Consider a rule of no liability: the injurer is never obligated to pay damages to the victim. How much care will each party take? Let us consider the injurer's choice of care. As before, the goal of the injurer is to minimise his expected cost of care. With no liability, the injurer's problem is to solve:

Just as we had in the unilateral care model, the solution to this problem is:

That is, under a rule of no liability, the injurer takes no care at all. Given this choice, what is the victim's best choice of care? The goal of the victim is to minimise his expected cost of care. Under a rule of no liability (and given the result that we derived above) the victim's problem is:

The solution, which we denote by x^{NL} must obey the first-order condition:

or:

These choices constitute a Nash equilibrium: given the choice of the other party, neither party wishes to change their behaviour. The question is: how do these levels of care compare to the efficient levels? Recall that at the efficient level of care, we have x* > 0. Since we assumed that x_{i }and x_{v} were substitutes, and since x^{NL} = 0 < x* , this means that with a rule of no liability the victim must be taking more care than the efficient level. That is, xN^{L} > x* . In the bilateral care model, a rule of no liability is inefficient: no care is taken by the injurer, and too much care is taken by the victim.

The result is best seen in Figure 5.3.1. Instead of expected social cost now being a curve, it is shaped like a cup. Imagine that you are looking down on the cup. Each curve in this diagram represents a different level of aggregate expected costs. These costs decrease as we move towards (x*, x*) (the bottom of the cup), and are minimised at the point (x*, x*), the efficient level of care.

With a rule of no liability, the injurer takes a level of care that is equal to zero. For the victim, the point x* no longer minimises his expected costs. The victim effectively now faces the total social costs, and must choose the level of care that minimises total social costs, subject to the constraint that x_{i} = 0. This means the victim chooses the point on the vertical axis for which total social costs are minimised. This occurs at the point xN^{L} > x* in Figure 5.3.1.