where f
3
t
i
5 Cp
i
t
i
2 Ct
i
. It should be noted that the induced probability of accidents characterized by 14 is
identical to that for the case of strict liability under nonbankruptcy also see footnote 4. That is because, regardless of the agents’ asset levels relative to liability, the agents
can escape from bankruptcy if the principal is held responsible for the entire liability.
III. Effects of Liability Rules on the Induced Probabilities of Accidents
In this section, we compare the levels of the induced probability of accidents associated with each liability-sharing rule to examine the impacts of liability-sharing rules on the
risks of the activities. The induced probabilities of accidents for each case are charac- terized by 5, 8, and 14, respectively. As analyzed above, when there is no likeli-
hood of the agents going bankrupt, the liability share for the agents adversely affects the level of the induced probability of accidents i.e., dp
i
t
i
du . 0. Thus, it is obvious that changes in liability sharing between the principal and the agents affect the induced
probability of accidents. Under bankruptcy of the agents, we examined jointly shared liability between the principal and the agents with L 2 A
i
and A
i
, respectively, as one of internalizing liability. Then the induced probability of accidents is adversely affected by
the agents’ asset levels i.e., dp
i
t
i
dA
i
. 0. The agents’ bankruptcy case, therefore, shows how changes in the agents’ asset levels, rather than liability-sharing rules, affect
the induced probability of accidents. In contrast to joint liability cases, it is not obvious what affects the induced probability of accidents for the case of strict liability of the
principal because condition 14 is neutral to both u and A
i
. Upon considering all liability-sharing rules examined above, general reasoning suggests that the induced
probability of accidents under nonbankruptcy is lower than that under bankruptcy. Nevertheless, given the three conditions 5, 8, and 14, it is not quite clear—
particularly when it is possible for agents to become bankrupt—which liability-sharing rule minimizes the induced probability of accidents, i.e., provides the safest environ-
ment for the activities.
Using p to denote the induced probabilities of accidents determined by the optimal contracts under nonbankruptcy, p
J
to denote joint liability under bankruptcy of the agents, and p
S
to denote strict liability of the principal, we obtain the following lemmas.
14
L
EMMA
1: p , p
J
L
EMMA
2: p . p
S
if uL2 ¶ uC 9t
i
u , uC 9pu p , p
S
if uC9t
i
u , uC9pu ¶ uL2 L
EMMA
3: p
S
, p
J
if A
i
2 ¶ uC9t
i
u , uC9p
S
u p
S
. p
J
if uC9t
i
u , uC9p
S
u ¶ A
i
2 Lemma 1 implies that the induced probability of accidents under nonbankruptcy is
always lower than that under bankruptcy when the principal and the agents are jointly responsible for compensating liability. Therefore, in the case that the agents face
bankruptcy, setting liability share as u A
i
L enables the agents to stay out of potential bankruptcy; thus, the induced probability of accidents will be lower. The essence of this
14
To distinguish the induced probabilities of accidents for the three different cases, a subscript for agent i is omitted. Instead, we use subscripts for joint and strict liability cases.
357 A. W
ATABE
lemma is to imply that an appropriate choice of a liability sharing rule not only prevents the agents from going bankrupt but also lowers the induced probability of accidents.
Lemma 2 indicates that the induced probability of accidents when liability is shared by the principal and the agents for 1 2 uL and uL, respectively, is lower res. higher
than when the principal is held strictly liable, if the marginal cost of risk reduction is less res. greater than half of the agents’ liability share. As above, the induced probability
of accidents decreases with the agent’s liability share if there is no likelihood of bankruptcy. When strict liability of the principal applies, u 5 0. And the condition
uL2 ¶ uC9t
i
u , uC9pu always holds for u 5 0. Thus, p . p
S
holds for u 5 0. As a result of Lemmas 1 and 2, therefore, strict liability of the principal provides the safest
environment for conducting activities when there is no likelihood of bankruptcy. Lemma 3 signifies that, given the possibility of the agents’ becoming bankrupt, the
induced probability of accidents when liability is shared by the principal and the agents for L 2 A
i
and A
i
, respectively, is lower res. higher than when the principal is held strictly liable, if the marginal cost of risk reduction is less res. greater than half of the
agents’ asset levels. Unlike the case of nonbankruptcy, strict liability of the principal may not necessarily provide the safest environment for activities.
To examine Lemma 3 in detail, consider the case in which the marginal cost of risk reduction exceeds half of the agents’ assets. In this case, strict liability of the principal
is preferred to joint liability. Strict liability of the principal is equivalent to u 5 0, and therefore, the second condition in Lemma 2 holds such that p . p
S
. And from Lemma 1, p , p
J
. Hence, when there is a likelihood of bankruptcy for the agents, strict liability of the principal provides the safest environment for the activities if A
i
2 ¶ uC9t
i
u. Meanwhile, if half of the agents’ assets exceeds the marginal cost of risk reduction, joint
liability is preferred to strict liability of the principal, i.e., p
J
, p
S
. Suppose that u is set sufficiently small but strictly above zero. Then, the agents’ bankruptcy is eliminated and
from Lemma 1, p , p
J
. Thus, if the condition for p , p
S
always holds whenever the condition for p
J
, p
S
holds, setting u , A
i
L results in p , p
J
, p
S
. This, however, never happens. To see this, when uC9t
i
u ¶ A
i
2 holds, p
J
, p
S
, and when uC9t
i
u , uL2 holds, p , p
S
. For the condition uC9t
i
u , uL2 to be satisfied whenever the condition uC9t
i
u ¶ A
i
2 holds, liability share u must be greater than A
i
L. But, this is contradictory when u , A
i
L for p , p
J
, p
S
. Consequently, when there is a likelihood of bankruptcy for the agents and uC9t
i
u ¶ A
i
2, liability sharing between the principal and the agents such as L 2 A
i
and A
i
provides the safest environment for activities. We summarize the results of the analysis shown in Lemmas 1–3 in the following
propositions: P
ROPOSITION
1: Under the optimal contract, if there is no likelihood of bankruptcy for the agents, the induced probability of accidents becomes lowest given strict liability of the principal.
P
ROPOSITION
2: Under the optimal contract, if there is a likelihood of bankruptcy for the agents, the induced probability of accidents becomes lowest for strict liability of the principal if A
i
2 ¶ uC9t
i
u. Otherwise, the induced probability of accidents becomes lowest for joint liability compen- sating for L 2 A
i
and A
i
by the principal and the agents, respectively.
IV. Social Welfare Effects of Liability Rules
