The model

Two parties are in a situation that might result in an accident. The victim is the party that will suffer harm; the injurer is the other party.

Discrete precaution levels. For reasons of simplicity, we assume that each party has a discrete choice between being careful and careless.

Varying precaution costs. Each party can prevent the accident at a cost that is lower than infinite. We denote the precaution costs as c and more specifically the precaution costs of the injurer as c_I and the precaution costs of the victim as c_V. The two costs may have different values, and the chance that the precaution costs have a certain value may differ between the parties. In other words, the distribution of the accident types with respect to precaution costs (accidents that are relatively inexpensive to prevent, accident that are relatively more expensive to prevent, ...) can have any form and is not necessarily the same for both parties.

In the appendix a computer simulation for different distribution curve shapes will be discussed. Precaution costs are said to be reasonable if they are lower than (or equal to) the harm (c<1). For instance if cI=0.25 and cV=1.3, we will say that the injurer can avoid the accident at a reasonable cost, while the victim cannot.

Harm normalized to one. For mathematical simplicity, we assume that if both parties are careless, an accident will certainly occur. This avoids the possible confusion between accident costs and expected accident costs (= accident costs times the probability of an accident if both are careless). This assumption is relaxed in the appendix. The harm that the victim suffers is denoted by h. For mathematical simplicity, we will assume that h always equals 1 (h=1). The normalization of h allows us to express the precaution costs as a percentage of the harm. For instance if cI=0.25 and cV=1.3, this means that the injurer can avoid the accident at a precaution cost that is 25% of the accident costs and that the victim can avoid the accident at a

precaution cost that is 130% of the accident costs.

Types of accidents. Alternative precaution family: the accident is prevented if at least one party is careful. Further division of the alternative precaution family: if only one of the parties can avoid the accident at a reasonable cost (cI<1 and cV>1, or cI>1 and cV<1), we will speak of a unilateral precaution accident. If each of the parties can avoid the accident at a reasonable cost (cI<1 and cV<1), we will speak of an alternative precaution accident (strictly defined). If none of the parties can avoid the accident at a reasonable cost (c_I>1 and c_V>1), we will speak of an efficient accident¹⁰.

(Least-cost) avoider, non-avoider. In the case of unilateral precaution accidents, we will refer to the party who can avoid the accident at a reasonable cost as the avoider and to the other party as the non-avoider. In other words, the avoider can be the injurer as well as the victim.

In alternative precaution cases (strictly defined), we will use the term least cost avoider for the party whose precaution costs are the lowest.

Courts cannot observe precaution costs. The crucial assumption of our model is that courts cannot observe precaution costs at all. Since they do not know c_I and c_V, they do not know whether they are dealing with a unilateral precaution accident, an alternative precaution accident (strictly defined), or an efficient accident. If it is a unilateral precaution accident, they do not know what party (the injurer or the victim) is the avoider and if it is an alternative precaution accident, they do not know who is the least cost avoider. Courts do not know the likelihood of the distribution of precaution costs either. They have no reason, for instance, to assume that injurers in general bear lower precaution costs than victims or vice versa.

Can courts observe precaution levels? As explained above, both parties have a discrete choice between being careful or careless. Can courts observe who has been careful and who has been careless? If accidents are of the alternative precaution type, this question is irrelevant. If a case is brought before court, this means that an accident has taken place and that both parties have been careless. If one party was careful, or both were careful, an accident would not have occurred.

Do courts know whether the accident is of the alternative precaution family or of the joint precaution family? In our analysis we assume that courts do know of what type the accident is. This is quite realistic an assumption. Even if courts have no information on the precaution

10 In the literature there is some confusion between the concepts unilateral precaution and alternative precaution.

In the archetype of unilateral precaution accidents, one party can avoid the accident at a cost < 1, while the other party’s precaution costs are infinite. In the archetype of alternative precaution accidents, each of the parties can prevent the accident at a cost < 1. Terminological confusion may arise when one party can avoid the accident at a cost < 1, while the other party’s precaution costs are > 1 but not infinite and when both parties’ precaution costs are > 1 but not infinite.

costs, they can in most cases figure out whether only either party (alternative precaution cases) or both parties (joint precaution cases) should have changed their behavior in order to avoid the accident. We focus the analysis only on the alternative precaution family.

Legal rules. We assume that courts have only 3 choices: (i) letting the injurer bear all accident costs, (ii) letting the victim bear all accident costs, or (iii) letting the parties share the accident costs. For (i) we will use the term simple negligence, for (ii) contributory negligence and for (iii) comparative negligence (CN), although courts can also use other labels and legal techniques to reach the same results. All-or-nothing rule (ANR) is the term that we will employ to refer to simple negligence and contributory negligence. For simplicity, we assume that under comparative negligence, the losses will be equally shared between the parties (a 50/50 division), although in appendix 4 we will look at the filtering characteristics of different sharing rules (such as 60/40, 70/30, ....). We will denote as d the damages paid by the injurer.

Under simple negligence and perfect compensation, the injurer pays d=1. Under contributory negligence, the injurer pays d=0. Under comparative negligence (with a 50/50 division), the injurer pays d=0.5.

4.2.A. Standard assumptions

In our model, we will make a number of additional assumptions that can be considered as standard in the economic literature on tort law:

(i) Only the tort law system generates an incentive stream. Criminal sanctions (a second incentive stream) do not exist. The incentives created by the tort law are not undermined by insurance (insurance is not available);

(ii) Parties are perfectly informed on each other precaution cost and on the harm (parties’ perfect information) and know exactly who will be declared responsible in the case of an accident and to what extent (perfect predictability of the judicial decisions, this assumption will be relaxed in section 4.3.F);

(iii) No under- or overcompensation by the court (perfect compensation, this assumption will be relaxed in section 4.3.D);

(iv) Parties never escape liability (the apprehension rate is 100% and there is no judgment proofness);

(v) Parties are rational and utility maximizing.

(vi) Since we solely focus on incentives and not on transaction costs or the costs of a suboptimal risk allocation, we assume transaction costs away and we assume that parties are risk neutral.

4.2.B. The graph

To clarify the analysis we use a simple graph, which shows on the y-axis the relative costs of

precaution (calculated as a percentage of the harm). As already explained, the harm h is normalized to 1. As a consequence, the costs of precaution are also normalized by the expected harm. For instance, if the expected harm is $100,000 and the precaution cost is

$70,000, in our normalized model the harm h is 1 and the precaution cost is 0.7.

The x-axis shows the ranked order of accidents, from the accidents which are (relatively) inexpensive to avoid to the accidents which are too expensive to avoid. Thus on the x-axis all accidents are ranked on the basis of their y-value. For mathematical simplicity, the ranked order of accidents is also normalized by the number of accidents to be avoided (those accident the precaution cost of which is smaller than the expected harm), so that x=1 means 100% of the accidents to be avoided. By definition, the precaution cost curve slopes upward (dy/dx ≥ 0): because of the ranked order of accidents, an accident cannot have a smaller y than the preceding one.

In figure 1, the precaution cost curve y(x) is a straight line, which means assuming a uniform distribution of accident types (there are as many accidents that can be prevented at 10% of the expected harm as at 20%, 30%,… 90%, 100%, …) so that the costs of precaution y=x (for further details see appendix 1).

y

1

a b 1 g x 0

Inefficient accidents Social

loss a

Social loss g

Social loss b

Standardized expected harm, h Standardized precaution cost, c

Efficient accidents

Inefficient Accidents:

h>c

Accidents “a” and “b” are inefficient accidents and should be prevented. If they occur there is a social loss equal to the difference

“expected harm - precaution cost”.

***

Efficient Accidents:

h<c

Accident “g” is an efficient accident. If it is prevented there is a social loss equal to the difference “precaution cost – expected harm”.

Figure 1. The basic unilateral-precaution model

4.2.C. The social welfare function

Efficiency requires the minimization of the social loss. If y<1 (c<h) the accident is inefficient;

if it occurs, the social cost is the difference between the harm and the costs of precaution. If y=1 it is indifferent. If y>1 (c>h) letting the accident occur is better than preventing it; if the accident is prevented, there is a social loss of a different type: too high a precaution cost. The social loss in that case is the difference between the precaution cost and the expected harm. In the graph the accidents to be avoided are in the region delimited by the x=1 line, out of that region a good rule is a rule that lets the accidents occur.

Accidents “a” and “b” in figure 1 are accidents to be avoided (c<h), while accident “g” is an efficient accident (c>h) and the legal rule should let it occur. Therefore, the social losses

“a” and “b” occur if the accidents “a” and “b” are not prevented, the social loss “g” occurs if the accident “g” is prevented.

4.3. Unilateral precaution (only one party can avoid the accident at a reasonable cost)