The efficiency of legal rules in the unilateral care model
The efficient level of care
Let us address the question of efficient legal rules in the unilateral care model. Efficiency here is interpreted in an ex ante sense - we seek to find the allocation of resources which minimises the expected aggregate costs of care. Let us denote such a level (assuming it exists) by x*. Note that in principle there might be more than one such efficient level of care, although our assumptions allow us to rule out such cases. Let wt > 0 be the dollar cost of providing one unit of care (that is, wi is the marginal cost of care). To find x* we need to solve:
Note that, in general, the efficient level of care, x* will not be zero (that is, it will always be efficient for the injurer to take some care). However, it is also true that in general it is never efficient to reduce the possibility of harm to zero. The necessary and sufficient first-order condition for this problem is:
This equation is of the form:
The left-hand side is the social marginal cost of care. The right-hand side is the marginal reduction in expected damage (or, alternatively, the expected social benefit of an extra unit of care). Thus, this is just a condition of the usual economic form 'provide care until the social marginal cost of the last unit of care equals the expected social marginal benefit of that last unit'. It follows that, as with any other externality problem in economics, the agents will respond to marginal incentives, rather than total incentives. The solution to this problem is shown in Figure 4.2.1.
Notice that at x*, the injurer still imposes a negative external cost on the victim, since the level of xt chosen by the injurer still determines both the total and marginal expected damages of the victim. But, in the language of Buchanan and Stubblebine (1962), the externality is no longer Pareto relevant: there are no trades between the injurer and the victim that could take place that would make the victim better off, without making the injurer worse off.
It is straightforward to verify (both analytically and diagrammatically, using a diagram exactly like Figure 4.2.1 below) that the efficient level of care increases (decreases) if the marginal benefit of care rises (falls) and that the efficient level of care decreases (increases) if the marginal cost of care rises (falls).
Figure 4.2.1 The efficient level of care