In the case of linear demand curves, welfare analysis in the Cournot model is straightforward, and simply requires computation of the sum of the consumer surplus and producer profits. To see how the efficiency of different rules can be compared, suppose that consumers underperceive risk. Let us compare a rule of strict liability and a negligence rule. If we take the linear demand function u'(Q) = P = a - Q and plug u"(Q) = -1 into equations (6.9), (6.11) and (6.14), we get:
where Q* is the efficient quantity. Since firms choose the efficient level of care under each legal rule, the welfare loss under each legal rule is simply the usual deadweight loss triangle of Cournot oligopoly, of Cournot oligopoly as shown in Figure 6.5.1. The first thing to notice is that the negligence rule may actually be efficient here, despite firms having market power and despite consumers not being able to accurately perceive risk. This happens if QNR = Q*, which is true if:
where HHI is the Herfindahl-Hirschman concentration index, which is defined as HHI = ^ s,2, and in this case is simply equal to:
The HHI is often used a measure of the degree of market power in the industry (although, as we show in Chapter 10, there are several problems with this interpretation). When markets are imperfectly competitive, there are two opposing welfare effects of a negligence rule when consumers underperceive risk. The first is the standard welfare loss from Cournot oligopoly, which is the output-reducing effect and is more severe, the fewer firms there are in the industry. The second is the outputenhancing effect, which in the absence of market power would result in an inefficiently high level of output and depends on the extent to which consumers underperceive risk. In equation (6.18), these two opposing (and otherwise welfare-reducing effects) offset each other exactly.
Another way to understand the effects of a negligence rule is to investigate what happens to welfare when there is a slight change in consumer misperceptions. Welfare under a negligence rule is (assuming a linear demand curve):
Under a negligence rule, the change in welfare when l changes by a small amount is therefore equal to:
Since < 0, this is positive as long as: dX
This expression has a very intuitive economic explanation. From equation (6.17), we know under a negligence rule that if l rises, QNR falls. Equation (6.20) simply says that this reduction in quantity will be welfare improving if a - QNR (which is the marginal consumption benefit of the last unit consumed), is less than the marginal social cost of that unit, which is c + H(x*) + wx*. Thus, a rise in welfare as a result of l rising can only occur if the quantity that is produced under a negligence rule exceeds the efficient quantity.
More generally, the total welfare loss under a negligence rule can be lower or higher than the welfare loss under a strict liability rule. The triangle deadweight loss from a rule of strict liability is:
whereas the triangle deadweight loss from a negligence rule is:
A strict liability rule is preferable if:
As the number of firms grows large, output increases and the left- hand side of this inequality grows smaller. Thus as the industry becomes less concentrated the inequality in (6.23) is more likely to be satisfied. Intuitively, when (6.23) holds the output-reducing effect is dominated by the output-enhancing effect, and a rule of strict liability is efficient. Moreover, as market concentration falls, the welfare loss from a strict liability rule falls, and the welfare loss from a negligence rule rises as output grows larger and exceeds the efficient quantity by an ever greater.
On the other hand, as market power increases and the number of firms declines, it is less likely that the inequality in (6.23) holds, and so more likely that a negligence rule becomes efficient. As market concentration rises, the welfare loss from underperception of risk and overconsumption becomes dominated by the output reducing effect under the negligence rule.