The Design of Liability Rules for Highly Risky Activities - Is Strict Liability the Better Solution?

Transcription

1 Marti Nell, Adreas Richter The Desig of Liability Rules for Highly Risky Activities - s Strict Liability the Better Solutio? Workig Papers o Risk ad surace Hamburg Uiversity No Jue 200 Tor zur Welt der Wisseschaft

2 Marti Nell, Adreas Richter 2 The Desig of Liability Rules for Highly Risky Activities s Strict Liability the Better Solutio? No 0 Jue 200 SSN Hamburg Uiversity, stitute of Risk ad surace, Vo-Melle-Park 5, D-2046 Hamburg, Germay, phoe: , fax: , marti.ell@rrz.ui-hamburg.de. Hamburg Uiversity, stitute of Risk ad surace, Vo-Melle-Park 5, D-2046 Hamburg, Germay, phoe: , fax: , richter@rrz.ui-hamburg.de.

3 The Desig of Liability Rules for Highly Risky Activities - s Strict Liability the Better Solutio? Abstract Strict liability is widely see as the most suitable way to gover highly risky activities, such as evirometally dagerous productio or geetic egieerig. The reaso which is usually give for applyig strict liability to these areas, is that ot oly efficiet care is supposed to be iduced but also a efficiet level of the risky activity itself. t is argued that, i case of o market relatioship betwee ijurers ad victims, this could oly be achieved through strict liability but ot via the egligece rule. this paper we show that the superiority of strict liability does o loger persist i a world of risk averse parties. Our results suggest that i terms of risk allocatio the egligece rule should be preferred for abormally risky activities, if isurace markets are imperfect. The reaso is that highly risky activities typically affect a large umber of idividuals, such that strict liability implies a quite ufavorable allocatio of risk. Therefore the egligece rule turs out to be superior, if a market relatioship betwee the parties exists, sice it icurs less cost of risk. f there is o market relatioship betwee ijurer ad victims, o clear result ca be derived. The paper cocludes with some remarks o the usefuless of upper bouds to a ijurer s liability as well as regulatios that exclude liability for uforeseeable losses. We argue that this kid of supplemet to a strict liability rule ca improve efficiecy. JEL-Classificatio: G22, K3

4 . troductio Negligece is the fudametal liability priciple i may coutries, for example i the Uited States ad Germay. Apart from this, typically strict liability is applied to certai highly risky activities. Germay some areas like evirometal liability, product liability or the liability for risks related to geetic egieerig are ruled by special laws, accordig to which these risks are subject to strict liability. the Uited States the courts decide upo whether i a specific case the activity is deemed as beig abormally dagerous 3 such that strict liability should be applied. The applicatio of the strict liability rule to very risky activities is justified by meas of differet lies of reasoig, depedig o whether a market relatioship betwee defedat ad victim does exist, as would be typical for product liability, or whether there is o such relatioship. the latter case, which ca be assumed to be true for most evirometal damages, a liability rule should ot oly set icetives for efficiet loss prevetio (care) but should also be able to iduce efficiet activity levels. Sice, whe the egligece rule is applied, a defedat is ot held liable if he exercises a level of care that equals or exceeds due care, he will ot take ito accout the remaiig risk ad will therefore exceed the welfare maximizig activity level. cotrast to this, strict liability would lead to optimal care ad cotrol activity i a efficiet way, sice a defedat would i ay case iteralize the etire liability risk. Thus, as the usual argumet goes, the egligece rule ad strict liability are equivalet with regard to loss prevetio icetives, but strict liability seems to be a better solutio i terms of cotrollig risky activities. f a market relatioship betwee defedat ad victim does exist, it is usually argued that i a world with homogeeous ad completely iformed victims strict liability ad the egligece rule would be equivalet i terms of settig icetives for loss prevetio as well as cotrollig the activity level. The reaso is that the remaiig risk at the level of efficiet loss prevetio is bore etirely by the victims, either directly i the case of the egligece rule or idirectly via the product price i the case of strict liability, implyig there would be o exteralities. 4 Reasos give for the supposed superiority of strict liability uder these circumstaces are the cosiderable problems of determiig efficiet care, i compariso to 3 4 See Restatemet (Secod) 59, 520 of Torts. See e.g. Lades/Poser 985, pp. 535.

5 2 other areas of liability law, 5 ad the fact that oly strict liability creates icetives to research ad develop ew security techologies. 6 The statemet, that the two liability regimes are equivalet with regard to cotrol of loss prevetio ad activity level if there is a market relatioship betwee the parties, or that strict liability is superior if there is oe, is correct oly if the parties are risk eutral. The stadard assumptio of risk eutrality is especially crucial withi the cotext of this paper, cosiderig that areas subject to strict liability usually bear extreme risks. additio to this, damages will very ofte cause losses to may victims, such that strict liability leads to a risk accumulatio while egligece spreads the risk. f, however, the parties are risk averse i reality, the results from a aalysis based o risk eutrality, might lead to substatial misjudgmets ad therefore icorrect policy recommedatios. This is due to the followig reasos: At first, as a additioal criterio for the ecoomic evaluatio of liability rules, risk allocatio effects come ito play as soo as risk aversio is cosidered. Additioally, uder these circumstaces, liability rules are ot ecessarily equivalet i terms of iducig care. Furthermore, if there is a market relatioship betwee ijurers ad victims, egligece leads to a better outcome, as it icurs lower risk-bearig costs tha strict liability. Fially, i the case of o market relatioship the superiority of strict liability with regard to cotrollig the extet of risky activities might ot persist, sice strict liability probably leads to a activity level too low i compariso to the welfare maximizig level. Cosiderig the impact the risk attitude has o the aalysis of liability rules for highly risky activities, the questio of a adequate assumptio has to be discussed i more detail: While it is widely accepted that idividuals are risk averters, firms are ormally cosidered as risk eutral. Sice i our cotext the ijurers typically are firms, the problem of adequately modelig risk prefereces for this case has to be examied more closely. A typical ratioale offered for the risk eutrality assumptio is that the shareholders hold well-diversified portfolios ad will thus aim to maximize the expected profit of the firm. 7 t follows immediately that this explaatio is oly valid for joit stock compaies, but ot for parterships. Furthermore, as is well kow, eve i perfect capital markets, a security s risk ca oly be completely diversified if there is o systematic compoet. Apart from this plety of evidece for imperfectess ca be See Rose-Ackerma 99. See e.g. Shavell 980, p. 2. See e.g. Doherty 985, p. 465, Shavell 987, p. 89, or Milgrom/Roberts 992, p. 87.

6 3 foud i real capital markets. Particularly as a cosequece of the trasactio costs icurred by a trasfer of shares, real ivestmet portfolios are usually isufficietly diversified. Eve if there is o systematic risk ad the risk arisig from a idividual ivestmet ca be elimiated etirely by meas of diversificatio, it must still be stated that etrepreeurial decisios are made by the maagemet, ad that it is ormally impossible for the owers to cotrol every sigle decisio. 8 The maagemet thus has a certai discretio i activity o the firm s behalf. t is a stadard result of agecy theory that maagemet s icome should deped o the firm s profit i order to give appropriate icetives. 9 The idividual maager caot perfectly diversify his profit-depedet icome. Therefore, some of the most ifluetial decisio makers will exhibit risk aversio, i particular if they are cofroted with the possibility of large losses. 0 Thus, eve for joit stock compaies, uder realistic assumptios cocerig the imperfect maagemet-shareholder relatioship ad the resultig icetive problems, risk aversio i firm behavior is a very plausible assumptio. This premise has empirical support as joit stock compaies buy isurace coverage at a substatial rate, which is most easily explaied by risk aversio. Aother argumet commoly used to justify the assumptio of risk eutral decisiomakig says the ivolved parties had the opportuity to buy isurace. perfect markets the isurace premium equals the expected losses from the cotract, ad isurace customers buy complete coverage. Therefore, a isured party would act like a risk eutral decisio maker. But agai, this idirect ratioale for the risk eutrality assumptio via isurace supply turs out to be of very limited value, as real markets demostrate that isurace does ot work i such a perfect fashio. surace coverage would ot usually be available at a actuarially fair rate, a fact that may be attributed to may differet reasos, such as trasactio costs ad i particular 8 This will usually at least be true for those stockholders who, for diversificatio reasos, ivest oly a small part of their ivestmet budget i the sigle firm. 9 See amog others Tirole 990, p See Shavell 987, p Naturally there is a multitude of possible additioal motives for corporate isurace demad. For example, Mayers ad Smith, 982, metio the reductio of expected trasactio cost of bakruptcy, advatages due to specializatio the isurace compaies might have i the area of loss hadlig particularly the hadlig of liability claims, as well as tax icetives. Grace ad Rebello, 993, explai corporate isurace demad alteratively as a sigalig behavior.

7 4 isurers risk aversio. This implies that, accordig to well kow results from isurace demad theory, ratioal decisio makers would ot cover their etire risk by isurace. 2 Additioal cost of risk allocatio arises if, for example, isurability problems lead to limitatios i the supplied coverage. Liability isurace usually covers losses up to a certai amout specified i the cotract. some areas of liability isurace these upper bouds for idemity paymets leave the isured party with a sigificat share of the risk. particular the very cautious settig of sums isured i evirometal liability isurace meas that a cosiderable risk remais with the isured. As we have see, i essece all explaatios for assumig risk eutrality are based o the assumptio of perfect capital or isurace markets. However, the presece of trasactio costs aloe is sufficiet to show that these markets are geerally imperfect. Therefore, the premise of risk aversio, as the empirically domiat risk attitude, seems to be more suitable. Thus, i the followig we will aalyze liability rules for highly risky activities uder the assumptio that both parties, ijurers ad victims, are risk averse. A aalysis of this kid must at first address the questio of why the activities which are subject to strict liability are cosidered highly risky. The mai reaso for this might be that damages caused by these activities typically affect a large umber of victims. For example, a defective pharmaceutical product ca give rise to health problems for may people. Similar cosequeces might be triggered if the productio of a commodity is commoly iflueced by certai stochastic factors. A sceario for the latter example could be that oe defect, which was ot disclosed at the time of productio, later affects a etire lie of productio. As prove i particular by recalls quite ofte observed i the automotive idustry, this problem is of cosiderable importace. Distict positive correlatio is rather obvious also i the evirometal liability area, as evirometally harmful emissios usually iflict damages for may idividuals. Fially, a extreme case of positive correlatio arises if liability claims are combied to go to trial as a class actio lawsuit. The high risk i the above-metioed examples does ot primarily result from a high loss potetial from the sigle claim, but from the possibility of a multitude of claimats. t is this kid of risk accumulatio that is typically subject to strict liability. 2 For a detailed discussio of optimal isurace decisios see amog may others Borch 960, Arrow 963, Borch 976, Raviv 979.

8 5 Liability rules for areas characterized by the potetial of loss accumulatio have ot bee a subject of extesive research i ecoomic literature so far. The assumptio usually employed is that damages would oly harm oe perso. The possibility of may idividuals beig affected by oe evet has ot bee cosidered i most of the law ad ecoomics research. 3 This seems to be surprisig at first glace, sice problems of evirometal ad product liability have i particular bee discussed heavily, ad they certaily are as was metioed above good examples for the dager of loss accumulatio. Nevertheless there is a simple explaatio for the eglect of the umber of victims as a importat factor: Takig it ito accout would ot chage the structure of the results, as log as the parties are assumed to be risk eutral. This ca be show as follows: Let us assume that a certai liability risk threates idetical potetial victims. For each of them the expected losses are E [, depedig o the ijurer s level of loss prevetio (x). Regardig the effects of loss prevetio measures we itroduce the usual assumptios: de[ < 0, dx d 2 E[ > 0 2 dx () f the parties are risk eutral, the optimal level of care miimizes the fuctio c( x) + E[ (where c (x) with c ( x) > 0, c ( x) 0 deotes the cost of loss prevetio). f the (uambiguous) global miimum is a iterior solutio, the miimum locus solutio of x is the de[ c ( x) = (2) dx As ca be see from (2), the optimal level of care icreases with the umber of victims. This result, however, is ot specific for the may victims problem, as ay icrease i risk that ca be modeled as a liear trasformatio of the expected losses affects i the same way. The oly effect of a icreasig, uder these circumstaces, is that for determiig the optimal loss prevetio level a larger extet of risk has to be cosidered. No other cosequeces have to be take ito accout. For example, it does ot matter whether x 3 See, however, Nell ad Richter (996), who explicitly take ito accout the implicatios the umber of victims has o the efficiet desig of liability rules.

9 6 may idividuals are i dager of sufferig comparably small losses or whether there is oly oe potetial victim facig the sum of these risks. Therefore, reducig the model to the aalysis of oe represetative victim does ot have ay sigificat ifluece o the results. f, o the other had, risk averse decisio makers are cosidered, the umber of victims becomes relevat. This is because, for evaluatig a liability rule, it is ot the icetive fuctio aloe aymore which is importat. As was stated earlier, oe also has to take ito accout the risk allocatio effects, ad thus particularly the impact of the umber of ivolved parties o optimal risk sharig. A iterestig questio is how the fact that the umber of risk bearers icreases with the risk affects the optimal liability rule. Crucial with respect to this is the iteractio betwee the umber of victims ad the ijurer s risk premium. This iteractio, agai, depeds o the correlatio betwee the sigle risks. As we are goig to aalyze the cosequeces of risk accumulatio, we cocetrate o the case of complete correlatio. Liability isurace agaist this kid of risk is either madatory or it is purchased volutarily to a sigificat extet. Thus, we will icorporate isurace supply i our aalysis. Reasoably, we assume imperfect isurace markets i this paper. The remaider of the paper is orgaized as follows: sectio 2 we ivestigate the efficiet liability rule for correlated risks whe the parties are risk averse ad the activity level is give exogeously. sectio 3, isurace is icluded i the aalysis. The activity level uder strict liability is the subject of sectio 4. The paper cocludes with a summary ad discussio of the mai aspects. 2. Optimal liability for correlated losses whe o isurace is available 2. Basic assumptios We cosider the case of oe (potetial) ijurer egagig i some activity that ivolves the risk of harmig victims, who are assumed to have idetical prefereces. 4 The amout of losses, L would be the same for every sigle victim, where L has a two poit distributio ( L, p, 0) with L 0 ad 0 < p <. Sice i the situatios which are to be ivestigated here > the loss distributio usually ca oly be iflueced by the potetial ijurer, but ot by the 4 this paper we do ot cosider the possibility of more tha oe ijurer ifluecig the risk, although this is a importat problem, i particular if oe is cocered with certai evirometal liability problems.

10 7 victims, we do ot cosider victims loss prevetio measures. With regard to the impact the ijurer s loss prevetio (x) has o the distributio of losses, the model is kept more geeral tha stadard law ad ecoomics models. Those models usually cocetrate o the case of loss prevetio reducig the loss probability. this paper, however, we allow for mitigatio measures which either reduce the probability or the extet of losses. The fuctio of the cost of care, c(x) is assumed to be covex, as was metioed earlier. Furthermore, a premise is added here that will be relevat i the cotext of large : Amog other thigs this paper will aalyze how results chage whe the umber of victims icreases. f the set of possible loss prevetio levels would be assumed to be ubouded, the result for may situatios would be as follows. Ay arbitrarily high prevetio level could be efficiet if oly is large eough. reality, however, usually there would be a maximum mitigatio level. Additioal mitigatio might be possible but remai without ay impact. Examples iclude istallig the most up to date filter plat for avoidace of harmful emissios or carryig out all kow tests before marketig a ew pharmaceutical product. For this reaso, we will use the assumptio of a maximum level of care, ijurer ca choose from the set [, x ]. 0 max x max such that the As was explaied before, this paper deals with risk averse decisio makers. Whe risk aversio is take ito accout, i geeral the parties levels of wealth become relevat, as for most utility fuctios the degree of risk aversio depeds o wealth. The degree of risk aversio iflueces the optimal liability rule. Thus, the use of utility fuctios which do ot show costat absolute risk aversio implies a wealth-depedet desig of the optimal liability rule. This agai is criticized with covicig argumets i literature. 5 To avoid these problems we assume utility fuctios with costat absolute risk aversio (CARA). The risk aversio coefficiets are deoted by α for the ijurer ad β for the victims, the utility fuctios are deoted by V ad U. Furthermore q ( 0 q ) is the share of a loss that has to be bore by the ijurer. This meas that we allow for a strict divisio of losses ( 0 < q < ) as a solutio as well as for boudary solutios, such as the egligece rule or strict liability. All relevat parameters are assumed to be kow by the ivolved parties, i particular by the courts. Furthermore, we assume that the ecessary differetiability requiremets are fulfilled i ay case, which meas especially that the order i which oe takes the expected 5 See e.g. Abraham/Jeffries 989. See also Miceli/Segerso 995 who criticize Arle 992, as the latter paper argues i favor of wealth depedet liability rules i a ot very cosistet way.

11 8 value ad the derivative ca be exchaged. First, we cosider the case of a exogeouslygive level of activity. 2.2 The social cost fuctio The expected utility of a ijurer with a liability share q coductig mitigatio at level x is E[ V ( W α ( W ( ) ) ( ) ) ] [ c x q L c x q L x = E e x] (3) α where W is the ijurer s iitial wealth. The sigle victim s expected utility is E[ V ( W V β ( W ( ) ) ( ) ) ] [ V q L q L x = E e x] (4) β (W deotes the victim s iitial wealth). V The certaity equivalet of the ijurer s fial wealth is give by CE( W e α ( W c( x) ) = l{ E[ e α = W c( x) l{ E[ e α q L ) α q L x]} x]} (5) For a sigle victim we get CE( W e V β ( L ) = WV l{ E[ e x]} (6) β We cosider mitigatio measures that affect either the probability or the size of loss. This is modeled such that either the probability of loss, p(x) is a strictly decreasig ad covex fuctio ad the size of loss, L is a costat, or that otherwise L = L ) ( L 0 ad L > 0 ( x ) with costat loss probability p. To keep thigs simple, however, we will restrict the derivatios to the geeral formulatio i the followig. The problems that require explicitly distiguishig betwee the two models are tackled i the appedix. Welfare will be measured by the sum of the parties certaity equivalets: < α q L β ( L W c( x) l{ E[ e x]} + [ WV l{ E[ e x]}] (7) α β

12 9 Sice WV fuctio of social cost ad W are ot affected by the liability rule, we will cocetrate o the C T α q L β ( L : = c( x) + l{ E[ e x]} + l{ E[ e x]} α β (8) = : R = : R V The social cost is the sum of the loss prevetio cost, the moetary equivalet of the ijurer s stochastic liability paymets ( R ), ad the correspodig value for the victims ( R ). The latter expressios will be called the parties idividual cost of risk i this V paper. We assume these cost fuctios to be strictly decreasig ad strictly covex i x for ay (positive) liability share. R < 0, x 2 R x 2 > 0 x 0, 0 < q (9) ad R V < 0, x 2 R V x 2 > 0 x 0, 0 q < (0) This meas the margial beefit from loss prevetio is positive ad strictly decreasig i x The optimal liability rule The optimal liability rule is a solutio to the optimizatio problem mi 0 x xmax, 0 q C T = c( x) + R + R V () As ecessary coditios for a iterior solutio we derive 6 This assumptio is due to purely techical reasos. t guaratees that certai problems with respect to the uiqueess of solutios are avoided. For the case of loss prevetio affectig the extet of loss, this assumptio is ot eeded. f, however, mitigatio reduces the probability of loss, the covexity of (ad R V, respectively) would ot be esured without additioal assumptios. R

13 0 R RV c ( x) = (2) x x ad R RV q = ) q q (3) implyig that the efficiet sharig of liability is defied by q β = α + β (4) The ijurer s optimal liability share decreases i, because R icreases i faster tha at a liear rate ad therefore stroger tha the sum of the victims costs. For, q teds to zero. Furthermore, the followig results ca be derived: Propositio : Uder the assumptios of this sectio the optimal mitigatio level icreases i. For a sufficietly large umber of victims the maximum level of care becomes optimal. Proof: See appedix A. Thus, for a give activity level, the egligece rule with due care x max is approximately efficiet. The ijurer fulfills the stadard of due care, ad risk allocatio would at least be approximately optimal. cotrast to this, strict liability yields the more usatisfactory ecoomic results the larger the umber of victims. The ijurer would choose sufficietly large, but his liability share would be oe, while the optimal value teds to zero. x max for 3. The impact of isurace supply So far, results have bee derived uder the assumptio that the parties bear the etire risk assiged to them by a liability rule. reality, however, potetial ijurers as well as potetial victims usually have the opportuity to buy isurace. this sectio we aalyze the way i which the supply of isurace coverage iflueces the desig of a optimal liability rule. To

14 avoid uecessary complicatios, we cocetrate o isurace beig available for the ijurers. f isurace markets were perfect, ecoomic actors could get rid of their etire risk at a premium that equals expected losses. cotrast to this, liability isurace cotracts i real markets limit the provided coverage, ad premiums ormally exceed the expected losses. For both of these reasos a share of the risk is typically kept by the isured. Therefore, i the followig we will aalyze how isurace supply affects the optimal liability rule if markets are imperfect. At first we will cocetrate o the effects of premiums exceedig the expected value of claims. surace premiums are assumed to be calculated as the sum of the expected losses ad a proportioal loadig. f, for example, the ijurer is assiged the whole risk L, the price of liability isurace would be Π [ d, = ( + m) d E[ (5) where m is the loadig factor ad d ( 0 d ) deotes the level of coverage. 7 We start by cosiderig isurace demad decisios for a give level of loss prevetio ad a give risk sharig. The optimal coverage the is determied as a solutio to α q ( d ) L mi c( x) + l{ E[ e x]} + ( + m) d q E[ 0 d α (6) = : R yieldig the first order coditio ( d, x, α q ( d ) L E[ L e x] + m) E[ = (7) α q ( d ) L E[ e x] ( From (7) we get for our model framework the well kow fudametal result, that was briefly metioed before: f the loadig factor m is positive, ratioal isurace customers choose a level of coverage d <. The optimal coverage icreases if ceteris paribus 7 As well as the other parties, isurers are assumed to have complete iformatio. This meas that, i particular, problems of moral hazard are ot discussed i this paper. our model the isurer is able to observe the isureds actios ad, thus, to directly tie the premium to the level of care.

15 2 the umber of victims icreases or the loadig factor decreases. Complete isurace coverage ( d = ) ca oly be optimal if m = 0. Let us assume from ow o that isurace is always worthwhile ( d > 0 ). cosider the followig miimizatio problem: 8 We mi 0 x xmax, 0 q, 0 d C T ( d, x, = c( x) + R ( d, x, + R + ( + m) d q E[ V (8) As a first order coditio for a iterior solutio we get R ( d, x, RV de[ c ( x) ( + m) d q = 0 x x dx (9) R ( d, x, + q R V + ( + m) d E[ = 0 q (20) ad also (7). Substitutig the explicit expressios for the partial derivatives i (20) ad usig (7) we get β q = (2) α ( d ) + β This meas q is icreasig i the isurace coverage. particular, the optimal ijurer s liability share is larger if liability isurace is available, compared to the case without isurace. this sese, the opportuity to buy isurace expads the ijurer s capacity, as log as the premium is ot prohibitively high. The more efficiet the risk allocatio device isurace works, the more risk would be bore by the ijurer. But oly if isurace is costless (m = 0) we derive q =. A iterestig questio is whether or ot, i the case with isurace, the optimal risk sharig still teds to the risk allocatio situatio of the egligece rule or whether this tedecy is possibly compesated by icreases i isurace coverage. fact, it ca be show 8 Formally, we esure through a additioal premise (see propositio 2) that, for sufficietly large, the level of coverage is positive i the optimal solutio.

16 3 that d teds to oe so fast that the optimal ijurer s liability share does ot coverge to zero, but remais above a certai level. Propositio 2: Uder the assumptios of this sectio ad if + m) E[ < E[ L e ( β L E[ e β L x] x] x (22) a positive q mi exists with the property that for ay umber of victims q q (23) mi Proof: See appedix B. Thus, if the ijurer has the opportuity to pass risk to a isurace compay at a costat rate, the strog icrease of its idividual cost of risk ad the impact o efficiet risk sharig are slowed dow. The efficiet liability rule uder these circumstaces does ot coverge to the egligece rule. We ow cosider the level of loss prevetio for very large : From (9) follows de[ L x ] c ( x ) > ( + m) q d (24) dx if x is a iterior solutio. Therefore de[ L xmax ] c ( x ) > ( + m) qmi d (25) dx As icreases the right had side of (25) grows without boud, such that for a sufficietly high umber of victims the maximum level of care will be optimal. f isurace premiums cosist of the expected value of losses ad a proportioal loadig, either strict liability or the egligece rule approximate the optimal solutio. stead, a liability rule that assigs a share of every victim s claim to the ijurer turs out to be approximately efficiet, give the latter fulfills the due care stadard. q mi x max

17 4 The compariso of strict liability ad the egligece rule depeds heavily o the loadig factor m. As the trasactio cost of isurace declies, strict liability gets more attractive i compariso to the egligece rule ad vice versa. However, the latter results o the approximately efficiet liability rule oly hold if there are o limitatios to the demad of liability isurace ad if the isurers are risk eutral ad therefore base calculated premiums o the expected losses oly. f, o the other had, isurers are risk averse, the price of isurace respectively i case of limited coverage the ijurer s risk premium grows faster tha at a liear rate. this case agai, the egligece rule, with a stadard of due care x max, would be approximately optimal for large umbers of victims. Sice upper bouds for the coverage are very commo i real liability isurace markets, there seems to be cosiderable evidece that the results of sectio 2 still hold eve if liability isurace is available. 4. Variable level of activity Oe argumet that is quite ofte stated i favor of strict liability i the cotext of highly dagerous activities is the fact that this rule would lead to a efficiet activity level. O the other had, egligece would, if there were o market relatioship betwee the parties, iduce the activity to be carried out at a excessive level. The reaso for this is that a ijurer would ot be held liable for damages as log as the stadard of due care is fulfilled. 9 But, as was metioed above, strict liability oly leads to a optimal activity level if we assume risk eutral parties ad/or perfect isurace markets. f, however, ijurers are risk averse ad isurace markets are imperfect, the iduced activity level is too low ad the extet of the uder-ivestmet i the risky activity icreases i the umber of victims. We wat to explai this iteractio i more detail. For that purpose we cosider a society that cosists of idetical idividuals. We assume that a risky activity ca be carried out at a level a which ca be varied cotiuously. As the focus here is o the problem of cotrollig the activity, it is assumed that the liability risk ca oly be iflueced through the activity level, but ot by meas of loss prevetio. Furthermore, we assume that the amout of potetial damages, but ot the loss probability, depeds o the level of a ( L = L ) with L ( a) > 0 ). The activity does ot icur ay other costs. For each idividual it yields utility Z(a) with Z ( a) > 0, Z ( a) < 0. The social optimum the is a solutio to ( a

18 5 max a [ Z( a) l{ p e γ γ L ( a) + p}] (26) coditio where γ is the idividuals risk aversio coefficiet. This gives the first order Z ( a) = p L ( a) e p e γ L ( a) γ L ( a) + p (27) t is assumed that the activity ca oly be carried out by oe idividual. f the activity is ruled by strict liability, the idividual s objective fuctio is the followig: G ( a) : = Z( a) l{ p e γ γ L ( a) + p} (28) A first order coditio for a iterior solutio is Z ( a) = p L ( a) e p e γ L ( a) γ L ( a) + p (29) f ceteris paribus the umber of victims icreases, the relevat decisio maker s margial cost (right had side of equatio (29)) icreases at a higher rate tha the margial retur of the activity, ad we derive da d γ L ( a) γ L ( a) p L ( a) e ( p) = < 0 γ L ( ) 2 ( ) a G a { p e + p} (30) The optimal level of activity strictly decreases i the umber of potetial victims, such that the differece betwee the activity level iduced by strict liability ad the socially optimal level icreases as the umber of victims icreases. Uder realistic premises cocerig the margial cost ad margial beefit fuctios we ca proceed from the assumptio that for large the risky activity is etirely preveted, eve if it is socially desirable accordig to (27) See e.g. Shavell 980, pp., ad Shavell 987, p This result is ot decisively affected by takig isurace markets ito accout, if these markets are imperfect. Uder these circumstaces strict liability would still iduce a isufficiet activity level. However, if a proportioal loadig o top of the expected value of losses is charged as the isurace premium ad if isurace coverage is ulimited, the margial cost does ot icrease without boud i, ad thus the activity level does ot ted to zero.

19 6 The result that strict liability for risks with the potetial of loss accumulatio leads to a activity level lower tha i the social optimum does ot deped o whether there is a market relatioship betwee ijurers ad victims or ot. the case of risk aversio ad if there is a market relatioship, the egligece rule is superior to strict liability, sice the total cost of risk bearig is lower: 2 f the potetial victims are the customers, risk premiums are iteralized via a reductio of their willigess to pay. f, however, there is o market relatioship betwee victims ad ijurers, oe caot, without additioal assumptios, make a geeral statemet about the compariso of strict liability ad egligece. 5. Cocludig remarks May coutries employ the egligece rule as their mai liability regime. Highly risky activities, however, are ofte govered by strict liability. The reaso usually give for applyig strict liability to these areas is that ot oly efficiet care is supposed to be iduced, but also a efficiet level of the risky activity itself. t is argued that, i the case of o market relatioship betwee ijurers ad victims, this could oly be achieved through strict liability but ot via egligece. Most activities which are cosidered very risky are characterized by the fact that they edager a large umber of potetial victims. Therefore, strict liability implies a quite ufavorable allocatio of risk, as the risk is ot spread but completely assiged to the ijurer. The hereby icurred secodary cost of risk allocatio i the sese of Calabresi has bee largely igored i the law ad ecoomics literature, by meas of assumig risk eutral idividuals or perfect isurace markets. The premise of risk eutral decisio makig as well as the assumptio of perfect isurace markets are empirically ot very well established. Therefore, the topic of this paper is the questio of whether strict liability remais the superior regime for highly risky activities eve if the parties are risk averse ad the isurace markets are icomplete. We have show that for a give level of activity ad a sufficietly large umber of victims, egligece is the better solutio, if o isurace is supplied. Strict liability, o the cotrary, turs out to be clearly suboptimal because of its risk allocatio effects. Takig isurace markets ito accout does ot affect these results substatially, if the available isurace coverage is limited. The same statemet holds if isurace premiums iclude a risk depedet loadig 2 the same way it follows that, due to the cost of risk, uder these coditios the risky activity uder the egligece rule is carried out at a lower level i compariso with the case risk eutrality.

20 7 due to risk aversio of the isurer. However, if the loadig does ot deped o the structure of the risk, but is calculated as a percetage of expected losses, either strict liability or the egligece rule is optimal. this situatio a liability rule is efficiet that makes the ijurer participate with a certai positive fractio (smaller tha oe) i every damage. We ca therefore coclude that strict liability caot be see as the superior liability rule for highly dagerous activities, if risk allocatio aspects are take ito accout. terms of risk allocatio the egligece rule should be preferred for activities with the potetial of loss accumulatio, if isurace markets show a substatial degree of imperfectess. With respect to cotrollig the level of the risky activity egligece turs out to be superior, if a market relatioship betwee the parties exists. This is because the egligece rule icurs less cost of risk. f there is o market relatioship betwee ijurer ad victims, o clear result ca be derived. We ca oly state that egligece iduces excessive use of the risky activity while strict liability leads to a activity level below the social optimum. For risks subject to strict liability the extet of a ijurer s share i the risk is very ofte limited by a upper boud. 22 Furthermore, rules are commo which exclude losses from a ijurer s liability, if there would have bee o way to prevet them accordig to most recet sciece fidigs or by applyig latest techology. 23 At first glace, these regulatios seem to be ecoomically questioable ad icompatible with the priciple of strict liability. particular the existece of upper bouds has bee criticized Our cosideratios, however, show that this kid of limitatio of strict liability is actually a way to improve efficiecy. The exclusio of uforeseeable losses, for example, ca be iterpreted as a egligece rule with a very restrictive stadard of due care: A ijurer is held liable if the loss was foreseeable. 22 See, for example, the Germa Evirometal Liability Law 5 (60 millio DM for bodily ijury ad the same amout for material damage), the Product Liability Law 0 () (60 millio DM), the Pharmaceutical Products Law 88 (200 millio DM respectively 2 millio DM i pesio paymets), ad the Geetic Egieerig Law 33 (60 millio DM). 23 Oe example is agai the Germa Product Liability Law ( ), accordig to which a defedat is ot held liable if it was impossible, accordig to recet research fidigs respectively by use of latest techology, to detect the defect at the time the product was put o the market. See also 84 of the Germa Pharmaceutical Products Law that assigs losses due to isufficiet istructios to the ijurer oly if, roughly speakig, these istructios do ot comply with the stadards of medical sciece. 24 See Faure/v.d. Berg The exclusio of uforeseeable losses i the sese that state of the art loss prevetio is applied, is see less oe-sided, sice by defiitio liability for this kid of losses does ot have a impact o behavior. favor of strict liability eve for these losses is argued, if i priciple the ijurer would have bee capable of fidig out about ukow dagers through research. O the other had, it has to be kept i mid that the dager of beig held liable for uforeseeable losses keeps ivestors from ivestig i the developmet of useful but dagerous activities.

21 8 Therefore, to avoid havig to compesate the victims all kow loss prevetio measures must be carried out, or i other words, the maximum level of care must be carried out. But this is, as we have show, the optimal stadard of a egligece rule if the umber of potetial victims is large. Furthermore, i the case of a upper boud of liability, the actual ijurer s share i the risk decreases with a icreasig umber of victims. This, agai, is exactly a feature we derived for the optimal regime to gover abormally dagerous activities. Thus, both kids of supplemets for a rule of strict liability seem to be useful tools to reduce iefficiecies this rule would have whe applied to areas characterized by the potetial of loss accumulatio.

22 9 Refereces Abraham, K. S., ad J. C. Jeffries Puitive Damages ad the Rule of Law: The Role of Defedat s Wealth. Joural of Legal Studies 8: Arle, J. H Should Defedats Wealth Matter? Joural of Legal Studies 2: Arrow, K. J Ucertaity ad the Welfare Ecoomics of Medical Care. America Ecoomic Review 53: Borch, K The Safety Loadig of Reisurace Premiums. Skadiavisk Aktuarietidskrift 43: Borch, K Optimal surace Arragemets. Asti Bulleti 8: Doherty, N. A Corporate Risk Maagemet: A Fiacial Expositio. New York. Faure, M., ad R. v. d. Bergh Liability for Nuclear Accidets i Belgium from a terest Group Perspective. teratioal Review of Law ad Ecoomics 0: Grace, M. F., ad M. J. Rebello Fiacig ad the Demad for Corporate surace. The Geeva Papers o Risk ad surace Theory 8: Lades, W., ad R. Poser A Positive Ecoomic Aalysis of Products Liability. Joural of Legal Studies 4: Mayers, D., ad C. W. Smith O the Corporate Demad for surace. Joural of Busiess 55: Miceli, T., ad K. Segerso Defiig Efficiet Care: The Role of come Distributio. Joural of Legal Studies 24: Milgrom, P., ad J. Roberts Ecoomics, Orgaizatio ad Maagemet. Eglewood Cliffs (New Jersey). Nell, M., ad A. Richter Optimal Liability: The Effects of Risk Aversio, Loaded surace Premiums, ad the Number of Victims. The Geeva Papers o Risk ad surace 2: Raviv, A The Desig of a Optimal surace Policy. America Ecoomic Review 69: Rose-Ackerma, S. 99. Tort Law i the Regulatory State. i: Schuck, P. H. (Ed.). Tort Law ad the Public terest: Competitio, ovatio, ad Cosumer Welfare. New York. Shavell, S Strict Liability versus Negligece. Joural of Legal Studies 9: -25. Shavell, S Ecoomic Aalysis of Accidet Law. Cambridge (Mass.). Tirole, J The Theory of dustrial Orgaizatio. Cambridge (Mass.).

23 20 Appedix A Proof of propositio : Cosider the miimizatio problem mi 0 x xmax, 0 q C T = c( x) + R + R V (3) where, for the case that mitigatio affects the loss probability, R α q L = l{ p( x) e + p( x)} α (32) ad, if mitigatio affects the extet of losses, R = l{ p e α α q L ( x) + p} (33) ( is determied i the same way). R V Firstly, it has to be show how the optimal level of mitigatio reacts o a ceteris paribus variatio of. Therefore, we substitute for q i (2). We derive α + β c ( x) = α β respectively p ( x) ( e p( x) e α β L α+β α β L α+β ) + p( x) (34) c ( x) = L ( x) e p e α β L ( x) α+β α β L ( x) α+β + p (35) With α β /( α + β) also the right had side i (34) ad (35) strictly icreases i. Sice furthermore the margial beefit of loss reductio decreases i x, the optimal mitigatio level icreases as grows. For sufficietly large we get x = x. f for all x were smaller tha x, there max max would have to be a x with x x. The, however, the left had side i (34) x max

24 2 ad (35) would ted to c (x), while obviously the right had side would grow without boud, implyig that for sufficietly large equality could ot be fulfilled (i cotradictio with the assumptio that x < x ). max q.e.d. Proof of propositio 2: Appedix B With β q = oe gets α ( d) + β α β α q ( d) = (36) β α + ( d) Usig (7) ad h : = ( d) yields L α+β / h E[ L e x] ( + m) E[ = (37) ad we fid E[ e α β α β L α+β / h x] α β β h α + β / h ad α β 0 0 α + β / h h (38) Cosiderig (22) it ca be see that for ay give level of care there exists a h(x), which solves (37). The hereby defied fuctio h(x) is cotiuous o the compact iterval [ 0, xmax ] ud thus assumes a maximum i this set. With h max : = max{ h( x) : x [0, xmax ]} q β β = : q α ( d ) + β α h + β = max mi (39) q.e.d.

25 For orders please cotact / Kotaktadresse für Bestelluge: Prof. Dr. Marti Nell Geschäftsführeder Direktor des stituts für Versicherugsbetriebslehre Vo-Melle-Park 5 D-2046 Hamburg Tel.: +49-(0) Fax: +49-(0) marti.ell@rrz.ui-hamburg.de