An Alternative to the Basic Causal Requirement for Liability Under the Negligence Rule

Proposition 1.

(a) Not participating in the activity is socially best when both

1. v < p _D h + c

   and

2. v < (p _R + p _D )h

hold.

1. Participating in the activity and taking care is socially best when

1. v > p _D h + c

   and

2. c < p _R h

hold.

1. Participating in the activity and not taking care is socially best when

1. v > (p _R + p _D )h

   and

2. c > p _R h

   hold.

Proof.

The claims are self-evident. For example, not participating is best when it is superior to participating and exercising care (meaning that (1) is negative) and superior to participating and not exercising care (meaning that (2) is negative). □

Proposition 2.

Suppose that the negligence rule with the negligence-based cause-in-fact requirement applies. Then if the party engages in his activity, he will be induced to take care when doing so is socially desirable but would not take care otherwise.

Proof.

If taking care is socially desirable and the party takes care, he will not be negligent so will not be found liable if harm occurs. Hence his utility will be

1. v – c.

   If he does not take care, he will be negligent and his negligence will be found a cause-in-fact of harm in state R. Hence his expected utility will be

2. v – p _R h.

   But since taking care is socially desirable, c < p _R h. Thus,

3. v – p _R h < v – c.

   Accordingly, the party would exercise care.

If taking care is not socially desirable, however, the party would never be found negligent and thus would not exercise care. □

Next, and supposing again that a party who engaged in an activity was negligent and that harm occurred, let us say that the party’s activity was a cause-in-fact of harm if the harm would not have occurred had he not participated in his activity. This will be the case in states R and D. We define the negligence rule with the activity-based cause-in-fact requirement to be the regime under which a negligent party would be found liable and would thus have to pay damages of h when harm occurred if and only if his activity was a cause-in-fact of harm. We then have

Proposition 3.

Suppose that the negligence rule with the activity-based cause-in-fact requirement applies. Then if the party engages in his activity, he will be induced to take care when doing so is socially desirable but would not take care otherwise.

Proof.

If taking care is socially desirable and the party takes care, he will not be negligent, so that his utility will be given by (9). If he does not take care, he will be negligent and his expected utility will be given by (2). But

1. v – (p _R + p _D )h < v – c

   because c < p _R h holds. Thus, the party would exercise care.

If taking care is not socially desirable, the party would never be found negligent and thus would not exercise care. □

Last, we consider the issue of participation in the activity.

Proposition 4.

Suppose that the negligence rule applies with either the negligence-based cause-in-fact requirement or the activity-based cause-in-fact requirement. Then

(a) a party’s decision whether to engage in the activity will be the same under both requirements;

(b) a party will always decide to engage in the activity when that would be socially desirable; but

(c) a party will sometimes decide to engage in the activity when that would be socially undesirable: if the exercise of care would be socially desirable, a party will inappropriately engage in the activity when v is in (c, c + p _D h); and if the exercise of care would be socially undesirable, the party will inappropriately engage in the activity when v < (p _R + p _D )h.

Proof.

(a) The decision whether to engage in the activity depends on a party’s utility if he does not engage in the activity and his utility if he engages in the activity. These utilities do not depend on which cause-in-fact requirement applies. In particular, if a party does not engage in the activity, his utility will be 0 regardless of the cause in fact requirement. And if a party does engage in the activity, Propositions 2 and 3 tell us that, regardless of the requirement, he will take care when that is socially desirable – and thus obtain v – c – and not take care when that is socially undesirable – and thus obtain v. The party will therefore behave identically under both requirements.

1. (b) When the activity is socially desirable, one case is where care is also socially desirable. From Propositions 2 and 3 we know that in this case, regardless of the cause-in-fact requirement, if the party engages in the activity, he will take care and not bear liability. Thus he will engage in the activity if v > c. And we know from Proposition 1 that since the activity is socially desirable, v > c + p _D h, implying that v > c so that the party will engage in the activity.

The other case is where care is not socially desirable. From Propositions 2 and 3 we know that in this case, regardless of the requirement, if the party engages in the activity, he will not take care and not bear liability. Thus he will engage in the activity if v > 0, which is true by assumption.

1. (c) We again consider two cases. One is where taking care is socially desirable. Then engaging in the activity is socially undesirable if v < c + p _D h. But in this case the party will bear no liability if he takes care; and since he has the option to do this, he will engage in the activity whenever v > c. Hence, we will observe inappropriate engagement in the activity when v is in (c, c + p _D h) under both cause-in-fact requirements.

The second case is where taking care is socially undesirable. In this case, engaging in the activity is socially undesirable when v < (p _R + p _D )h. But because there would be no liability in this case, the party will engage in the activity so long as v is positive. Hence, regardless of the cause-in-fact requirement there will be engagement in the activity that is socially undesirable when v < (p _R + p _D )h. □

Proposition 5.

Suppose that the negligence rule with the negligence-based cause-in-fact requirement applies with probability q, that the exercise of care is socially desirable, and that a party engages in his activity. Then the party will be induced to take care only when q exceeds a critical value q _N = c/(p _R h); when q equals q _N , the party will be indifferent between taking care and not; and when q is less than q _N , he will not take care.

Proof.

If taking care is socially desirable and the party takes care, he will not be negligent so will not be found liable if harm occurs. Thus, his utility will be (9). As observed in the proof of Proposition 2, we know that if a party engages in the activity and is negligent, his expected liability will be (10), meaning that when law enforcement is imperfect, his expected liability will be v – qp _R h. Thus the expected utility of a party who engages in the activity and takes care minus his expected utility if he does not is

1. v – c – (v – qp _R h).

   Accordingly, if q = c/(p _R h), (13) will be zero and a party who engages in the activity will be indifferent between exercising care and not; if q < c/(p _R h), the party will not take care; and if q >c/(p _R h), the party will take care. Note that c/(p _R h) < 1 because (6) is assumed to hold. □

Proposition 6.

Suppose that the negligence rule with the activity-based cause-in-fact requirement applies with probability q, that the exercise of care is socially desirable, and that a party engages in his activity. Then the party will be induced to take care only when q exceeds a critical value q _A = c/[(p _R + p _D )h]; when q equals q _A , the party will be indifferent between taking care and not; and when q is less than q _A , he will not take care.

Proof.

If taking care is socially desirable and the party takes care, he will not be negligent so will not be found liable if harm occurs. Thus, his utility will be v – c. As observed in the proof of Proposition 3, we know that if a party engages in the activity and is negligent, his expected liability will be (2), meaning that when law enforcement is imperfect, his expected liability will be v – q(p _R + p _D )h. Hence, the expected utility of a party who engages in the activity and takes care minus his expected utility if he does not is

1. v – c – (v – q(p _R + p _D )h).

   Therefore, if q = c/[(p _R + p _D )h], (14) will be zero and a party who engages in the activity will be indifferent between exercising care and not; if q < c/[(p _R + p _D )h], the party will not take care; and if q > c/[(p _R + p _D )h], the party will take care. Furthermore, c/[(p _R + p _D )h] < 1 because (6) is assumed to hold. □

From the previous two propositions, we see that q _A < q _N . This leads to the next conclusion.

Proposition 7.

Suppose that the negligence rule applies with probability q, that the exercise of care is socially desirable, and that a party engages in his activity. Then the party will be induced to exercise care more often under the activity-based cause-in-fact requirement than under the negligence-based cause-in-fact requirement. In particular, the party will be negligent under the negligence-based cause-in-fact requirement but not under the activity-based cause-in-fact requirement whenever q is in the interval (q _A , q _N ).

Finally, let us consider engagement in the activity under the two cause-in-fact requirements.

Proposition 8.

Suppose that the negligence rule with either of the cause-in-fact requirements applies with probability q. Then

1. a party will always engage in his activity when doing so is socially desirable;

2. a party will sometimes engage in his activity when doing so is socially undesirable;

3. a party will undesirably engage in his activity at least as often under the negligence-based cause-in-fact requirement as under the activity-based cause in fact requirement—if the party undesirably engages in his activity under the activity-based cause-in-fact requirement, he will also do so under the negligence-based cause-in-fact requirement; and

4. there exists a range of situations in which the exercise of care is socially desirable and under the negligence-based cause-in-fact requirement, a party will socially undesirably engage in the activity and fail to take care, whereas under the activity-based cause-in-fact requirement, he will socially desirably engage in the activity and take care.

Comment: Parts (c) and (d) imply that parties will undesirably engage in the activity more often under the negligence-based cause-in-fact requirement than under the activity-based cause-in-fact requirment.

Proof.

(a) It was shown in Proposition 4(b) that a party would always engage in his activity when q = 1, meaning that his achievable utility is positive. When q < 1, the maximum utility of a party who engages in the activity must be at least as high as it was when q = 1, and thus must be positive. Therefore, he will engage in the activity.

1. (b) The proof of Proposition 4(c) applies when q < 1.

2. (c) Suppose that under activity-based cause-in-fact requirement, the party undesirably engages in the activity. There are then two possibilities—the party exercised care; or the party did not exercise care. It will be shown that under each possibility, the party would undesirably engage in the activity under the negligence-based cause-in-fact requirement.

3. (i) The party exercised care: Because the hypothesis is that under the activity-based cause-in-fact requirement engaging in the activity was socially undesirable and care was exercised, we know in particular that v < c + p _D h.

Because the party chose to take care, we know that doing so must have been socially desirable, that is, that c < p _R h. The logic is that if taking care was not socially desirable, the party could not be found negligent for failure to take care. Hence he would not have taken care, a contradiction.

Because the party exercised care and thus could not have been found liable, and because he engaged in the activity, we know that v > c.

Now let us use the above inferences to show that under the negligence cause-in-fact requirement, the party would also have undesirably engaged in the activity.

First, since v > c, we know that under the negligence-based cause-in-fact requirement, the party would choose to engage in the activity: The party has the option to exercise care and avoid liability. Thus, if he engages in the activity and takes care, he will be better off than not not engaging in the activity. (Although we cannot infer from what was just said that he will choose to take care, we do know that since he has the option to do so, if he acts optimally it must involve engaging in the activity.)

Second, let us show that the party’s engagement must be socially undesirable. One possibility is that the party engages in the activity and takes care. But since we know v < c + p _D h, doing so would be socially undesirable. The other possibility is that the party engages in the activity and does not take care. In this case, engagement in the activity will be undesirable if v < p _D h + p _R h. But this condition must hold because we showed that v < c + p _D h and c < p _R h.

1. (ii) The party did not exercise care: Because the hypothesis is that under the activity-based cause-in-fact requirement engaging in the activity was socially undesirable and care was not exercised, we know in particular that v < (p _R + p _D )h.

The inference we draw from the hypothesis that the party did not take care depends on whether care was not or was socially desirable. If care was not socially desirable, there would be no liability for failing to take care. Hence, the party would engage in the activity as long as v is positive. If care was desirable, however, the party would be negligent and thus face expected liability of q(p _R + p _D )h under the activity cause-in-fact requirement. And because the party did not take care, we infer that c > q(p _R + p _D )h. Finally, because the party chose to engage in the activity, we infer that v > q(p _R + p _D )h.

Now we can verify that under the negligence-based cause-in-fact requirement, the party will also undesirably engage in the activity. Suppose first that care is not socially desirable. In that case, the party will have no reason to take care and thus will engage when v > 0. Thus the party will engage in the activity and that will be socially inappropriate given our assumption that v < p _R h + p _D h.

Next suppose that care is socially desirable. We had inferred above that c > q(p _R + p _D )h. This implies that c > qp _R h, meaning that under the negligence-based causal requirement, the party will choose not to take care. And we had also inferred that v > q(p _R + p _D )h. This implies that v > qp _R h, meaning that the party will engage in the activity under the negligence-based requirement. And since our hypothesis is that v < (p _R + p _D )h, engaging in the activity will be socially undesirable.

1. (d) Suppose that the exercise of care is socially desirable given engagement in the activity, that is, suppose that c < p _M h.

We know from Propositions 5 and 6 that if q lies in the interval (q _A , q _N ) = (c/[(p _R + p _D )h], c/(p _R h)), then the party will exercise care under the activity-based cause-in-fact requirement but will not do so under the negligence-based cause-in-fact requirement.

Now consider any q in the interval just given and any v in the interval

(c + p _D h, (p _R + p _D )h). We claim that for such q and v, a party will undesirably engage in the activity and fail to take care under the negligence-based requirement, whereas he will desirably engage in the activity and take care under the activity-based requirement. The claims about the taking of care are true, as explained in the prior paragraph.

Regarding the claim that the party will undesirably engage in the activity under the negligence-based cause-in-fact requirement, we know that if he engages in the activity, he will be negligent, and thus his expected liability will be qp _R h. But since q < c/(p _R h), we have qp _R h < c. Thus, expected liability is less than c. But v > c in the interval. Hence the party will engage in the activity. That will be socially undesirable because in the interval v < (p _R + p _D )h.

Finally, regarding the claim that the party will desirably engage in the activity under the activity-based causal requirement, we know that if he engages in the activity, he will take care and thus his expense will be c. But, again, v > c in the interval, so the party will engage in the activity. Further, that will be socially desirable since v > c + p _D h in the interval. □