1984), Lambert (1983), Allen (1985) and Rogerson (1985) have also presented multi-period insurance contracts under moral hazard. Dionne (1983), Dionne and Lasserre (1984, 1985), Cooper and Hayes (1983), as well as Henriet and Rochet (1984) have studied this type of contract under adverse selection. TRB21:5-G

428

MARCELBO"ff~Rand GEORGESDlot,rr,~

expected information. The expected actions of the agents can be represented by ft(O~-'(f, g)) and by gt(O'2-1(f, g)) for t = 2, 3 . . . . and by f t and gl for t = 1. Expected benefits of agents (h9 take the following random values: with f/~i = f(O~ , - ,- t) - p(g,(O2 t ",- l ))l insurance l h'2 U(S - f'(O[-l)) - C(g'~(0'2-1)) without insurance

{-

hi = 0 hi (1 - p(g~(O'2-1)))U(S)

+ p(g'~(O'2-1))U(S - 1) - C(g'x(O'2-~)). If we now assume that each agent maximizes his welfare in the long run, the question remains as to which strategies (f, g) will produce the highest level of welfare. Rubinstein and Yaari proposed a no-claims discount strategy for the insurer which consists of the following, where N(t) is the set of periods where 0' = 1 and tN(t)l is its cardinal measure: fl = p,~, /-

l

1

if~

f'+l(Oi) = ] P**

Of <- p(x**)l +

E

Ot IN(t)l

sEN(t)

Pk

otherwise,

where P*~' = p(x*~')l; IN(t)] is the number of periods during which the individual was insured, IN(t)l = X~=~ 0~; P~ is a premium such that

U(S - Pk) < (1 - p(x*))U(S)

+ p(x*)U(S - 1),

where x* is the optimal level of prevention without insurance; otleett)lis a statistical tool sufficiently high to avoid penalizing too often unlucky drivers with a level of prevention activities equal to x** but sufficiently low to motivate individuals to choose x *~'. In other words, the insurer, at t = 1, sets the premium at a level which corresponds to x *~' and, subsequently, will continue to charge this same premium if the insured's observed average claim [(1/]N(t)l) ~,,e~m 0~'] is less than the expected claim corresponding to x** plus a statistical margin of error. It is possible to prove that this type of contract is enforceable and that it will yield an average utility level equal to that obtained under full information about prevention activities, i.e. the insured will choose full insurance and level x *~ of prevention activities [see Rubinstein and Yaari (1983) for more details], if the discount rate is nil and if the number of periods is infinite. However, Radner (1981, 1985) demonstrated that one may improve the welfare of individuals under moral hazard by the use of multi-period insurance contracts, even when the number of periods is finite and the discount rate is positive. But one must keep in mind that this only represents a second best solution. This strategy can be adapted to our initial model, that is, can take into account demerit points, license revocations/suspensions, and past accidents as well as other individual characteristics i which could influence the probability of an accident. Knowing or having estimated the probability of accident p(O~, i) conditional on the information vector Oi and observed individual characteristics i, we would obtain fl = p,,,

•

1

p(Ol, i) < p,.+~' + a!Nml

f'+l(O~) = l P'+*

LPk

otherwise.

where i represents observable individual characteristics (age, sex, residence . . . . ), p(O], i) is the probability of having an accident during period (t + 1) based on observed driving experience

The economicsof road safety

429

and the other observable factors, calculated by using the estimated conditional probability function: p** is the probability of having an accident based on level x** of prevention activities and on the other observable factors, as calculated by using the estimated conditional probability function; and P** = p**l. If we now take into account externalities associated with road safety activities, the Pareto optimal level of prevention with insurance is no longer equal to x** but to x ÷* > x**. It would still be possible to apply in this case the previous type of contract described. The probability p** as well as the corresponding premium would have to be redefined by using x ÷* as the efficient level of prevention activities [see Boyer and Dionne (1985a) for more details]. The long-term insurance contract becomes: f l = pi+,, /-

f'+l(O~) = I P/+*

( Pk

p(O~, i) < p,+* + cx!toni sEN(t)

otherwise.

This type of contract should allow for a Pareto optimal level of prevention with moral hazard and externalities if the number of periods is sufficiently large. Theoretical extensions to the above analysis would be the calculation of the optimal finite number of periods required in order to obtain a solution which is close to the Pareto optimal solution, and the definition of a specific premium schedule which would approximate the theoretical results discussed previously. These extensions will not be presented here. The study of the empirical foundation of this type of model is also important. It consists of: (1) verifying whether or not previous accidents, demerits points and license revocations/ suspensions accumulated up to period t as well as other observable factors such as individual characteristics over period t + 1 do in fact significantly affect the probability of accidents in t + 1; and (2) estimating whether or not past experience accumulated up to period t as well as environmental factors when the accidents occur and other observable factors over period t + 1 do significantly affect the probability that an accident in t + 1 is of a given type. Recently, we have focussed our research on the analysis of the above empirical propositions. We selected two random samples, one consisting of approximately twenty-thousand drivers and the other of twenty-thousand accidents. Data is for the Province of Qutbec. To our knowledge, the samples are unique in that they are the only ones which group together information taken from four different sources: driving permits, previous accidents, demerit points and revocations/suspensions. We obtained for every driver, information such as age, sex . . . . . the number of demerit points accumulated since August, 1980, the number and types of car accidents since August, 1980, and any record of revocations/suspensions of the driving permit since August, 1981. More than sixty variables were constructed from these samples. Using this data and the Probit, Logit and Ordinary Least-squares regression models, we verified the first above-mentioned empirical proposition. Indeed, we showed that past experience up to July, 1982 is very significant in explaining the probability of accidents over the period from August, 1982 to July, 1983. Moreover, individual characteristics are also significant.t Using these preliminary results, premiums tables were calculated and were compared with actuarial tables which were derived from the optimal Bonus-Malus system (Seal, 1969; Lemaire, 1985) and the same set of data. Detailed results are now available (Boyer and Dionne, 1985b, 1986). Results concerning the second empirical proposition are forthcoming. CONCLUSION In this paper we have presented a theoretical framework for the analysis of road safety in different contexts characterized by the following factors: the presence or the absence of exter-

)These variablescan be proxiesforriskexposure.Howeverourdataset does not allowus to verifysuchconjectures. More research is neededon this issue.

430

MARCELBOYERand GEORGESDIOh'NE

nalities, moral hazard, taxes (subsidies), government regulation, liability rules, liability insurance and multi-period insurance contracts. The main results are the following: • A Pareto optimal solution for insurance coverage and road safety is characterized by a full insurance coverage o f risks and a social level of prevention which takes into account externalities between drivers. • Without asymmetrical information, such a Pareto optimal solution can be obtained with or without a fault system for negligence if an adequate rating system is set up in order to induce individuals to take into account externalities. Taxes and subsidies can also be efficient. Direct government regulation of road safety is another way to reduce inefficiencies due to externalities. However, this type of intervention will not generally lead to a socially optimal level of road safety under asymmetrical information. This negative result also applies for traditional measures used in order to reduce external effects such as taxes (subsidies). • Under asymmetrical information, two mechanisms were examined in some detail: fault for negligence and multi-period insurance contracts. It was shown that one-period liability insurance contracts (assuming that the legal system can observe the individual's level of road safety activities when accidents occur) or multi-period no-fault insurance contracts (assuming an infinite horizon with no discounting) based in part on the individual's past driving record can give individually rational self-protection activity levels which are socially efficient in presence o f both moral hazard and externalities. Under less stringent assumptions, these contracts can give second-best solutions. More research is needed to compare these solutions. Acknowledgements--This paper is a synthesis and an extension of two articles published in French by the same authors

(1984, 1985a). We would like to thank R. Abramson for editorial assistance.

REFERENCES Allen F. (1985). Repeated principal-agent relationships with lending and borrowing. Econommics Lett. 17, 2% 31. Arnott R. and Stiglitz J. (1983). Moral hazard and optimal commodity taxation. Working paper # 500, Queen's University. Arnould R. J. and Grabowski H. (1981). Auto safety regulation: an analysis of market failure. BellJ. Economics 12, 2%49. Boyer M. and Dionne G. (1983a). Riscophobie et 6talement a moyenne constante: analyse et applications. Actualitd Econornique 59, 208-230. Boyer M. and Dionne G. (1983b). Variation in the probability and magnitude of loss: their impact on risk. Can. J. Economics~Revue Canadienne d'Econornique 16, 411-419. Boyer M. and Dionne G. (1983c). The riskiness of equivalent governmental policies. Working paper 8319, Department of Economics, Universit~ de Montrral. Boyer M. and Dionne G. (1984). Srcurit6 routi6re: efficacitr, subvention et rrglementation. L'Actualit~ Econornique 60, 200-222. Boyer M. and Dionne G. (1985a). Srcurit6 routirre: responsabilit6 pour n6gligence et tarification. Can. J. Economics~Revue Canadienne d'Econornique 18, 814-830. Boyer M. and Dionne G. (1985b). Moral hazard and experience rating: An empirical analysis. Publication # 427, Centre de recherche sur les transports, Universit6 de Montrral. Boyer M. and Dionne G. (1986). La tarification de l'assurance automobile et les incitations /~ la srcurit6 routirre: une 6tude empirique. Revue Suisse d'Econornie et de Statistique 122(3), 293-322. Boyer M. and MacKaay E. (1981). The concept of adequate information: market failures and their corrections. Working paper # 8126, Department of Economics, Universit6 de Montrral. Brown J. (1973). Toward an economic theory of liability. J. Legal Studies 2, 323-350. Cook P. J. and Graham P. A. (1977). The demand for insurance protection: the case of irreplaceable commodities. Quart. J. Economics 61, 143-156. Cooper R. and Hayes B. (1983). Multiperiod Insurance Contracts, Yale University. Crandall R. W. and Graham J. D. (1984). Automobile safety regulation and offsetting behavior: some new empirical estimates. Amer. Economic Rev. 74, 328-331. Dehez P. and Drrze J. H. (1982). State-dependent utility, the demand for insurance and the value for safety. (Edited by M. W. Jones-Lee). The Value of Life and Safety. North-Holland, Amsterdam. Diamond P. (1974). Single activity accidents. J. Legal Studies 3, 107-164. Dionne G. (1981). Le risque moral et la s61ection adverse: une revue critique de la iittrrature. L'Actualit~ Economique 57, 193-225. Dionne G. (1982). Moral hazard and state dependent utility function. J. Risk and Insurance X L l X , 405-423. Dionne G. (1983). Adverse selection and repeated insurance contracts. Geneva Papers on Risk and Insurance 8, 316-333. Dionne G. and Lasserre P. (1984). Adverse selection and finite-horizon insurance contracts. Euro. Econ. Rev. (forthcoming).

The economics of road safety

431

Dionne G. and Lasserre P. (1985). Adverse selection, repeated insurance contracts and announcement strategy. Rev. Economic Studies LII, 719-723. Dr~ze J. H. (1983). Decision theory with moral hazard and state-dependent preferences, Universit~ de Louvain (CORE). Gandry M. (1984). DRAG, un ModUle de la Demande Routi~re des Accidents et de leur Gravit6, Appliqu6 au Qu6bec de 1956 ~ 1982. Publication ~ 359, Centre de rech~'che sur les transpo~s, Universit6 de Montr6al. Gauvin J. L. (1974). Rapport du Comit6 d'Etude sur l'Assurance Automobile. Editeur officiel du Qu6bec, Qu6bec. Graham J. D. and Garber S. (1984). Evaluating the effects of automobile safety regulation. J. Policy Anal. Management 3, 206-224. Green J. (1976). On the optimal structure of liabihty laws. Bell J. Economics 7, 553-574. Henriet D. and Rochet J. C. (1984). The logic of bonus-penalty systems in automobile insurance: A theoretical approach. Laboratoire d'tconomttrie de l'Ecole Polytechnique, Paris. Holmstrom B. (1979). Moral hazard and observability. Bell J. Economics 10, 74-91. Hoy M. (1982). Categorizing risks in the insurance industry. Quart. J. Economics XCVII, 321-337. Hoy M. (1984). The impact of imperfectly categorizing risks on income inequality and social welfare. Can. J. Economics XVII, 557-569. Joksch H. C. (1976). Critique of Sam Peltzman's study: the effects of automobile safety regulation. Accident Analysis and Prevention 8, 129-137. Kunreuther H. and Pauly M. (1985). Market equilibrium with private knowledge: an insurance example. J. Public Economics 2,6, 269-288. Laffont J. J. (1976). La thtorie 6conomique de l'auto-protection. Revue Economique 4, 561-588. Lambert R. A. (1983). Long-term contracts and moral hazard. Bell J. Economics 14, 441-453. Landes E, M. (1982). Insurance liability and accidents: a theoretical and empirical investigation of the effect of no-fanlt accidents. J. Law and Economics/OgV, 49-65. Landes E. M. (1982). Compensation of automobile accidents injuries: is the tort system fair. J. Legal Studies 11, 253-259. Lemaire J. (1985). Automobile Insurance: Actuarial Models. Kluwer Nijhoff Publishing, Boston. Lipnowski I. and Shilony Y. (1983). The design of a tort law to control accidents. Math. Soc. Sci. 4, 103-115. MacAvoy P. W. (1976). The regulation of accidents. (Edited by H. Manne and R. L. Miller). Auto Safety Regulation. Law and Economics Center of the University of Miami, Miami, Florida. Nelson R. (1976). Comments of Peitzman's Paper on Automobile Safety Regulation. (Edited by H. Manne, and R. L. Miller). Auto Safety Regulation. Law and Economics Center of the University of Miami, Miami, Florida. Oi M. Y. (1973). The economics of product safety. Bell J. Economics 4, 3-39. Outreville J. E (1984). The impact of the government no-fault plan for automobile insurance in the province of Quebec. J. Risk and Insurance Ll, 320-336. Peltzman S. (1975). The effects of automobile safety regulations. J. Political Economy 83, 677-725. Polinski A. M. and Shavell S. (1979). The optimal tradeoff between the probability and magnitude of fines. Amer. Economic Rev. 69, 880-891. Radner R. (1981). Monitoring cooperative agreements in a repeated principal-agent relationship. Econometrica 49, 1127-1148. Radner R. (1985). Repeated principal-agent games with discounting. Econometrica 53, 1173-1199. Robertson L. S. (1977). A critical analysis of Peltzman's 'the effects of automobile safety regulation'. J. Economic Issues XI, 3. Rogerson W. B. (1985). Repeated moral hazard. Econometrica $3, 69-77. Rubinstein A. and Yaari M. E. (1983). Repeated insurance contracts and moral hazard. J. Economic Theory 311, 79-97. Seal H. L. (1969). Stochastic Theory of a Risk Business. John Wiley, New York. Shavell S. (1978). Theoretical issues in medical malpractice (Edited by S. Rottenberg), The Economics of Medical Malpractice. American Enterprise Institute, Washington, D.C. Shavell S. (1979). On moral hazard and insurance. Quart. J. Economics XCIII, 541-562. Shavell S. (1980). Strict liability vs negligence. J. Legal Studies 9, 1-25. Shavell S. (1982). On liability and insurance. Bell J. Economics 13, 120-132. Shavell S. (1984). Model of the optimal use of liability and safety regulation. Rand J. Economics 15, 271281. Townsend R. M. (1982). Optimal multiperiod contracts and the gain from enduring relationships under private information. J. Political Economy 90, 1166-1187. Witt R. C. and Urrutia J. (1983). A comparative economic analysis of tort liability and no-fault compensation systems in automobile insurance. J. Risk and Insurance L, 631-669.