Consider our various legal rules in this setting. First, consider a no liability rule. Firms in industry i have no incentive to take any care under this legal rule. Marginal costs are therefore equal to c, and equilibrium output satisfies:
so that output exceeds the efficient level.
In industry v, marginal costs are equal to c + QfLH(0), which exceed the efficient level. The competitive equilibrium quantity satisfies:
and output is less than the efficient level. The no liability rule is not efficient.
Now suppose there is a rule of strict liability. The profit of the representative firm in industry i is:
Since firms are faced with the full social costs of their actions, they choose the efficient level of care for the level of Qv that is chosen, and also (conditional on Qv) choose the efficient level of output. However, since under a strict liability rule firms in industry v are fully compensated for all harm, the profit of the representative firm in industry v is:
Setting price equal to marginal cost in industry v yields Pv = u' ) = c.
This results in a level of production of v that is too high. The reason is that entry and production in industry v is partially responsible for creating harm, but under strict liability firms in industry v face none of these costs. In turn, this results in underproduction of good i. For the same reasons, a rule of no liability has a similar outcome: competitive equilibrium in both markets involves over production of good i but underproduction of good v.
Finally, consider a negligence rule. Suppose the due standard is set at the efficient level, x*. To avoid liability, each firm (for the same reasons discussed in section 22.214.171.124) will choose to take the efficient level of care. But then this will lower the marginal cost in industry i to c + w;x*, which is less than the level needed to ensure efficient production and consumption in that market. Hence, the competitive equilibrium will involve overproduction in market i, and will again result in underproduction in market v.
In section 126.96.36.199 we saw that second-best negligence rules existed, which involved setting due standards of care higher than xi*. Is that the case here? Suppose that the due standard of care is set at some level above the due standard, say at x; > x*. Then as long as the due standard of care is not set too high, firms in market i will choose to take the level of care xt to avoid liability. This will increase marginal costs, which reduces production in market i. In turn, this reduces the marginal harm in industry v. Both effects result in a welfare improvement over the negligence rule when the due standard of care is set at xi*.