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 xt = x* to minimise his costs.

Similarly, if the injurer is taking care of x*, the victim now faces costs of wvxv + H (x*, xv). These costs are equal to total social costs less wjx*.

Thus, the victim total social costs, subject to the constraint that xt = x*. Diagrammatically, the victim must choose the point on the line xt = x* with the lowest total social costs. But this is just the point xv = x*. That is, the victim solves:

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.

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