A negligence rule

Now consider the following negligence rule: the court sets due standards of care, z1 and z2 for firms 1 and 2, and applies these due standards as follows:

  • • If a firm meets its due standard of care, it is not liable.
  • • If either firm is the only firm which does not meet its due standard of care, it will be held liable for the total amount of the victim's losses.
  • • If both firms do not meet their due standards of care, they will both be held liable for some pre-specified portion of the victim's losses, with firm 1 paying a fraction s1 of the losses, and injurer 2 paying a fraction s2 of losses, with s1 + s2 = 1.

Assuming that firms cannot collude among themselves, does this negligence rule induce either of the firms to take an efficient level of care? Assume that z1 = x* and z2 = x* are the due standards of care. First, note that no firm will choose a level of care that exceeds their due standard, since this simply adds extra costs of care without any offsetting benefit. Thus, the firms have only two classes of strategies: either meet the due standard (i.e. xt = x*) or not meet the due standard (i.e xi < x*). The individual payoffs from each strategy for each player depend on what the other player does. We can represent the players expected costs from these two classes of strategies in Table 10.2.1.

Let us find the Nash equilibria of this game (there may be more than one). Consider each of the cells in Table 10.2.1.

Table 10.2.

1 Costs under a negligence rule

Firm 2

Meet

Don't Meet

Firm 1

Meet

(wx*, w 2 x*)

[ wrf, W2 X2 +p( x*, X2)h]

Don't

Meet

[ wx +p( xv x*)h, w2 x*]

[ wx + sp( xv x2)h, w W2 x2 + S2P( xv x2)h ]

• {Meet, Meet} is a Nash equilibrium.

To show this, we just need to show that {Meet} is a best response to {Meet} for each player. So, consider player 1, and suppose player 2 chooses {Meet}. Under the negligence rule, firm 2 does not have to compensate the victim, and we are effectively back in a unilateral care model, with p(x1)h = p(x1, x*)h and x* fixed. Firm 1's cost function is:

Since x* minimises w1x1 + p( x1, x*)h, setting the due standard at x* creates a jump in 1's cost function at exactly the right place, and his costs are minimised by choosing x1 = x*.

Thus, x1* is a best response to x2*. Similarly, it is straightforward to show using the same kind of reasoning, that x2* is a best response to x2*. Thus, {Meet, Meet} is a Nash equilibrium.

• {Meet, Don't Meet} and {Don't Meet, Meet} are not Nash equilibria.

Suppose they were. Then this would imply, for example, that {Don't Meet} would be a best response to {Meet}. But this contradicts the result we just established - that {Meet} was the unique best response to {Meet}. So {Meet, Don't Meet} and {Don't Meet, Meet} cannot be Nash equilibria.

• {Don't Meet, Don't Meet} is not a Nash equilibrium.

If {Don't Meet, Don't Meet} is a Nash equilibrium, this would mean that {Don't Meet} would be a best response to {Don't Meet} for each player. This would require:

for firm 1 and:

for firm 2. Adding these two conditions yields:

So if {Don't Meet, Don't Meet} is a Nash equilibrium, the condition in (10.4) must hold. Since s1 + s2 = 1, the condition is equivalent to the existence of an x1 and an x2, with x1 Ф x* and x2 Ф x*, and with:

But w1x[1] + w2x[1] < w1x[1] + w2x[1] + p(x[1],x[1])h, so if equation (10.5) were to hold, it must also be true that if {Don't Meet, Don't Meet} is a Nash equilibrium, there must exist x1 and x2, with x1 Ф x[1] and x2 Ф x[1], such that

But this cannot be true, since (x[1], x[1]) minimises w1x1 + w2 x2 + p (x1, x2) h. Thus, {Don't Meet, Don't Meet} cannot possibly be a Nash equilibrium.

We can also say something about the incentives for mergers and the likelihood of mergers under this legal rule. Since this legal rule induces efficient outcomes, there would be no gains to the firms if they merged. Together with the conclusions regarding strict liability, we can conclude that mergers should be more likely under a strict liability rule than in the presence of a negligence rule.

  • [1] A victim (v), who is harmed by an employee of the company if anaccident occurs, but who cannot himself take care.
  • [2] A victim (v), who is harmed by an employee of the company if anaccident occurs, but who cannot himself take care.
  • [3] A victim (v), who is harmed by an employee of the company if anaccident occurs, but who cannot himself take care.
  • [4] A victim (v), who is harmed by an employee of the company if anaccident occurs, but who cannot himself take care.
  • [5] A victim (v), who is harmed by an employee of the company if anaccident occurs, but who cannot himself take care.
  • [6] A victim (v), who is harmed by an employee of the company if anaccident occurs, but who cannot himself take care.
  • [7] A victim (v), who is harmed by an employee of the company if anaccident occurs, but who cannot himself take care.
  • [8] A victim (v), who is harmed by an employee of the company if anaccident occurs, but who cannot himself take care.
  • [9] A victim (v), who is harmed by an employee of the company if anaccident occurs, but who cannot himself take care.
  • [10] A victim (v), who is harmed by an employee of the company if anaccident occurs, but who cannot himself take care.
 
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