In the appendix we will simulate the effect that comparative negligence and all-or-nothing rules have on the social loss under non-uniform distributions of the costs of precaution. We will use a more complex model than the one employed in section 1. The computer simulations were implemented in the MATHEMATICA environment.
We will refer to the graph in figure 1.
For each accident we consider the precaution cost c=[0, ∞) needed to avoid it and the expected harm h=(0, ∞); “h” is given by “Hp”, where “p” is the probability that, if the avoider is careless, an accident will occur and “H” is the harm;
Calculate for each accident y=c/h (0≤y<∞), the standardized precaution cost, and rank all the accidents on the base of the value of y; if y<1 (precaution cost < expected harm) the accident is socially undesirable; if y>1 (precaution cost > expected harm) it is socially advantageous that the accident occurs; if y=1 it is indifferent.
Let x=N/Ns, Ns = number of accidents with y≤1 (accidents which should not occur), N = position of the accident in the raked order described above, notice that x=1 when y=1, x and y are standardized measures for the precaution cost and the ranked order of the accidents; for simplicity we will consider x as a continuous variable, even if in fact it has discrete values;
Consider a function y=f(x), which provides a relationship between the standardized precaution cost and the ranked order of the accidents; dy/dx≥0 (because of the ranked order, an accident cannot have a smaller y than the preceding one); f(0)=0 the first accident to avoid is the one with the smallest y (c infinitely smaller than h); f(1)=1 the last accident to avoid has y=1 (c=h);
We will test the model by employing the function y=xq, q=(0, ∞).
By employing this model, we will calculate the social loss under ANRs and the social loss under CN as functions of q, and then we will compare them in order to test whether or not CN yields smaller social losses than ANRs for any possible value of q. This will provide us with a criterion to choose between ANRs and CN in unilateral precaution cases described in the previous sections.
The results of that comparison will be plotted. We will use a graph showing on the x-axis the value of q and on the y-axis the value of the difference between the social loss under ANRs and the social loss under CN. The value depicted on the y-axis is to be interpreted as the comparative advantage of CN in reducing the social loss due to a judicial lack of information for different precaution cost distributions.
Because the central case is the uniform distribution of precaution costs, given by q=1, the graphs will be centered on this value. To the right-hand side of the graph, q>1-distributions will be shown (1<q<∞), and to the left q<1-distributions will be presented (0<q<1) using the inverse of the values indicated on the right-hand side (see figure 8). This solution accounts for the fact that, with respect to the central uniform distribution of accidents (the straight line in figure 7), q=2-distribution is the mirror image of q=½-distribution, q=4-distribution is the mirror image of q=¼-distribution, and so forth.
Y
q=1 Y
q<1
Y
q>1 Figure 7. Different shapes of the function y=xq.
Perfect compensation
We will now refer to figure 2. The social loss is given by the sum of the differences h-c for each single accident that the legal rule under analysis fails to prevent.
For ANRs, we denote as “A1” the area of the dotted triangle in figure 2, and as ½A1 the social loss due to ANRs (50% of the accidents will occur).
∫
For CN, we denote as A2 the gray area in figure 2; in this case the social loss is equal to the whole area A1, because all the accidents between x=xm and x=1 occur.
∫
In order to choose between ANRs and CN, we need to determine which rule yields the smaller social loss. For this purpose, we can compare ½A1 (social loss under ANRs) and A2 (social loss under CN). CN is preferable over ANRs if ½A1-A2>0. ANRs are preferable over CN if ½A1-A2<0. The two rules are equivalent if ½A1-A2=0. Let L(q) denote the former difference.
So, CN is preferable over or equivalent to ANRs if:
0
Figure 8 shows the value of the function ½A1-A2=L(q) for different values of q, i.e. for different precaution cost distributions. The social loss under CN is smaller than the social loss under ANRs if L(q) is positive; in this case CN performs as a filter and yields a smaller social loss than ANRs. A
-10 -5 5 10
negative value of L(q) indicates than the loss under CN is greater than the loss under ANRs and thus ANRs are the best solution. From figure 8 it is clear that CN is better than ANRs for any value of q>½.
ANRs have a better performance only for 0<q<½. However, the gain in social loss saving (given by the vertical distance between the curve and the x-axis) of comparative negligence is largely superior than the one due to ANRs. It can be also noticed that the social loss does not depend on the number of accidents each rule prevents. ANRs always prevent 50% of the accidents to avoid, while CN prevents a variable number of accidents depending on the value of q.
The graph on the left-hand side of figure 9 shows the previous statement. The greater q, the larger the number of accidents which will be prevented. On the contrary, the number of accidents prevented under ANRs is constant. The graph on the right-hand side of figure 9 shows the shape of the precaution cost distribution when q=½. In this case CN prevents only 25% of the accidents and the social loss is equal to the social loss under ANRs.
The number of accidents prevented under CN (xm) can be calculated as follows:
Figure 9 shows also that CN tends to prevent more accidents than ANRs when q>1. The following table resumes the previous results.
q>1: n° of accidents under CN < n° of accidents under ANRs social loss under CN < social loss under ANRs
q=1: n° of accidents under CN = n° of accidents under ANRs social loss under CN < social loss under ANRs
½<q<1: n° of accidents under CN > n° of accidents under ANRs social loss under CN < social loss under ANRs
q=½ : n° of accidents under CN > n° of accidents under ANRs social loss under CN = social loss under ANRs
q<½ : n° of accidents under CN > n° of accidents under ANRs social loss under CN > social loss under ANRs
Appendix 2. Overcompensation
In this section we will relax the assumption of perfect compensation. Under ANRs, on the one hand, if the court makes the victim bear the cost of the accident, errors in compensation do not have any effect on the incentive stream simply because no compensation takes place. On the other hand, if the court
25% 75%
Figure 9: Number of accidents prevented under CN and under ANRs
always targets the injurer, overcompensation will create an extra incentive for him, while undercompensation will shift part of the incentive stream towards the victim. In other words, the court could minimize the effect of errors by always targeting the victim. Under CN, the court always directs the incentive stream towards both parties, and the effect of errors cannot be eliminated in the same way.
We will run two analyses. First we will compare ANRs targeting the injurer (for example, simple negligence) to CN. Second, we will compare ANR targeting the victim (for example, contributory negligence) to CN.
In both cases D (the damages that the injurer has to pay) are no longer equal to H (the harm that the victim suffers), but D=H(1+e), where e is the error committed by the court while estimating H, consistently also the expected damages change, d=h(1+e). The errors are systematic and predicted by the parties. The court can overcompensate or undercompensate the victim for the harm.
overcompensation: e>0
perfect compensation: e=0
undercompensation: e<0.
For overcompensation we denote:
overcompensation with small errors 0<e<1
overcompensation with large errors e≥1 (D is twice or more than twice as much as H) Overcompensation - small errors (0<e<1) – ANRs targeting the injurer
Under ANRs targeting the injurer, overcompensation of the harm increases the incentive stream directed toward the injurer and induces the injurer to take overprecaution. The injurer will avoid an accident if the expected damages (d) are greater than the precaution costs (c), even if c>h. In this case there is a loss given by the difference c-h (zone II in figure 10). Because we assume that the court always targets the injurer and that only in 50% of those cases the injurer is the avoider, that extra social loss arises only in 50% of the cases. Therefore, the social loss is equal to the social loss under perfect compensation plus ½ of the stripped triangle in figure 10.
Also in the case of overcompensation, under CN the court over-valuates the damages, but the effect is lower because CN contemporarily targets the victim and the injurer with half the maximum incentive stream, so that the injurer pays D=H(1+e)/2, and the victim bears the remaining H(1+e)/2.
The incentives for the injurer increase: the injurer will pay D to the victim, so he will be careful as far as the precaution cost = d (which is greater than h/2). The incentives for the victim are undermined: the victim will expect to bear just h-d = h - h(1+e)/2 = h(1-e)/2 (which is smaller than h/2).
In figure 10’s right-hand side graph: the injurer will be careful till the point z (more than under perfect compensation, xm), while the victim will be careful only till the point s (less than under perfect compensation, xm). Before s both will be careful, after z both will be careless. The result is that under overcompensation 100% of the accidents in the gray area and 50% of the accidents (only the injurer is careful) in the dotted area will occur.
It is to be noticed that the analysis is exactly the same as for undercompensation. The only Figure 10. Overcompensation - small errors
ANRs CN
difference is that in that case the victim will be over-careful and the injurer under-careful, ceteris paribus.
In order to test our model in the case of small errors in overcompensating damages, we compare the total (ordinary loss + additional loss due to errors) social loss under ANRs and the total social loss under CN. If the difference “social loss under ANRs targeting the injurer (L1)” – “social loss under CN (L2)” > 0 then CN yields to a smaller social loss in the case of errors.
( ) ( ) ( )
( ) 0Figure 11 shows the performance of CN for any value of q and for different values of e (0<e<1, e=.1, e=.5, e=.9), and shows that CN is still to be preferred on ANRs because yields a smaller social loss. The result becomes clearer when the errors become larger.
Overcompensation - large errors (e≥1) – ANRs targeting the injurer
We now consider the case in which large overcompensation (e≥1) occurs. That is, the injurer has to pay damages that are (twice or) more than twice as much as the harm suffered by the victim. The analysis differs from the cases already examined only with respect to CN. With respect to ANRs the analysis is exactly the same as in the case of small errors.
Under CN the analysis is completely different. In the big dotted triangle 50% of the accidents occur, because only the injurer will be careful, as the victim is entirely compensated for the harm. The small stripped triangle depicts the loss due to over-precaution of the injurer, regarding again 50% of the accidents.
The injurer will be careful as far as c<h(1-e)/2 (precaution cost < expected damages to pay to the victim), which corresponds to the point xf on the x-axis. The victim does not have any incentive to be careful because he will recover the whole harm from the injurer, because (1-e)/2<h. If only the injurer
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Figure 11: CN against ANRs targeting the injurer – Overcompensation - small errors
e=.1
e=.5
e=.9
is careful 50% of the accidents will occur both in zone I and in zone II. However, the social loss in zone I is given by the difference h-c (accidents we should prevent), while the social loss in zone II is given by the difference c-h and is due to the injurer’s overprecaution (accidents which should occur). If loss under ANRs (L1) - loss under CN (L2)> 0, then CN is more efficient than ANRs in minimizing the
social loss in the case of large errors.
( )
Figure 12. Overcompensation - large errors CN
Figure 13: CN against ANRs targeting the injurer – Overcompensation - large errors
( ) ( )
Figure 13 shows even a clearer pattern than in the previous case. The function L(q) is always positive for any value of q and any value of e (we have tested e=1.1, e=3, e=5). That means that CN always yields a smaller social loss.
Overcompensation - small errors (0<e<1) – ANRs targeting the victim
We now consider the case in which under ANRs the court always targets the victim. In this case no problem of over or undercompensation arises, because, by making the victim bear the entire cost of the accident, the court does not need to estimate the harm. However, errors still affect the performance of CN in the same way as we have already seen in the former two sections.
Therefore, in order to test the relative efficiency of the two rules, we will compare ANRs without errors and CN with errors. We will start with small errors in overcompensating damages (0<e<1).
We can still refer to figure 10. The loss due to ANRs targeting the victim is just the big dotted triangle (if the court targets the victim errors in compensation have no effect) in the left-hand side graph. The loss under CN is the same as in the case tested above for overcompensation, small errors (see figure 10, the right-hand side graph). If the difference “social loss under ANRs targeting the victim (L1)” – “social loss under CN (L2)” > 0 then CN yields a smaller social loss in the case of errors.
( ) ( )
It is to be noted that small errors tend to transform a CN rule in an ANR, so that if the error is equal to 1 there is no difference between the two rules. That is why, even if we introduce small errors CN is still better than ANRs (as shown in the graph) and the difference between the two rules slowly disappears as e becomes closer to 1.
Overcompensation - large errors (e≥1) – ANRs targeting the victim
We have already said that if the error is equal to 1, CN and ANRs targeting the victim give the same results. We are now going to test the effect of large errors in overcompensation. Again ANRs targeting the victim are not affected by errors. On the contrary, under CN the injurer will be over-careful and the victim will not be careful at all. As we have already explained, the loss is depicted by the two dotted triangles in the right-hand side graph of figure 12.
If “social loss under ANRs (L1)” – “social loss under CN (L2)”> 0, then CN is more efficient than ANRs as it yields a smaller social loss in the case of large errors.
( )
Figure 14: CN against ANRs targeting the victim – Overcompensation - small errors
e=.1e=.5 e=.9
e=.9999
In this case ANRs are always better than CN. The graph shows that the value of the function L(q) is negative for any value of q. When the court entitles the victim to recover more than twice as much as the harm that he actually suffered, CN becomes worse than ANRs targeting the victim.
Concluding remarks for overcompensation
The simulation shows that CN generally yields a smaller social loss than ANRs also in the case of overcompensation. In the case of small errors (both with respect to ANRs targeting the injurer, equation a2, and figure 11, and ANRs targeting the victim, equation a4, and figure 14) the superiority of CN is due to the filtering effect. In the case of large errors CN is also superior to ANRs targeting the injurer because of a lighter over-deterrence effect. On the contrary, compared to ANRs targeting the victim in the case of large overcompensation CN is no longer the best rule. This is due to the fact that large errors erode CN’s filtering effect. However, because this is the only case, we can conclude for a general superiority of CN.
Appendix 3. Undercompensation
If the court underestimates the harm suffered by the victim there is an effect on the incentive streams that the tort-liability system gives both to the victim and to the injurer.
For undercompensation we denote:
undercompensation with small errors -½<e<0;
undercompensation with large errors -1<e≤-½ (D is half or less than half H, if e=-1 the rule becomes equivalent to no liability).
L(q)
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-0.5 -0.4 -0.3 -0.2 -0.1 0.1
2 4 6 8 10 101 81 61 41 21
q
Figure 15: CN against ANRs targeting the victim – Overcompensation - large errors
e=1.1
e=3 e=5
Undercompensation – small errors (-½<e<0) – ANRs targeting the injurer
If the court errs in setting the damages that the injurer has to pay, and if those damages are set at a lower level than under the perfect-compensation assumption, D=H(1+e), and e<0. The incentive stream that the injurer faces is undermined in the same measure, d=h(1+e). However, undercompensation creates an additional incentive stream for the victim. This is due to the fact that he has to bear the portion of the harm not internalized by the injurer (H-D=H-H*[1-e]=H-H+e=e; thus the victim bears a portion of the harm equal to the error that the court makes).
Table 1. ANRs targeting the injurer. Undercompensation, small errors
avoider Zone I Zone II Zone III
1. injurer careful careful careless
2. victim careful careless careless
no accidents 50% of the accidents 100% of the accidents The effect of errors in (under)compensation is actually ambiguous: on the one hand, the injurer is less careful, on the other, the victim has an incentive to be careful also in the case of the injurer being liable. The table shows the parties’ behavior and its effect on the number of accidents. In zone I (in figure 16) both the victim and the injurer have an incentive to avoid the accident because both bear a portion of the harm which is greater than the precaution cost. There is no social loss. In zone II only the injurer has an incentive to be careful. In zone III neither the victim nor the injurer has an incentive to be careful, because the precaution cost exceeds the portion of the harm that each party has to bear.
Under CN, the case of undercompensation is the same as the case of overcompensation. The only difference is that in this case the victim is over-careful and the injurer under-careful, thus also the points s and z are inverted. The analysis already made in the case of overcompensation with small errors can be employed for the cases of both undercompensation with small errors and undercompensation with large errors.
CN is preferable if “social loss ANRs (L1)” – “social loss CN (L2)”>0. We denote as AII, AIII (for ANRs) and Ai, Aii (for CN) the social-loss areas corresponding to the zones II and III and to the zones i and ii in figure 16. For each area we calculate a fraction consistently to table 1.
2 0
Figure 16. Undercompensation - small errors y
xq In the case of undercompensation the result is ambiguous. CN mantains an advantage for small values of e, while when e increases ANRs targeting the injurer seems to be better. The reason behind this result is that a systematic undercompensation and the fact that the injurer is always held liable make ANRs close to CN. When the error is equal to ½ (the victim bears half the harm) ANRs (with errors) give exactly the same result as CN (without errors). If we introduce errors also under CN the performance becomes worse.
Undercompensation, large errors (-1<e≤ -½) – ANRs targeting the injurer
We now turn to the case of large errors in undercompensation (-1<e≤-½), where the damages that the
We now turn to the case of large errors in undercompensation (-1<e≤-½), where the damages that the