# A negligence rule

Let us now consider a negligence rule. Recall that in the unilateral care situation, we modelled a negligence rule as a choice by the court of z, the level of care of the injurer. We will not consider all possible negligence rules here. We will only consider a negligence rule which sets the due standard at *z =* x*, the efficient level of care. Suppose that this is the rule. We will show that the efficient levels of care (x*, x*) are a Nash equilibrium. In other words, under a negligence rule, given that the victim chooses x*, the injurer's best response is to choose x*. And, given that the injurer chooses x*, the victim's best response is to choose x*.

To see this, suppose that the victim is choosing a level of care equal to x*. Under a negligence rule where *z =* x*, the injurer's expected costs are:

Since *w _{{}x* < w_{{}x* + H*(

*x*,x*), the injurer chooses

_{{}*x*to minimise his costs.

_{t}= x*Similarly, if the injurer is taking care of x*, the victim now faces costs of *w _{v}x_{v} + H* (

*x*, x*

_{v}). These costs are equal to total social costs less w

_{jx}*.

Thus, the victim total social costs, subject to the constraint that *x _{t} =* x*. Diagrammatically, the victim must choose the point on the line

*x*with the lowest total social costs. But this is just the point

_{t}= x**x*x*. That is, the victim solves:

_{v}=

The first-order condition is:

But this is exactly the same first-order condition that we had at the efficient level of care. Thus, the victim also behaves efficiently. Therefore in the bilateral care model, a negligence rule in which the due standard for the injurer is set at *z =* x*, is efficient.