Adverse incentives from improved technology: traffic safety regulation in Norway.

I. Introduction

Much of the literature on safety regulation by economists have pointed out adverse incentive effects. It does, however, seem misguided to believe that effects like these were unknown to regulators before they were pointed out by economists. One example of a popular regulatory measure that takes incentive effects explicitly into consideration, is the use of road-bumps in residential areas. In this case the regulator incurs a positive cost to make the technological environment worse for drivers. If speed were unchanged after the regulation, accident probabilities would go up due to reduced control of the cars. The regulators believe, however, that the physical deterioration of the driver environment will reduce speed to the extent that accident rates go down. It might be a disturbing feature of this line of thought that the same regulators may advocate a technical improvement of a stretch of highway (better surface, wider road etc.) in the interest of accident prevention. In this case the regulator must believe that the ceteris paribus technical improvement of the driving environment will dominate a possible adverse incentive effect.

It may well happen that the technological nature of these two types of regulation--or the behaviorable responses--differs to such an extent that it is not illogical to advocate both the measures cited above. The example illustrates that for purposes of optimal regulation, it is not only necessary to establish that adverse incentive effects may occur, but also to say something about when they are likely to occur--and when not. The theoretical contribution of the present analysis is to provide one step in this direction, within a unified framework that captures some earlier considerations made in the literature as special cases.(1) The theoretical analysis provides an apparatus for evaluating the effects of regulation in different circumstances with different predictions concerning incentive effects. These theoretical categories have identifiable empirical counterparts. The predictions are tested against a Norwegian data set. The data consist of a pooled cross section and yearly time series of automobile drivers, motorcyclists, and pedestrians that have been killed or injured in Norwegian counties, 1980-1986.

The next section contains a theoretical discussion of the behavioral responses to regulation. It is shown how the interaction between prevention technologies and attitudes toward risk determines behavior. Individual accident prevention varies greatly--and not always monotonically--with different attitudes towards risk. The impact of changing risk attitudes on prevention is also sensitive to the type of risky situation that is considered. In spite of the great variability in individuals' incentives to take precautions, it is, however, possible to identify simple properties of the prevention technology that makes us able to predict the direction of the incentive effects regardless of the underlying preferences.

The empirical analysis in section III is embedded in the reduced form of the theoretical model. Although the data set is quite aggregated, it has interesting features that makes it better suited to reveal potential external effects and moral hazard than in much of the earlier literature. One such feature is unique information on actual seat-belt use compiled by the national police task-force for traffic control in Norway. The empirical analysis suggests the presence of adverse incentive effects to technical and legal regulation. One of several results in this category is that increased use of seat-belts by drivers has a significant adverse effect on other road users.

II. The Agents' Response to Regulation

The starting point of the analysis is a simple model of self-protection as introduced by Ehrlich and Becker |7~. The probability that an agent will experience a well-defined accident is a function of the level of care taken by the agent (c), and a vector describing the physical and technological environment (x). This relationship is modelled as a decreasing, convex prevention function

p = p(c,x), p |is an element of~ (0,1~. (1)

The agent's efforts to prevent accidents might also have an impact on the severity of the accidents should they occur. In the case of automobile accidents, taking care by reducing speed will reduce the probability of a collision, and also reduce the damage should such a collision occur. On the other hand, if the potential accident considered is driving over a cliff and falling down, reduced speed will only reduce the probability of the accident. The height of the cliff will determine the potential damage, regardless of the speed of the car as it started to fall. This shows that also the physical environment (x), might affect the loss from an accident for the agent. These considerations give rise to a nonegative, nonincreasing, convex loss function of the form

L = L(c,x). (2)

Losses are measured in monetary terms. This has the advantage that attitudes towards risk are easily defined when only monetary elements enter the utility functions additively. A potential problem is that an analysis of traffic accidents should accommodate the possibility of health losses. Viscusi and Evans |20~ have argued forcefully against use of the monetary equivalent of ill health formulation to analyze health losses. Their argument is directed against formulations with a fixed monetary equivalent. The monetary equivalent can, however, be derived from more primitive concepts in the vein of Cook and Graham |5~. It follows from the analysis in Risa |15~ that such derivations implies that the loss function should be decreasing in prevention costs (c), given that the Viscusi and Evans |20~ assumptions regarding the health state dependent utility functions hold. The loss function in the present formulation can therefore also be interpreted to contain elements of monetized health losses.

Risa |16~ contains a discussion of how a principal can regulate accident risks among multiple agents with external effects in prevention activities. That analysis motivates investigation of incentive effects from regulation. Here, optimal regulation will not be discussed. The focus is limited to a positive investigation of the expected incentive effects from regulation. The main analytical departure from Risa |16~ is the introduction of the endogenous loss function. Different technical assumptions on this function will identify different risk-taking regimes, and different regulation schemes, that correspond to empirical counterparts in traffic safety regulation.

A (representative) individual has a gross wealth of Y. Disposable wealth without an accident, but with accident prevention, is |y.sup.0~ = (Y - c). In the accident state, disposable wealth equals |y.sup.1~ = (Y - c - L(c, x)). Superscripts 0 and 1 denote the favorable and unfavorable state of the world, respectively, throughout the paper. The optimization problem for the individual facing a well defined potential accident is to choose a level of care that maximizes expected utility, given a physical environment.

|Mathematical Expression Omitted~

The first order necessary condition for a maximum is:

|Mathematical Expression Omitted~,

where subscripts denote (partial) derivatives with respect to care (c), and disposable wealth. The second order sufficient condition for a maximum is:

|Mathematical Expression Omitted~.

The properties and predictions of this model depend to a large extent on the limiting assumptions that are made concerning the partial derivatives of the p and L functions. We consider three regimes that are special cases of the model above.

Regime 1: Technical assumption: |p.sub.c~ |is less than~ 0, |L.sub.c~ = 0.

Individual precautions influence accident probabilities, but not the loss.(2) Empirical counterparts to this regime:

Care and attention by pedestrians to prevent being hit by a car.

Reduced speed by drivers, if the potential accident is driving over a cliff.

Regime 2: Technical assumption: |p.sub.c~ |is less than~ 0, (1 + |L.sub.c~) |is less than~ 0.

Individual precautions influence both the size of the loss and the probability of the accident. The marginal loss reduction in case of the accident is greater than the marginal value of care.(3) Empirical example:

Reduced speed to prevent the adverse effects of collisions.

Regime 3: Technical assumption: |p.sub.c~ |is less than~ 0, |L.sub.c~ = -1. Individual precautions influence the probability of the accident and the size of the loss. The loss reduction corresponds exactly to the costs of taking care.(4) Regime 3 is primarily represented as a technical benchmark in the discussion to follow. It can, however be given an empirical interpretation of paying attention to prevent accidents by drivers and pedestrians.(5)

It is straightforward to check that (5) is satisfied for risk averse individuals given Regimes 2 and 3. This guarantees a unique solution to the optimization problem.(6) Risa |16~ discusses (reasonable) restrictions on the prevention technology that are sufficient to secure uniqueness in Regime 1.

Risk Aversion and Risk Taking

We want to consider how different attitudes towards risk, and different physical environments influence the agents' care to prevent accidents. It was already noted by Ehrlich and Becker |7~ that the incentive to self-protect may vary non-monotonically with changing attitudes towards risk. More recently, this point has been discussed by Dionne and Eeckhoudt |6~, and Briys and Schlesinger |3~. In Risa |16~ it is shown that increasing risk aversion first increases, and then decreases, the optimal level of care in Regime 1. The reason is that taking care to reduce the probability of an accident also reduces disposable wealth should the accident occur. Strong risk averters will "save" on prevention costs to improve the accident state, even if this increases the probability of the accident. This effect is not present in Regime 3.

To illustrate some of these effects, we have calculated the optimal level of care (c*), and the corresponding optimal accident probability (p*), based on (4), given explicit U, L, and p functions. The calculation is based on a CAMRA (Constant Absolute Measure of Risk Attitudes) utility function of the form: U = -|e.sup.-Ry~, where R is the measure of risk attitudes, and y is disposable wealth. The prevention function is of the following form: p = |(2+c).sup.-1~. The loss function in Regime 1 is: L = 100, and in Regime 3: L = 100 - c. The optimally chosen level of accident probability as a function of risk attitudes in Regime 1 and 3 is illustrated in Figure 1.

Accident prevention increases monotonically (and optimally chosen probabilities of the accident decrease) with increasing risk aversion, as one would expect, in Regime 3. In Regime 1, however, accident prevention first increases with increasing risk aversion, and then decreases. We have argued that in traffic accident prevention, taking care in the form of paying attention etc. may be better covered by the predictions of Regime 3 than Regime 1. If this is the case, we will expect higher prevention activities for more risk averse agents. We will now turn to consider the effects of the physical environment on prevention activities.

The Moral Hazard of Technological Improvements

Given that (5) is satisfied, (4) defines the optimal level of care, c*, as a function of x. Invoking the implicit function theorem, the comparative static properties of this function are found by simple differentiation of (4):

|Mathematical Expression Omitted~

Again, subscripts are partial derivatives, and |V.sub.cc~ is as defined in (5). We will consider two distinct regulatory schemes:

Regulation Scheme 1: Technical assumption: |L.sub.x~ |is less than~ 0, |p.sup.x~ = 0. Regulation of the potential loss, but not the probability of the accident.(7) An example of this is mandatory seat belt laws. Wearing the seat belt does not protect against collisions or other accidents, but against the adverse consequences of the accidents (i.e., the losses).

Regulation Scheme 2: Technical assumption: |L.sub.x~ = 0, |p.sub.x~ |is less than~ 0. Regulation of accident probabilities, but not the potential loss. This corresponds to erecting a guardrail to prevent cars from driving over a cliff. The main reason for this simplification is to make the exposition clearer. The qualitative results are not changed if a nonzero effect on losses is allowed.

Adverse incentive effects can be measured by the sign of dc*/dx, and dp*/dx = |p.sub.c~(dc*/dx) + |p.sub.x~. If dp*/dx |is greater than~ 0, an adverse incentive effect to regulation is large enough to swamp the direct regulation effect |p.sub.x~. In Regulation Scheme 1 defined above, there is no direct TABULAR DATA OMITTED regulation effect on accident probabilities by assumption, even for the regulated agents. The only effect on accident probabilities for all agents will therefore be the incentive effect in this case. By using the definitions above, simple manipulation of (6) makes identification of sufficient conditions for adverse incentive effects under Regulation Scheme 1 (mandatory seat belt laws) possible. The results are summarized in Table I.

The table shows that incentive problems to regulation will always be present in Regime 3, given risk averse agents in the accident state in Regime 2, and in most cases in Regime 1.(8) We have argued that an empirical interpretation of Regime 3 is the case with primarily psychic prevention costs in the sense of being alert, and paying attention. If the behaviorable response to forced wearing of seat belts is of this type, then Table I predicts not only reduced care and attention, but increased accident risk, always, in response to regulation. The corresponding incentive effects for Regulation Scheme 2 are summarized in Table II.(9)

It is not surprising that regulation affecting accident probabilities directly, makes predictions regarding the final accident risk more difficult than in the case of Regulation Scheme I. However, it is interesting that the elasticity condition in the first column of Table II identifies sufficient conditions for adverse incentive effects regardless of the underlying preferences, and for all regimes considered. (If the elasticity condition is satisfied, |p.sub.cx~ must be positive.) This implies that we have identified simple properties of the prevention technology that makes predictions concerning incentive effects in response to regulation stable over the regimes considered, and over different individual preferences. Given that prevention functions observe the properties in column I of Table II, this would make analysis on an aggregate level involving different types of risk, and different preferences, more appealing. Risa |16~ discusses empirical interpretations of the elasticity conditions. However, without empirical knowledge about the properties of the prevention function, it is difficult to predict theoretically whether a measure like improving the technical standard of a strech of road will increase or decrease the accident level. This result is in contrast to the case of regulation of seat belt use, where stronger theoretical predictions can be made.

III. Empirical Analysis

Individual panel data containing information on accidents, risk attitudes, seat-belt use, and the technological environment surrounding each individual would be preferable for empirical estimation purposes. Such data are not available, and this analysis follows a long tradition in exploiting more aggregate data.(10) The available data are a pooled yearly time series and cross section of 18 Norwegian counties. Using counties as empirical observational unit amounts to accepting a "representative individual" approach to the problem at hand. One does not have to consult an insurance company to realize that this approach has serious shortcomings in the analysis of traffic accident behavior. Figure 1 in the theoretical analysis above, shows that different individuals facing the same technological and regulatory environment may differ dramatically in their care-taking behavior. However, the theoretical analysis identifies conditions under which changes in the technological environment, or the stringency of regulation, lead to the same qualitative changes in individual behavior, regardless of their previous care level, and the regulatory regime considered. Although this theoretical background makes aggregation somewhat more palatable, the results from the empirical analysis should be interpreted with care.

The data on accidents with injuries (and death) are based on the police reports of road traffic accidents that are published yearly by the Central Bureau of Statistics of Norway. The data base contains detailed information as to where the accident happened and what category of road user was injured. The number of accidents for a particular group of road users in a county and year is divided by the population at risk to control for different county sizes. For drivers of automobiles, motorcycles, and mopeds, the population at risk is approximated by the respective number of registered vehicles. Pedestrian and bicycle injuries are measured on a per capita basis. In the seven years represented by 1980-1986 we have this information for all 18 counties in Norway, excluding Oslo. Within each of the 18 counties we separate the material according to whether the accident happened within a densely populated area or on the highway. It is assumed that the mechanisms and factors explaining accidents may vary outside densely populated areas as compared to within. Let |A.sub.kt~ represent accidents per population at risk in county k, time period t, for a particular group of road users, within, or without densely populated areas. The basic model assumes that the accidents are generated through the equation

|A.sub.kt~ = |Beta~|prime~|X.sub.kt~ + ||Epsilon~.sub.kt~~ (7)

where |X.sub.kt~ represents a vector of characteristics, |Beta~ is a parameter vector, and ||Epsilon~.sub.kt~ is a zero mean random error.(11) Based on our theoretical considerations, the vector of characteristics will contain elements from five groups:

* seat belt use

* technical standard of automobiles

* technical standard of roads

* density of automobiles in an area

* attitudes towards risk.

Some of the variables capturing these considerations do not change rapidly over time. It is therefore possible that they may be correlated with permanent intercounty differences that stem from other sources. To control for this, and the potential bias from import and export of trips between counties, we have also estimated a fixed effects specification of the model, where the accidents are assumed to be generated through the equation

|A.sub.kt~ = |Gamma~|prime~|X.sub.kt~ + ||Kappa~.sub.k~ + ||Tau~.sub.1~ + |e.sub.kt~ (8)

where |Gamma~ is the parameter vector, ||Kappa~.sub.k~ and ||Tau~.sub.t~ are county and year effects, respectively, and |e.sub.kt~ is the zero mean random error. In the fixed effects specification, the dummy variables may capture some of the intercounty differences in accident generation that actually originates from the more theoretically based variables in the characteristics vector. Therefore, results from estimation of both the models (7) and (8) are reported.

The data on seat belt use is compiled directly from the files of the special police task force for traffic control. This unit has been enforcing the mandatory seat belt law with extensive controls inside and outside of densely populated areas in all 18 counties. The fact that the same national task force has conducted controls with the specific aim of enforcing seat belt laws all over the country should rule out possible regional differences in the collection of data. The percentage of drivers controlled and not fined in an area is interpreted as the actual percentage of drivers wearing seat belts. This variable is labelled BELTS 1 and BELTS 2 in the analysis. All empirical variables marked 1 and 2 refer to events and behavior outside, and inside, densely populated areas, respectively. It may happen that the observed level of seat belt wearers is too high using these observations due to some drivers being able to slip the belt on if they detect the control in advance. Our main concern is to analyze differences between regions, and these differences will probably not be affected by this possible bias. Variable definitions and descriptive statistics for the variables are presented in Table III.

Most of the variables in Table III need no further comment. The empirical counterpart of the theoretical considerations regarding risk attitudes is not straightforward, however. Lacking relevant measures of risk attitudes, we have resorted to the crude indicators of age and sex. In line with Oi |13~ we hypothesize that middle-aged individuals, and women are more risk averse than other sections of the population. Thus, the variables MIDAGE and WOMEN are interpreted as risk attitude indicators. Due to the large difference in accident risk for heavier motorcycles where special drivers licenses are required, and mopeds that can be driven by anyone over 16 years of age, separate regressions have been run for these groups.

Results

We have run eight regressions for both models (7) and (8), one for each of the first eight variables listed in Table III. To correct for a possible inconsistency in the standard errors due to heteroscedasticity, the standard errors of the results are reported according to the method proposed by White |21~. The results for accidents outside densely populated areas are reported in Table IV, and the corresponding results for accidents within densely populated areas are reported in Table V. The variables capturing seat belt use, the technical standard of cars, and risk attitudes have the same definitions in Table IV and V. The technical standard of the roads within densely populated areas is measured by the PATH variable. The presence of physically separated pedestrian and bicycle paths is a phenomenon that is mostly present in recent road developments inside densely populated areas. A high incidence of paths is therefore an indication not only of safety measures TABULAR DATA OMITTED for pedestrians and bicyclists, but also an indication of good road standard for other road users. Outside densely populated areas the density of automobiles are measured as the relationship between registered automobiles and the total strech of roads in the county. Inside densely populated areas the aggregate length of roads is unknown. Automobile density is therefore measured as registered automobiles per capita. In the estimates of the basic model (7), proximity to Oslo, the share of rural population, and the composition of towns are included. These variables are not included in the fixed effects model (8), because they do not vary over time, and are therefore perfectly correlated with the county dummies.

The most striking result in Table IV is the implied adverse effect from wearing seatbelts. The BELTS 1 variable is positive and significant in the regressions of the basic model for all groups. In the fixed effects model it is also positive for all groups, but only significant for motorcyclists and mopedists. This result conforms well with our theoretical predictions. It is unusual to find a positive relationship between seat belt use and injured automobile drivers. In contrast to most previous studies that have analysed fatalities, the present data contain information on a broader set of injuries, including even minor ones. Therefore, the adverse incentive effect from the seat belts even on the drivers is captured, while the beneficial effect is hidden in the unobserved severity of the injury. There is also a significant adverse effect from seat belt use on all other road users within densely populated areas in the basic model regression. This effect fails to be significant in TABULAR DATA OMITTED the fixed effects model, and the (insignificant) point estimates for AUTO 2 and MC 2 are negative. The negative sign for the drivers could be explained by the lower speed within densely populated areas. With lower speed the seat belt may be able to protect from even minor injury, such that fewer accidents among drivers are reported.

Another interesting feature of the results is the performance of the indicators for automobile standard and road standard. In Table IV, the basic model regression for AUTO 1 have significant coefficients for the variables NEW, WRECK, and SURFACE that suggest the presence of compensating behavior. In the fixed effects model, only the negative coefficient of the rate of sub standard cars remains significant. On the other hand, the fixed effects model for AUTO 2 reported in Table V yields significant coefficients for both NEW, WRECK, and PATH, which all suggest compensating behavior. It can also be noted that some of the results are mixed. This applies both to results regarding road standard, and the indicators for risk attitudes, which should come as no surprise based on our theoretical discussion. The variables MIDAGE and WOMEN are such rough TABULAR DATA OMITTED indicators of risk attitudes, however, that it is not possible to speculate whether it is Regime 1 effects illustrated in Figure 1 that produce the mixed results.

It is not surprising that the regressions concerning automobile drivers have the highest overall explanatory power. This results from the fact that all the variables that are included in the regressions have a direct impact on the drivers, while the impact on other road users is only indirect. The reported Durbin-Watson statistics are quite low in some of the basic model regressions. The data consist of a wide cross-section, and a short time series, which is organized according to cross-sectional unit. This leads us to conclude that low values of d.w. may be an indication of model misspecification in the cross-section rather than an indication of dynamic effects. Introduction of the dummies in the fixed effects model increases all the d.w. indicators.

Even if some of the results are mixed, the theoretical prediction that technological improvements in automobile safety standards may be counteracted by behaviorable responses to the extent that accidents may increase with better prevention technologies, is not weakened by the empirical analysis.

IV. Concluding Remarks

The main point of this paper has been to investigate the existence of offsetting behavioral responses to technological improvements to reduce traffic accident risk. In this sense the paper has a positive focus in explaining and predicting behavior. The theoretical analysis showed that the level of individual prevention activities is highly sensitive to different attitudes towards risk, and different types of risky situations (Regimes). Another (and more comforting) result is that individual changes in behavior in response to regulatory changes can be unambiguously predicted, given simple restrictions on the prevention technology, without strong restrictions on the underlying preferences, even in Regulation Scheme 2. In Table II adverse incentive effects are present if |p.sub.cx~ is large (remember that the elasticities are negative). This has a straightforward explanation. The utility value of a technological improvement to an arbitrary agent can be found by application of the envelope theorem to the agent's maximized expected utility function. This exercise yields (dV*/dx) = -(|U.sup.0~ - |U.sup.1~)|p.sub.x~. When |p.sub.xc~ is positive, it is possible to increase the utility value of the regulatory change by decreasing c. If the cross partial is large, the behaviorable response to regulation should be expected to be large, because there is more to gain. The road-bump example in the introduction can be interpreted along these lines. The reason why behaviorable responses to road-bumps are great is that road-bumps are only a problem to the driver if speed is high (c is low). The utility cost to the agent from the deteriorated technological environment can be decreased by reducing speed (increasing c).

All these theoretical results hinge on the applicability of the expected utility approach to traffic safety analysis. Blomquist |2~ contains an updated discussion of potential problems with this approach. The present theoretical results are interesting in this perspective, since they place new empirical restrictions on the outcomes of expected utility maximization. Given empirical estimates of the properties of prevention functions, strong predictions regarding the incentive effects of technological regulation could be made. In the discussion of the total safety effects of safety devices like four wheel drive, studded tires in winter, etc., estimation of prevention functions (maybe on test tracks) might provide enough information to predict large scale behavior. In this sense, the present formulation of the theory of individual accident prevention, might provide enough structure to formulate empirical hypotheses that could in principle falsify the expected utility approach in this field.

The present empirical analysis belongs to tradition in this field of literature in employing aggregate data. However, the data at hand are better suited to reveal adverse incentive effects to regulation than much of the previous literature. One reason is that the data base is broader than usual, containing information on accidents leading to death or injury. When the emphasis is on detecting counteracting behavior it is an advantage to have information on a broader category of accidents than only the fatal ones. In our data base the fatalities account for only about 4% of the total registered accidents. Another advantage is the unique large scale data on seat-belt use. As was shown in the theoretical analysis, regulation of seat belt use is an area where the predictions on final accident risk, incentive effects taken into consideration, are less ambiguous than in other cases. On this background it is particularly interesting to note that the seat belt variable performed well in the empirical analysis. Despite the obvious theoretical problems of aggregation in this field, it seems reasonable to claim that the Norwegian data contain interesting regularities that suggest the presence of counteracting behavior to automobile safety regulation.

Appendix

To prove incentive effects in Regulation Scheme 2, an interior solution is assumed. (If (5) is not satisfied, the agent will be in a comer solution with no care taken. In this case there is no incentive effect.) The proofs for the two elasticity conditions in Table II regarding Regime I can be found in Risa |16~.

Proof of the elasticity condition yielding dc*/dx |is less than~ 0, Regime 2: Rewrite (4) and (6) slightly:

|Mathematical Expression Omitted~

(4') can be reformulated as follows:

|Mathematical Expression Omitted~

Define the left-hand-side of (*) to be r. Then the following must hold for (4') to hold:

r |is less than~ -|p.sub.c~/p. (**)

The relevant comparative statics are found by defining (6') (and bearing in mind that |L.sub.x~ = 0 in Regulation Scheme 2):

|Mathematical Expression Omitted~

where |V.sub.cc~ |is less than~ 0 as defined in (5). Using the definition above, a necessary and sufficient condition for dc*/dx |is less than~ 0 is:

r |is less than~ -|p.sub.cx~/|p.sub.x~. (***)

In an interior solution (**) must hold. In that case a sufficient (but not necessary) condition for (***) to hold is:

-|p.sub.cx~/|p.sub.x~ |is greater than or equal to~ -|p.sub.c~/p

which easily transforms to the elasticity condition in Table II.

Proof of |p.sub.cx~ |is less than~ 0 |implies~ dc*/dx |is less than~ 0, Regime 3.

The expression for the comparative static effect based on (6) provided the simplifications implied by Regulation Scheme 2, and Regime 3, reduces to:

|Mathematical Expression Omitted~

Inspection of this expression concludes the proof.

1. The discussion of automobile safety regulation and incentives was initiated in a seminal paper by Peltzman |14~. Viscusi |17; 18; 19~ has written a series of papers on safety regulation in different arenas. Surveys of automobile safety regulation can be found in Crandall et. al. |4~ and Blomquist |1~. Much of the literature on safety regulation--and particularly the present study--draws on the conceptual framework of Ehrlich and Becker |7~.

2. This corresponds to a standard model of self protection in the terms of Ehrlich and Becker |7~. The properties of this regime have been analyzed in some detail in Risa |16~.

3. This corresponds to the model proposed by Viscusi |17~.

4. This can be considered a special case of "pure" self protection. It is possible to self protect without an adverse effect on the disposable wealth in the accident state. In Risa |16~ it is shown that the conflict between reduced probability of the accident state, and reduced wealth in this state, in the optimization problem characterized by Regime 1, produces surprising incentive effects. These particular effects are ruled out in Regime 3.

5. Regime 3 corresponds to a situation where the costs of care only are borne in the non-accident state. This can be the case if prevention costs are non-transferable between states. If care only amounts to paying attention, this obviously represents a utility cost with a potential monetary value when it takes place. It does not, however, reduce the disposable wealth of a patient in hospital when the accident actually has happened. In this sense the prevention costs of strain and attention are fundamentally different from investment costs that have direct consequences for disposable wealth in the accident state.

6. Casual empiricism suggests that the existence of risk lovers with respect to traffic safety cannot be ruled out. If the inequality in (5) is reversed, the agent is in a comer solution, and c = 0.

7. In the terms of Ehrlich and Becker |7~ this could be called forced self protection.

8. Let R denote the average measure of absolute risk aversion. Then |Mathematical Expression Omitted~. The first order condition (4) demands that R |is less than~ -|p.sub.c~/p for an interior solution to exist at all.

9. The proofs are relegated to an appendix.

10. It is typical for the literature that some of the most thorough recent analyses on the effects of seat belt legislation on accidents found in Harvey and Durbin |10~, Evans and Graham |8~, and Garbacz |9~ use aggregated data.

11. The true population at risk will depend not only on the number of potential drivers in a county, but also on the net import of trips mae by drivers from foreign cunties. If the net import of trips between counties are imperfectly accounted for, the classical assumptions on the error term may not hold.

References

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