Settlement and the Strict Liability-Negligence Comparison Abraham L. Wickelgren UniversityofTexasatAustinSchoolofLaw Abstract Because injurers typically have better information about their level of care than do victims, the negligence rule increases the degree of asymmetric information between the parties, impeding settlement. Furthermore, the defendant s private information about its level of care creates a mixed-strategy equilibrium under the negligence rule in both a screening and a signaling model of settlement. This paper shows that as a result, expected accident costs are strictly greater under negligence than strict liability unless the injurer is sufficiently judgement-proof. That is, expected accident costs are lower under strict liability if the defendant is only somewhat judgement proof, while negligence has lower expected accident costs if the defendant is very judgement proof. 1 Introduction This paper revisits the comparison between strict liability and negligence rules for unilateral accidents when injurers have limited wealth. The traditional account, starting with Shavell (1986), says that while both strict liability and negligence rules can induce first best levels of care if injurers are not judgement proof, negligence produces superior outcomes if injurers do not have sufficient wealth to pay for the full harm that they cause. If the negligenge standard is fixed at the cost-minimizing level of care, Shavell (1986) shows that the negligence rule can induce first best levels of care as long as the injurer s level of wealth is not too small. If it is, then negligence and strict liability are equivalent. If the due care standard can be adjusted based on the injurer s level of wealth, Ganuza and Gomez (2008) show that negligence can improve on strict liability whenever the injurer is judgement proof (has less wealth than necessary to pay for the full harm it causes). Thus, the general conclusion has been 1
that the negligence rule is more robust to the judgement-proof problem than strict liability, so unless one includes court errors or activity levels in the model, the negligence rule performs better than strict liability. These comparisons of strict liability and negligence, however, have been done under the assumption that all cases go to trial. That is, they have not modeled the settlement game that follows the suit by the victim (plaintiff) against the injurer (defendant). This paper does model the settlement game, and, importantly, assumes that at the time of settlement the injurer s level of care is private information. That is, the basic economic model of behavior is a slight generalization of Spier (1997). 1 She shows that because settlement occurs under asymmetric information under the negligence rule, the injurer must randomize between taking due care and being negligent. If the defendant were never negligent, then the plaintiff would assume this and so the case would always settle at an amount independent of the defendant s actual care level, providing the defendant with no incentive to take care. If the defendant were always negligent, then (assuming the due care standard is not too high) by taking due care and going to trial the defendant would pay only its legal costs and its costs of care, which would be less than the amount the plaintiff would be willing to settle for. While Spier considers the implication of this for optimal damages and litigation subsidies, she does not analyze the effects of this mixed strategy equilibrium on the strict liablity-negligence comaprison with judgement proof injurers. This paper does that and shows that, assuming negligence standards are set minimize accident costs given the injurer s level of wealth, strict liability results in lower expected accident costs unless the injurer is substantially judgement proof. That is, under both the screening and signaling models of settlement, there is a critical level of wealth, strictly below the level necessary for strict liability to induce optimal precaution, so that strict liability results in strictly lower expected accident costs if the injurer s wealth exceeds this critical level while negligence results in strictly lower expected accident costs if the injurer s wealth is less than this critical level. This suggests that if the injurer is likely to have better information about its level of precaution than the victim, that strict liability is preferrable to negligence in a much broader range of circumstances than the traditional account suggests. In addition to comparing strict liability and negligence, the paper also analyzes the properties of the equilibrium under the negligence rule. 1 It is a slight generalization in that I allow for a continuum of care choices rather than a binary choice and because I consider both a screening and a signaling settlement game. The model is also related to Ordover (1978) and Hylton (1990), who do not consider settlement but do consider the incentive to sue under negligence. 2
Interestingly, these properties are nearly identical in the screening and signaling cases. In the both the screening and signaling models, the injurer randomizes between complying with the due care standard (which I refer to as compliance) and taking some lower level of care (being negligent). A negligent injurer, however, still has some incentive to take care because doing so reduces the probability of an accident. Effectively, theinjurertakescareasifhe(iwillrefertotheinjurerasheandthe victim as she) is strictly liable as long as his level of care is less than the due care standard. Howevever, because the victim does not know whether or not the injurer was negligent, the negligent injurer does not always pay for the harm he causes. In the screening game, the plaintiff mixes between making a low settlement offer (equal to the defendant s litigation costs) that the defendant accepts whether he is negligent or not and making a high offer that only the negligent defendant accepts. In the signaling game, there is actually no signaling. The defendant makes the pooling offer of zero and the plaintiff mixes between accepting this zero offer and going to court. Because the negligent injurer might be able to settle at only his litigation costs in the screening game or settle at zero in the signaling game, his level of care when negligent is inefficiently low. This negligent level of care, however, is greater the greater the due care standard and increasing in the defendant s litigation costs. In the screening model it is independent of the level of damages (that the defendant actually would pay), while in the signaling model is decreasing in the level of damages (that the defendant actually would pay). The probability that a case does go to court (given that there was an injury) is decreasing in the damages and both parties litigation costs. It is increasing in the due care standard. While the fact that the probability of a trial is decreasing in litigation costs is not surprising, the fact that it is decreasing in damages is. In a standard litigation game where the extent of asymmetric information is exogenous (as in Bebchuk (1984)), larger damages increase the probability of trial. Here, however, the result is reversed becasue larger damages require the plaintiff to be less aggressive about going to court in order to maintain the defendant/injurer s indifference between taking due care and being negligent. It is also worth noting that a higher due care standard results in more trials (as long as the standard is not so high that the defendant never complies). Thus, in choosingthesociallyoptimalduecarestandard,onemustconsiderthis tradeoff. In the negligence equilibrium, the injurer sets his probability of complying with the due care standard so that, if there is an injury, the plaintiff is indifferent between making a high or low settlement offer (in 3
the screening game) or accepting the zero offer or going to court (in the signaling game). This results in the probability of compliance being increasing the level of damages (that the defendant actually would pay) and decreasing in the litigation costs of both parties. In addition, the probability of compliance is increasing in the due care standard if care costs are quadratic. The reason is that changing the due care standard changes the strength of the signal of the defendant s precaution that an injury provides. On the one hand, if the due care standard is higher, an accident suggests it is less likely the defendant took due care. On the other hand, a higher due care standard also increases the defendant s level of care when he is negligent, which reduces the probability of an accident. Under quadratic costs, it is possible to show that this first effect always outweighs the second. Thus, to keep the posterior probability (after observing an accident) of compliance constant, the prior probability of compliance must increase. Shavell (1980) first compared strict liability and negligence. Following Shavell s (1986) comparison of strict liability and negligence with judgement proof injurers, there have been a number of other papers that have considered the issue in a variety of different settings. Dari- Mattiacci and De Geest (2005) look at how different precaution technologies affect the judgement proof problem. Dari-Mattiacci and De Geest (2006) analyze when judgement proofness might lead to over-precaution rather than under-precaution. Dari-Mattiacci and Mangan (2008) and Stremitzer (2010) analyze the interaction between the judgement proof problem and causation rules. None of these papers have considered the possibility of settlement and asymmetric information. Innes (1999) does not consider settlement but models a situation in injurers have private information about the distribution of damages they may cause, though not about their level of care. Kornhauser and Revesz (1990) examine the effects of judgement proofness on joint and several liability. The plan of the paper is as follows. Section 2 considers the screening model. Section 3 anlayses the signaling model. It is worth noting that much of the analysis in section 3 is quite similar to that of section 2. Section 4 concludes. Most proofs are in the appendix. By ALEA, I will have completed some numerical simulations that will provide a quantitative sense of exactly how judgement-proof an injurer must be before negligence is superior to strict liability but those numerical results are not in this draft. 2 Screening Model There is one potential injurer (he) and one potential victim (she). In period 0, the social planner chooses the legal rule, either the injurer is 4
strictly liable for any injuries he causes who chooses the probability that his actions injure the victim, or the injurer is liable for the victim s injuries if and only if the injurer s level of care is below some given level, (the negligence rule). If the social planner chooses the negligence rule, then it can also choose. In period 1, the injurer chooses his level of care,, whichidefine in terms of the probability that the injurer s action does NOT cause a harm of to the victim (I will call this a safety probability). The cost of choosing a safety probability is ( ); 0 00 0 I assume that these costs are non-monetary (they do not reduce the wealth of the defendant). In period 2, if the victim is hurt, she files suit against the injurer and makes him a take it or leave it settlement offer of. At this time, is the defendant s private information. In period 3, the injurer accepts or rejects the offer. If he accepts, then he pays to the victim. If he rejects, the case goes to court. In court, both sides incur legal costs of for the defendant and plaintiff, respectively. Under strict liability, the victim (plaintiff) winsandthe injurer (defendant) pays to the plaintiff. is the damages the defendant actually pays; they cannot exceed the defendant s wealth after trial, ( is the defendant s wealth prior to trial). That is, the defendant may be partially judgement proof. Under negligence, the court perfectly observes, so the plaintiff wins if and only if 2 If the plaintiff wins, then the defendant pays the plaintiff Under strict liability, it is straight-forward to solve the model by backward induction. Since information is symmetric and the plaintiff makes a take it or leave it offer, she will offer and the defendant will accept a settlement of = + in period 2. Given this, in period 1 the defendant chooses to minimize (1 )( + )+ ( ) The defendant will choose implicitly defined by 0 ( )= + The social planner seeks to minimize total expected social costs, which are the sum of precaution costs, expected injury costs, and expected litigation costs. Under strict liability, since there is complete information and all cases settle, total social costs are simply: =(1 ) + ( ) Let be the that minimizes result. This leads us to the following Lemma 1 Ifthesocialplannercanchooseany, then it 2 We assume that the negligent defendant pays the full harm. Under the de jure rule, this would only be the case if the negligence caused the plaintiff s harm. Stremitzer (2010) shows that, in practice, once a defendant is found negligent courts typically assume that the negligence caused the harm. 5
can obtain the first best safety probability with zero litigation costs under strict liability whenever Proof. If = then the defendant s cost are identical to social cost, hence he will choose =. Since all cases settle under strict liability, there are no litigation costs. This is feasible so long as or Q.E.D. Lemma 1 is just a slight variation of Shavell s (1980) result that strict liability can obtain the first best level of care in the absence of legal costs. Here, there are legal costs, but because there is symmetric information under strict liability, legal costs are never incurred because the case always settles. Because the plaintiff has the opportunity to makeatakeitorleaveitoffer, one must set damages below harm so that the amount of the settlement, damages plus the defendant s legal costs, equal the harm. Under negligence, I know from Spier (1997) that there is no pure strategy equilibrium. If the defendant were always non-negligent, then the plaintiff would offer = and the defendant would accept. In this case, the defendant s optimal level of care is given by 0 ( ) = Unless the due care standard is such that 0 ( ) the defendant would choose to be negligent, making this not an equilibrium. This would imply that the negligence standard was far below the socially optimal level as long as ). Similarly, if the defendant were always negligent, the game would look just like the strict liability game. This would not be an equilibrium if (1 )( + )+ ( ) (1 ) + ( ) (which is always satisfied unless which would imply that the negligence standard was high enough that it was effectively a strict liability regime). Thus, in the negligence regime, I will define implicitly by (1 )( + )+ ( )=(1 ) + ( ) and say that there is a negligence regime if and only if. In the mixed strategy equilibrium, let be the probability the plaintiff offers = +. With probability 1 she offers = Then, I can determine the optimal safety probability given that the defendant is negligent,, as follows: =argmin(1 )( ( + )+(1 ) )+ ( ) is implicitly given by: The plaintiff must then choose between and That is: 0 ( )= + (1) to make the defendant indifferent (1 )( ( + )+(1 ) )+ ( )=(1 ) + ( ) (2) 6
or = ( ) ( ) ( ) (3) (1 ) I know that 1 since The defendant must choose its probability of taking due care,, such that the posterior probability the defendant took due care given that there was an accident,, makes the plaintiff indifferent between offering = + (which only a negligent defendant accepts) and offering = (which both types accept). That is, = ( )+ (1 )( + ) or = ( + + ) 3 I can then determine the prior probability of taking care by noting that a prior of generates a posterior (1 ) (1 ) ( + )(1 )+ (1 ) of = ( + (1 )+(1 )(1 ) + ) Thus, = I now have the following result, which is just a slight generalization of the Spier (1997) result that there must be negligence in equilibrium. Lemma 2 If the optimal safety probability under strict liability is less than one, then the probability that the defendant takes due care under negligence is strictly less than one. Proof. If 1 then 1 which implies that 1 since + 0 If =1 then since 1 the defendant will choose = rather than = 1 so even if = 1 the defendant will be negligent in equilibrium. Q.E.D. Since there is no pure strategy equilibrium and I sometimes have trials, the definition of social costs is slightly different under negligence: ( ) =(1 ){(1 ) + ( )}+ {(1 )( + ( + ))+ ( )} If the defendant is negligent (probability 1 ), then the case will settle so social costs will be expected injry costs ((1 ) ) plus the costs of precaution when negligent ( ( )). If the defendant takes due care (probability ), then expected injury costs are (1 ) Thecasewill not settle if the plaintiff makes the high settlement offer, so expected litigation costs are ( + ) if there is an injury. Precaution costs from taking due care are ( ). 3 This analysis assumes that the plaintiff cannot drop the suit after the settlement offer has been rejected. Accounting for this, as in Nalebuff (1987), turns out to make the screening case very similar to the signaling case. The equilibrium with the smallest amount of negligence has the defendant always rejecting the plaintiff s offer, then the plaintiff mixes between going to court and not. This is exactly what happens in the signaling case. That said, introducing some possibility of court error would often be sufficient to make the threat to go to trial after settlement offer is rejected credible. 7
There are two important features of the negligence social cost function relative to the strict liability one. First, there is a positive probability of legal costs under negligence because the negligence rule creates asymmetric information that impedes settlement. Second, because the negligence rule creates a mixed strategy equilibrium, it makes it impossible to set a negligence standard (less than )thatleadstobehavior that minimizes accident and precaution costs (even if one ignored litigation costs). This is because the social cost function is convex is. So, even though one might be able to set such that the average care minimized expected injury and precaution costs, the actual probability distribution of care levels would lead to higher levels of the sum of expected injury and precaution costs. Before proceeding to comapre strict liablity and negligence, it is interesting to examine the properties of the mixed strategy equilibrium under the negligence regime. The following proposition describes how the probability of compliance, the level of care in the event of negligence, and the probability the plaintiff chooses to make a high settlement offer vary with the level of damages, the litigation costs, and the due care standard. Proposition 1 (A) The defendant s optimal safety probability given that he is negligent is independent of the level of damages and the plaintiff s litigation costs, increasing in his litigation costs, and increasing in the due care standard. (B) The probability that the plaintiff makes a high settlement offer is independent of her legal costs, decreasing in the damages and the defendant s legal costs, and increasing in the due care standard. (C) The probability of a trial givenaninjuryisdecreasinginthedamagesandbothparties litigation costs and, increasing in the due care standard. (D) The probability the defendant complies with the due care standard is increasing in the damages and decreasing the legal costs of both parties. If ( ) = 2 then the probability of compliance is also increasing in the due care standard. Proof. See Appendix. Using from (3) in (1), one can see that the optimal safety probability if the defendant is not taking due care is independent of the level of damages. This is because the probability the defendant makes the high settlement offer decreases as damages increase to keep the incentive to take care (given negligence) constant. This is necessary to keep the defendant indifferent between taking due care and being negligent. The plaintiff s litigation costs do not affect the defendant s incentives to take 8
due care or be negligent, so they do not affect either or Increasing the defendant s litigation costs increases his incentive to avoid an accident, leading to an increase in Because it also makes taking due care more attractive (for the same reason), must decrease to maintain indifference between due care and negligence. Increasing the due care standard makes being negligent more attractive, so must increase to compensate. This increase in leads to an increase in. If there is an injury, there is a trial whenever the defendant was non-negligent and the plaintiff makes the high settlement offer. This is simply ( + + ) Using part (B), then the results in part (C) follow from this condition. For part (D), recall that in the mixed strategy equilibrium, the probability that the defendant complies with the due care standard is set to ensure that the plaintiff is indifferent between making the low settlement offer and settling with probability one and making the high settlement offer and only settling with a negligent defendant. Thus, when the damages increase or legal costs decrease the relative benefit ofmakingthe high offer increases, meaning that the probability of compliance must increase to reduce the attractiveness of the high offer. If the due care standard increases, this changes the strength of the signal that the fact of an injury provides about whether the defendant was negligent or not. The higher due care standard suggests it is less likely that the defendant complied with the standard given that there was an injury. On the other hand, because the defendant s optimal safety probability increases when he does not take due care with the increased due care standard (from part (A)), it is less likely that there would be an injury given that he was negligent. As long as this latter indirect effect is not too strong, which is always the case with quadratic costs, the increased due care standard will lead to an increase in the ex ante probability of compliance to keep the posterior probability of compliance given an injury constant. One can now move on to the comparison of strict liability and negligence. The necessity of a mixed strategy equilibrium under negligence means that, even if one ignores litigation costs, strict liability produces lower expected accident costs (defined as the sum of expected injury and precaution costs) than any negligence standard even if the injurer s wealth is somewhat less than. Negligence produces lower accident costs only if the injurer is substantially judgement proof. The following proposition makes this claim more precise. Before stating it, I provide the following definition. Definition Let total expected accident costs, the sum of expected injury costs and precaution costs, be defined by the function ( ) = (1 ) + ( ) under strict liability. Let ( ) = ((1 ) + 9
( )) + (1 )((1 ) + ( )) define total expected accident costs under the negligence standard There are two different functions for accident costs because under strict liability the defendant plays a pure strategy equilibrium, while under negligence the defendant plays a mixed strategy equilibrium. The negligence accident cost function is only a function of the negligence standard because (1) and (3) show that is determined by and = (1 ) ( + )(1 )+ (1 ) Proposition 2 Let be defined implicitly by ( ( )) = ( ) such that ( ) If 000 0 and at =( ) 3 there exists a negligence standard with lower expected accident costs than strict liability, then there exists a ˆ [ + ) such that expected accident costs are lower under strict liability than under any negligence standard if and only if ˆ Proof. See Appendix Proposition 2 says that the traditional view that negligence is superior to strict liability in inducing judgement proof injurers to exercise optimal or second best optimal precaution is not robust to settlement underasymmetricinformation. Whileitisstillthecasethatlowerwealth levels tend to favor negligence over strict liability, when one takes into account settlement, strict liability generates strictly lower accident costs than any negligence standard even if the injurer s wealth is strictly less than the harm to the victim. Negligence can only produce a distribution of care levels with lower expected accident costs if the injurers wealth is less than some critical value that is strictly less than the harm the victim suffers. This critical level of wealth is at least as big as the level of wealth for which the largest precaution the negligence rule can induce (which is above the level that minimizes accident costs) with positive probability produces the same expected accident costs as is produced by strict liability with this level of wealth. At this level of wealth and with this negligence standard, the injurer mixes between taking excessive care and taking insufficient care (which is the same as he would take under strict liability), both of which produce the same level of expected accident costs. If this negligence standard minimizes expected accident costs, then this is the critical level of wealth. If it does not minimize expected accident costs, then the critical level of wealth is greater than this because at this wealth level there is a lower negligence standard that produces accident costs that are lower than they are under this maximum standard which has accident costs equal to those of strict liability. Notice that while this lower negligence standard will reduce expected 10
accident costs when complied with, it will be complied with less (, the probability of compliance, is increasing in ) and expected accident costs when the standard is not complied with are greater ( is increasing in ). It is also worth noting that Proposition 2 compares strict liability and negligence on the accident cost dimension, not the total social cost dimension. That is, it does not consider legal costs. If one were to consider legal costs, than in this model strict liability would be preferrable to negligence at even lower levels of wealth. The reason is that under negligence, if the defendant is non-negligent and the plaintiff makes the high settlement offer, the defendant rejects the offer and the case goes to trial, resulting in legal costs of + Under strict liability, however, because there is no asymmetric information, every case settles. So, there are never any legal costs. I do not provide an analytic comparison of total social costs under strict liability and negligence, however, because it is much more difficult toprovideanalyticconditionsfortheduecarestandardthatminimizes total social costs than for the due care standard that minimizes accident costs. Section 4, however, provides a numerical analysis that compares strict liability and negligence on the total social cost dimension. 3 Signaling Model The signaling model is identical to the screening model except that in period 2, if the victim is injured, she files suit but now the injurer (defendant) makes a take it or leave it settlement offer to the victim (plaintiff). Under strict liability, there still is symmetric information, so now the defendant will offer and the plaintiff will accept a settlement of = in period 2. Given this, in period 1 the defendant chooses to minimize (1 )( )+ ( ) The defendant will choose q implicitly defined by 0 (q ) = (underscored variables and functions will refer to variables in the signaling model where they are different from the screening model). The social planner seeks to minimize total expected social costs, which are the sum of precaution costs, expected injury costs, and expected litigation costs. Under strict liability, since there is complete information and all cases settle, total social costs are simply: =(1 ) + S C q (q ) Let be the that minimizes S C result. This leads us to the following Lemma 3 Ifthesocialplannercanchooseany, then it 11
can obtain the first best safety probability with zero litigation costs under strict liability whenever + + Proof. If = + then the defendant s cost are identical to social cost, hence he will choose q =. Since all cases settle under strict liability, there are no litigation costs. This is feasible so long as + or + + Q.E.D. Lemma3differs from Lemma 1 in that the defendant s level of wealth must be greater to achieve the first best when the defendant makes a take it or leave it offer because the defendant captures the settlement surplus of + in this case. Also, notice that in this case damages must exceed harm to counteract the fact that settlement leads to the defendant to pay less than the damages it would owe in court. Under negligence, there is also no pure strategy equilibrium in a signaling model. If the defendant were always non-negligent, then the plaintiff would accept =0(or not file suit). In this case, the defendant would choose to be negligent, making this not an equilibrium. Similarly, if the defendant were always negligent, the game would look just like the strict liability game. This would not be an equilibrium if (1 q )( )+ (q ) (1 ) + ( ) (which is always satisfied unless q which would imply that the negligence standard was high enough that it was effectively a strict liability regime). Thus, in the negligence regime in the signaling model, I will define implicitly by (1 q )( )+ (q ) =(1 ) + ( ) and say that there is a negligence regime if and only if. In the signaling model, let be the probability the defendant took due care given that there was an accident. First, note that there cannot be any completely separating equilibrium. If there were, then after receiving a low offer signaling that the defendant was non-negligent, the plaintiff would not want to sue. Thus, a negligent defendant would want to deviate and also make this low offer. Now, consider a partially separating equilibrium in which the non-negligent defendant offers zero and the negligent defendant mixes between offeringzeroand For a negligent defendant to be indifferent between these two offers, the plaintiff, after receiving an offer of zero must reject and go to court with probability + To ensure the plaintiff is indifferent about going to court after receiving an offer of zero, the posterior probability that the defendant is non-negligent given an offer of zero must be So, if the probability the defendant took due care given that there was an accident is then a negligent defendant must offer zero with probability Notice that this means that ( andintheequilibrium )(1 ) withthemostcompliancewiththeduecarestandard, =,thereis 12
a complete pooling equilibrium in which negligent defendants offer zero with probability one. From here one, I will focus on this maximum-compliance pooling equilibrium in which there is no signaling at the settlement stage. In this case, the plaintiff must still be indifferent between filing suit and not so that the injurer will mix between being negligent and not. So, the posterior probability that the defendant took due care still must be r = This implies that the prior probability of compliance is ( p = )(1 q ) ( Once again, the higher the due care standard, )(1 q )+ (1 ) the higher the probability of compliance. If the probability that the plaintiff rejects a settlement offer of zero and goes to court is then the optimal safety probability given that the defendant is negligent in the signaling model, q, is: q is implicitly given by: q =argmin (1 ) ( + )+ ( ) 0 (q )= ( + ) (4) The plaintiff must then choose to make the defendant indifferent between q and That is: (1 q ) ( + )+ (q )=(1 ) + ( ) (5) or = ( ) (q ) (1 q )+ ( q ) + (6) I can now show that Spier s (1997) result holds in signaling model as well as in a screening model. Lemma 4 If the optimal safety probability under strict liability is less than one, then the probability that the defendant takes due care under negligence is strictly less than one. Proof. If 1 then 1 which implies that p 1 since 0 If =1 then since q 1 the defendant will choose =q rather than =1 so even if =1the defendant will be negligent in equilibrium. Q.E.D. I now proceed to describe the comparative statics under the negligenceruleinthesignalingmodel. 13
Proposition 3 (A) The defendant s optimal safety probability given that he is negligent is independent of the plaintiff s litigation costs, decreasing in the level of damages and increasing in his litigation costs and the due care standard. (B) The probability of a trial given an injury is independent of her legal costs, decreasing in the damages and the defendant s legal costs, and increasing in the due care standard. (C) The probability the defendant complies with the due care standard is increasing in the damages and decreasing the legal costs of both parties. If ( ) = 2 then the probability of compliance is also increasing in the due care standard. Proof. See Appendix. The comparative statics under the signaling model are nearly identical to what they are under the screening model. The main differences are, first, that higher damages reduce the safety probability given negligence in the signaling model while they have no effect in the screening model. This is because increases in damages increase the relative advantage of compliance with the due care standard, requiring a reduction in the plaintiff s probability of going to court (which is equivalent to the probability of a trial given an injury) to compensate. Unlike in the screening model, however, this reduction in the probability of trial reduces the expected cost of taking due care as well, thus to maintain indifference, the expected cost of being negligent must fall, which reduces the optimal level of care in negligence. Second, in the signaling model the probability of trial given injury depends only on the plaintiff s probability of going to court, it does not depend on the probability of negligence. Thus, parts (B) and (C) in Proposition 1 can be combined into part (B) in Proposition 3. Otherwise, the comparative statics are thesameinthetwomodels. The definition of social costs under negligence in the signaling model is: S C =(1 p ){(1 q )( + ( + ))+ (q )}+p {(1 )( + ( + ))+ ( )} In the signaling model, because there is a pooling equilibrium, the probability of litigation depends only on the probability of an injury. It does not depend separately on whether the defendant was negligent or not. Once again, notice that while strict liability does not create asymmetric information, there is settlement with probability one under strict liability while there is a positvie probability of legal costs under any negligence rule. Also as in the screening model, the mixed strategy equilibrium under the negligence rule again makes it impossible to set a negligence standard (less than ) that leads to behavior that minimizes accident and precaution costs (even if one ignored litigation costs). 14
These similarities between the two models means that the accident cost comparison between strict liability and negligence is very similar in the signaling model as it was in the screening model. Proposition 4 Let be defined implicitly by ( ( )) = (q ) such that ( ) q If 000 0 and at = 3+ there exists a negligence standard with lower expected accident costs than strict liability, then there exists a ˆ [ + + + ) such that expected accident costs are lower under strict liability than under any negligence standard if and only if ˆ Proof. See Appendix Proposition 4 is nearly identical in form to Proposition 2. The differences are that the maximal due care standard is a slightly different function of damages than in the screening model and the safety probability under strict liability is different than in the screening model (for any given level of damages). In addition, the condition on damages for which the negligence standard has lower expected accident costs is different reflectingthefactthatthedefendant hasthebargainingpower in the signaling model while the plaintiff has it in the screening model. Lastly, the upper bound for the cutoff wealth level is higher in the signaling model reflecting the fact that damages must be higher to get the first best under strict liability in the signaling model due to the defendant having all the bargaining power in the signaling model. That said, the importance of Proposition 4 is that it reinforces the result in Proposition 2 that if the defendant has private information about his level of care, then strict liability has advantages over negligence given the availability of settlement. This general result holds regardless of which party makes the settlement offer. The details of exactly how judgement-proof the defendant must be in order for the negligence rule to result in lower accident costs than the strict liability rule, however, are different in the two models. 4 Numerical Analysis Propositions 2 and 4 demonstrate that strict liability can result in strictly lower accident costs than negligence if the defendant s wealth level is not too small. These results, however, do not provide any sense of where the cutoff level of wealth lies in each of the two models. Similarly, as mentioned above, the analytic results only examine the difference in accident costs generated by the optimal negligence standard and the optimal strict liability standard. They do not examine the difference in total social costs expected accident costs plus expected litigation costs 15
under the optimal strict liability and negligence rules. In this section, I report the results of numerical simulations of both the screening and signaling models to address these issues. In the numerical analysis, I used a simple quadratic costs function, ( ) = 2 Thus, = + in the screening model and 2 q = 2 in the signaling model. Similarly, I can obtain explict solutions for and (the defendant s care level when negligent and the probability the plaintiff makes the high settlement offer) in the screening model by solving (1) and (3) with this quadratic cost function. I can also obtain explict solutions for q and (the defendant s care level when negligent and the probability the plaintiff goes to court) by solving (4) and (5). (The algebraic details for the simulations are in the appendix.) Once I have explict solutions for these variables, I can obtain expressions for total accident costs and total social costs under strict liability and negligence in both the screening and signaling models that depend only on and. Unfortunately, these expressions (under the negligence rule) are sufficiently complicated that one cannot obtain analytic solutions for the due care standard that minimizes either accident costs or social costs. Instead, I proceed by simulation. Without loss of generality, I normalize =1(all parameters are then fractions of the harm from an accident). I then randomly draw values for the defendant s wealth, the legal costs of the plaintiff and defendant, and the effectiveness of care (the parameter). Using these values (and = { 1}), I then have Mathematica find the due care standard that numerically minimizes the expressions for accident and social costs (and the associated minimized costs) under the negligence rule and compare this to the costs under strict liability. I made 10,000 parameter draws for both the screening and signaling models. For the screening model, I drew wealth from a uniform distribution between zero and one (so the defendant s wealth was always less than the harm). I drew from a uniform distribution between 1 2 and 3 2 The minimum of 1 2 was chosen to ensure that in the first best there would be an accident with positive probability. The maximum of 3 2 guarantees that the first best probability of an accident is less than 2 3. For both the defendant s and the plaintiff s legal costs, I drew them from a uniform distribution between zero and half of the defendant s wealth. This means that legal costs are less in cases in which the defendant s wealth is less. I chose this both to ensure that damages for negligence was at least half of the defendant s wealth and because both sides should rationally spend less on legal costs when there is less at stake because the defendant has less wealth to pay damages. 16
For the signaling model, I drew wealth from a uniform distribution between 0 3 and 1 3 In the signaling model, the defendant can settle at rather than at (as in the screening model). This means that under-deterrence is a bigger problem under strict liability for judgement proof defendants. As a result, I found that negligence always resulted in lower costs than strict liability for very low levels of wealth. In addition, because there could be under-deterrence under strict liabilty even if the defendant s wealth exceeds the harm, it was necessary to examine defendant wealth levels above one. I, again, drew from a uniform distribution between 1 2 and 3 2 For both the defendant s and the plaintiff s legal costs, I drew them from a uniform distribution between zero and one-third of the defendant s wealth. I had to choose a lower level of legal costs relative to wealth in the signaling model because every defendant can settle at To get an equilibrium in which negligence is different from strict liablity, this amount must be less than the costs of a non-negligent defendant proving its non-negligence. If the plaintiff and defendant s legal costs are equal, this means that the defendant s wealth must exceed three times each party s legal costs, hence the upper limit on legal costs in the simulation. 4.1 Screening Model Results To get a sense of how judgement proof the defendant must be for negligence to be preferrable to strict liability, I computed the fraction of cases in the simulations for which the optimal negligence regime generated lower levels of accident costs and social costs than strict liability for various regions of defendant wealth. When I refer to the optimal negligence regime, this means that I have chosen the due care standard that minimizes the measure of cost that I am comparing. That is, when comparing accident costs, for example, I find the due care standard that minimizes accident costs under the negligence rule. This will not be the same due care standard that would minimize social costs (accident costs plus litigation costs) for the same set of parameters. In general, because higher due care standards generate more litigation, the optimal due care standard for minizing social costs for any given set of parameter values is lower than the optimal due care standard for minimizing accident costs. The following figure displays the results of the simulation in the screening model in graphical form. Figure 1 shows that, not surprisingly, the fraction of cases in which negligence produces lower costs than strict liability is decreasing in wealth. 4 4 For the social cost curve, there is one place where the relationship is not monotonic, but that is almost certainly an artifact of the fact that there were only 500 draws in the wealth regions around this point. 17
Negligence v. Strict Liability bywealth--screening Model Fraction Negligence Costs Lower 1.0 0.8 Accident Costs 0.6 Social Costs 0.4 0.2 0.2 0.4 0.6 0.8 Wealth Figure 1: Figure 1 Also not surprising is that negligence is more likely to result in lower accident costs than social costs for any level of wealth. Again, because there is perfect information with strict liability, there are no legal costs in the strict liability regime (all cases settle). Since negligence generates legal costs, it can generate greater social costs even if it reduces accident costs relative to strict liability. For accident costs, we see that once the defendant s wealth reaches about 70 percent of the harm he causes, strict liability has lower accident costs than negligence more than 50 percent of the time. For social costs, this 50 percent threshold is reached at a wealth level of about 45 percent of the harm. It is important to remember, however, that these percentages are not drawn from any empirically-validated distribution of the parameters. Rather, they are dependent on the distributional assumptions in the simulations regarding legal costs and the effectiveness of care. Thus, the percentages are given more to illustrate that strict liablity can generate lower costs than negligence even if the defendant s wealth is substantially below the harm caused. In order to get a more (though, certainly not entirely) distributionfree look at how the various parameters affect the relative costs under negligence and strict liability, I used the data generated by the simula- 18
tions to perform a regression of the difference in costs on the parameters of the model. A simple linear regression of the form = 0 + 1 + 2 + 3 + 4 + had an R-squared of almost 82 percent when using accident costs for the dependent variable, so while interaction terms and quadratic terms were significant when included in other models, I will discuss the simple regression model because its results are easiest to interpret and it does a very good job of explaining the simulation results (including all interactions and quadratics only increased the R-squared to almost 87 percent). The regression equation for accident costs is: = 0 18 + 0 18 +0 04 +0 12 +0 06 The largest p-value among for these coefficients (for the coefficient on )islessthan2 10 47 This indicates that increasing the damages the defendant will pay (which is equivalent to increasing his wealth for constant legal costs), increasing his legal costs, increasing the plaintiff s legal costs, and increasing the cost of precautions all tend to make strict liability more desirable than negligence for reducing accident costs. Notice that the positive coefficient on suggests that the more hazadarous the activity the more desirable is strict liability for reducing accident costs, which is roughly consistent with the legal doctrine that makes defendant s strictly liable for ultra-hazardous activities. I performed a similar regression analysis with social costs as the dependent variable. Once again, the simple model had an R-squared of about 80 percent, while a model that included all interaction terms and quadratic terms had an R-squared of about 90 percent. In the interest of ease of interpretation, I report the results of the simple model: = 0 15 + 0 21 +0 13 +0 10 +0 06 The results are quite similar to the results for accident costs. The largest p-value among for these coefficients (for the coefficient on ) is less than 10 325 Thequalitativeeffects of all the parameters are the same in the social cost regression as in the accident cost regression. Legal costs matter more since they directly affect the social cost of the negligence rule (which has trial with positive probability). The constant term is smaller, which reflects the fact that the strict liablity-negligence comparison is more favorable to strict liablity for social costs than accident costs. 4.2 Signaling Model Results In the signaling model, the negligence rule generated lower costs than strict liability a much larger fraction of the time for any given level of 19
Negligence v. Strict Liability by Wealth--Signaling Model Fraction Negligence Costs Lower 1.0 0.8 0.6 0.4 Accident Costs Social Costs 0.2 0.4 0.6 0.8 1.0 1.2 Wealth Figure 2: Figure 2 wealth than in the screening model. This is mostly due to the fact that the signaling model gives all the bargaining power in settlement to the defendant, while in the screening model the plaintiff had the bargaining power. As a result, under strict liability, in the signaling model the judgement-proof defendant does not lose all his wealth in the settlement, rather he settles for his wealth minus the legal costs of both parties. This reduces the deterrent effect of strict liability. The transfer of bargaining power does not weaken deterrence as much under negligence because taking due care eliminates liability entirely (except for the legal costs of having to prove the one took due care if the plaintiff does not drop the case), providing a much stronger incentive to meet the due care standard. Thus, as one can see in figure 2, even at the highest level of wealth, the negligence results in lower accident costs than strict liability just over half the time. Strict liability results in lower expect social costs more than half the time when the defendant s wealth exceeds 75 percent of the harm from the accident. Of course, the exact percentages in this figure depend on the distributional assumptions made in the simulations. As in the screening model, I again used the data generated by the simulations to perform a regression of the difference in costs on the parameters of the model. A simple linear regression of the form = 0 + 1 + 2 + 3 + 4 + had an R-squared of over 87 percent when using accident costs for the dependent variable in the signaling model, so while interaction terms 20
and quadratic terms were significant when included in other models, I once again focus on the simple regression model because its results are easiest to interpret. The regression equation for accident costs in the signaling model is: = 0 22 0 10 0 06 +0 16 +0 06 The largest p-value among for these coefficients (for the coefficient on ) is less than 10 181 One difference in the regression models between the screening and the signaling case is that in the signaling model I used wealth instead of damages as an independent variable. I did this because, unlike in the screening model data, damages are not simply wealth minus the defendant s legal costs. Since the data includes wealth that exceeds harm, sometimes the defendant s damages are capped at harm. Thus, to capture the effect of wealth, I used it directly as an independent variable. There are two main differences between the regression results in the signaling and screening models. First, the constant term and the other coefficients clearly indicate that (as is consistent with figure 2), negligence fares better in the signaling model for any given set of parameters. Second, for the accident cost regression, the sign on the legal cost variables is reversed. This is due to the fact that larger legal costs in the signaling model reduce the amount the defendant pays in a settlement under strict liability. As a result, larger legal costs weaken the ability of strict liability to deter in the signaling model. I performed a similar regression analysis with social costs as the dependent variable. Once again, the simple model had an R-squared of over 87 percent. In the interest of ease of interpretation, I report the resultsofthismodel: = 0 18 + 0 05 +0 11 +0 11 +0 07 The largest p-value among for these coefficients (for the coefficient on ) is less than 10 114 The main difference from the accident cost regression is that the legal cost coefficients are now positive. This is because larger legal costs increase the social costs of the negligence rule since cases go to trial under the negligence rule but not under the strict liability rule. This effectturnsouttobemoreimportantthan theweakened deterrent effect. The constant term is smaller, which reflects the fact that the strict liablity-negligence comparison is more favorable to strict liablity for social costs than accident costs. 21