Precaution Incentives in Accident Settings

Tim Friehe Precaution Incentives in Accident Settings

GABLER EDITION WISSENSCHAFT Ökonomische Analyse des Rechts Herausgegeben von Professor Dr. Peter Behrens Professor Dr. Thomas Eger Professor Dr. Manfred Holler Professor Dr. Claus Ott Professor Dr. Hans-Bernd Schäfer (schriftführend) Universität Hamburg, Fakultät für Rechtswissenschaft und Fakultät für Wirtschafts- und Sozialwissenschaft

Die ökonomische Analyse des Rechts untersucht Rechtsnormen auf ihre gesellschaftlichen Folgewirkungen und bedient sich dabei des methodischen Instrumentariums der Wirtschaftswissenschaften, insbesondere der Mikroökonomie, der Neuen Institutionen- und Konstitutionenökonomie. Sie ist ein interdisziplinäres Forschungsgebiet, in dem sowohl Rechtswissenschaftler als auch Wirtschaftswissenschaftler tätig sind und das zu wesentlichen neuen Erkenntnissen über Funktion und Wirkungen von Rechtsnormen geführt hat. Die Schriftenreihe enthält Monographien zu verschiedenen Rechtsgebieten und Rechtsentwicklungen. Sie behandelt Fragestellungen aus den Bereichen Wirtschaftsrecht, Vertragsrecht, Haftungsrecht, Sachenrecht und verwaltungsrechtliche Regulierung.

Tim Friehe

Precaution Incentives in Accident Settings With a foreword by Prof. Dr. Laszlo Goerke

GABLER EDITION WISSENSCHAFT

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Dissertation Universität Tübingen, 2008

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Foreword Law and economics has been established as an important sub-discipline of economics. Looking at the field, it is undisputed that the economics of tort law has been the subject of much study early on and continues to be. The analysis in that realm is centered on the internalization of external effects by the means of liability law, i.e. the allocation of a liability burden possibly depending on the behavior of parties involved in an accident. In the by now standard framework, introduced by path-breaking contributions such as Calabresi (1970) and Brown (1973), the outcome with regard to care-taking (and possibly the level of activity) under several liability rules, which are actually observed in practice, is compared to what is socially desirable. The objective of society usually is assumed to be wealth maximization. The set of results which may be called the central theory show that liability rules can indeed induce first-best behavior by parties, as long as several core assumptions hold. After the central theory of the economics of tort law had been settled, contributors to the literature started to test the robustness of the conclusions obtained when these core assumptions are varied. The existent literature on the economics of tort law is rich and diverse. Yet, without doubt, there are still numerous questions in the field which need to be answered and therefore require scholarly attention. The present book rightfully goes along that path. In a collection of chapters, different subjects are examined from a theoretical standpoint. One central aspect taken up in this book is that of imperfect information. Taking the set of all potential accidents into account, the level of harm suffered by victims varies to a large extent and often courts as well as injurers, and, possibly, even victims are not sure about the level of harm suffered in a particular accident. The usual approach courts take is trying to assess the precise magnitude of harm in order to fully compensate the victim. In this regard, courts often rely on expert testimony. An alternative way to deal with this variation in harm levels is to set compensation that is to be paid by the injurer equal to the expected harm. If the injurer cannot anticipate the precise harm level, imposing expected harm as compensation level can imply the same behavioral incentives as the costlier assessment of precise harm on every occasion (Kaplow and Shavell 1996). This V

can hold because the assessment of harm after the accident has occurred cannot improve the information available to injurers ex ante. This book, for one, investigates whether this result will continue to hold if the injurer as well as the victim can decide on precaution, and, for another, contributes to the literature by establishing that the expected harm measure can dominate accurate compensation even in the absence of assessment costs. This possibility arises due to behavioral implications the use of expected harm has for victims. If victims can decide on the value of the object put at risk, for instance the value of the car used in traffic, accurate compensation can imply inefficient victim choice because the accident eventuality is disregarded by victims. This feature indeed favors the use of the average measure. Furthermore, the book entails a chapter that gives an analysis of victim incentives to misrepresent harm suffered if compensation levels are contingent on specific due care levels for victims. It is established that courts’ asking for a higher level of victim care as a prerequisite of the compensation of a high level of harm can in some instances suffice to obtain the first-best outcome. Variation in harm levels implies uncertainty for the injurer if he cannot anticipate the victim type. The book also considers another kind of uncertainty. There are accident contexts in which individuals cannot be sure whether, in an eventual accident, they will be the injuring or the injured party. The car traffic setting is again exemplatory for such contexts. Individuals in such settings are imperfectly informed with regard to the role in an accident. The author discusses similarities between the case with unilateral harm and role-type uncertainty, and the case with bilateral harm. It is established that role-type uncertainty creates bilateral harm in expectation terms, which may create incentives very aligned with those in the case of bilateral harm. The author also discusses possible shortcomings of tort law in internalizing interdependencies between the injurer and the victim. The possibility of injurers having assets less than the harm done, the so-called judgment-proofness problem, is an important matter in this realm. Injurers whose assets fall short of harm do not take the harm level into account when they decide on cost-efficient precaution (Shavell 1986). The contribution in this regard is the result that potentially judgment-proof injurers may actually take more

VI

care than injurers with sufficient assets. This is in contradiction to the discussion in the literature heretofore which has always argued that, if care costs are conceived of as nonmonetary, for example, due to effort of cautious driving, then potentially judgment-proof injurers will choose less care than other injurers. The argumentation in this chapter relies on injurers being risk averse, an assumption less often applied in the literature on the economics of tort law. Moreover, another reason for the imperfect functioning of liability rules with regard to inducing first-best behavior is considered. Injurers might invest in activities which cause lower expected liability but do not change expected harm, which the author labels avoidance activities. In the context of car traffic, for instance, one might think of hit-and-run accidents or the attempt to falsely prove a contributory negligence of the opponent. The opportunity for injurers to decide on an individually optimal mix of avoidance and care in the pursuit of minimal individual costs allows the comparison of the absolute and relative performance of liability rules from a totally new perspective. All in all, this book offers, in a very readable fashion, the derivation of several results which are interesting and of great importance for the further development of the economics of tort law. Laszlo Goerke References Brown, J.P. (1973). Toward an Economic Theory of Liability. Journal of Legal Studies 2: 323-350. Calabresi, G. (1970). The Costs of Accidents: A Legal and Economic Analysis. New Haven: Yale University Press. Kaplow, L. and S. Shavell (1996). Accuracy in the Assessment of Damages. Journal of Law and Economics 39: 191-210. Shavell, S. (1986). The Judgment Proof Problem. International Review of Law and Economics 6: 45-58.

VII

Acknowledgments Unquestioningly, I am tremendously indebted to Laszlo Goerke. He enabled the pursuit of my research interests, was always ready to hand if a piece of advise was needed, opened up the means to participate in the research community, be it via the critical questioning of proposed research or the provision of financial means to meet and discuss with other scholars at conferences of interest, and furthermore went out of his way if some other kind of support was required. The best I can hope for is that I may one day be able to reproduce the environment created by him. Furthermore, I thank Eberhard Feess who acted as second referee and additionally gave support along the way. Willi Kohler completed the commission by chairing the oral examination, for which I am grateful. With regard to guidance at critical junctions, I would like to express my gratitude to Martin Kolmar. This thesis would not be were it not for the unrelenting questioning and support by Florian Baumann. He, the co-author of one of the papers this dissertation comprises, made a lasting impression on the book and me, for which I am very much indebted. Along the way, I had the luck to be the colleague of Nikolai St¨ ahler, Marcus Jansen, and Mario Mechtel. The time spent together, both within and outside of the ivory tower was, and I hope will continue to be, very enjoyable. I would like to thank Thorsten Nestmann for being around as a colleague, flatmate, and companion. In addition, I wish to thank Salvatore Barbaro, Sven Drebes, Magnus Hoffmann, Axel Mießner, Thomas Mohr, Christoph Moser, Normann M¨ uller, Matthias Oschinski, Dorothee Schmidt, Jan Schumacher, Sascha Thielmann, and Michael Wedow for making my time at the University of Mainz and the University of T¨ ubingen one to remember. Last but not least, I would like to thank my family for their constant encouragement and support. Tim Friehe

IX

Table of Contents Chapter 1: Introduction

1

1.1

General Introduction

1

1.2

Contribution

2

Chapter 2: The Economics of Tort Law: Basics and Selected Core Themes

7

1

Introduction

7

2

Basic Economics of Tort Law

9

3

Extensions to the Basic Model

15

3.1

Heterogeneity

17

3.2

Uncertainty

27

3.3

Administrative Costs

33

3.4

Risk Aversion

38

3.5

Judgment-Proofness

43

3.6

Bilateral Harm

49

References

52

Chapter 3: On the Incentive Effects of Damage Averaging in Tort Law

64

1

Introduction

65

2

The Model and Analysis

66

2.1

The Model

66

2.2

Analysis

66

XI

2.3

Discussion

70

2.3.1

Victim Incentives to Comply in the Case of Damage Averaging

70

2.3.2

Information on Type and Magnitude Is Costly

73

3

Conclusion

73

References

74

Chapter 4: On the Superiority of Damage Averaging in the Case of Strict Liability

75

1

Introduction

76

2

The Model

77

3

The Analysis

79

4

Conclusion

81

References

82

Chapter 5: Screening Accident Victims

83

1

Introduction

84

1.1

Motivation and Main Results

84

1.2

Relation to the Literature

87

2

The Model, Benchmark, and Common Court Practice

88

2.1

The Model

88

2.2

The Benchmark

90

2.3

The Common Court Practice

91

XII

3

Screening of Victims

93

4

Screening Victims with Variable Activity

102

5

Conclusion

105

Appendices

106

References

114

Chapter 6: A Note on Judgment Proofness and Risk Aversion

117

1

Introduction

118

1.1

Motivation and Main Results

118

1.2

Relation to the Literature

119

2

Model and Analysis

120

2.1

Non-Monetary Care

121

2.2

Monetary Care

124

3

Conclusion

128

References

129

Chapter 7: On the Similarity of Bilateral Harm and Unilateral Harm with Role-Type Uncertainty

131

1

Introduction

132

2

The Model

134

3

Bilateral Harm and Unilateral Harm with Role-Type Uncertainty

137

Compared 4

The Special Case of Dharmapala and Hoffman

XIII

140

5

Role-Type Uncertainty Can Enable Efficiency

141

5.1

Equilibrium in Efficient Care

142

5.2

No Care Equilibrium at Inefficient Care Levels

144

5.2.1

Consistent Victim Probabilities

144

5.2.2

Victim Probabilities Do Not Have to Be Compatible

146

5.3

Discussion

149

5.4

An Example

150

5.5

Role-Type Uncertainty and other Liability Rules

152

6

Conclusion

154

Appendices

156

References

163

On Avoidance Activities After Accidents

165

Chapter 8:

1

Introduction

166

2

The Model

168

3

Analysis

169

3.1

Strict Liability versus Negligence

169

3.2

Uncertain Due Care Standard

171

3.3

Second-Best Considerations on Negligence

174

3.4

Avoidance and Punitive Damages

177

4

Conclusion

181

References

182

Chapter 9: Conclusion

184

XIV

Chapter 1 1

Introduction

1.1

General Introduction

This book deals with law. Generally, law stipulates rights and duties and thereby often effects the coordination of individual behavior in highly complex societies. In relation to a famous setting, we would judge the need for law to be far less apparent if Robinson were the sole inhabitant of an island. As soon as Friday enters the stage, however, the potential for conflict arises and this obviously allows for a valuable contribution of law. Law can be analyzed from a multitude of standpoints and by using a variety of different methods. This book is a contribution to the law-and-economics literature. The discipline of law and economics uses the economic approach to predict behavioral consequences of norms. In addition, in a normative sense, it offers insight into the content and structure of norms that attain outcomes which serve some given objective. The objective usually considered is efficiency. Thus, law and economics helps to evaluate existing law and to design law inducing efficiency. In doing so, the economic analysis of law comprises analyses on both traditional areas of law, such as torts, contracts, and property, as well as other areas of law. This book contributes to a single branch of this burgeoning field of research, namely the economic analysis of tort law. This area is of undisputed empirical importance. For instance, Dewees et al. (1996) report that 25 percent of all Americans are injured every year and that the costs of injuries in the year 1985 have been estimated to be $182 billion (in 1988 dollars). The analysis of tort law from an economic stance has attracted scholarly attention early on (Calabresi 1970), but still leaves many important questions unanswered. This latter fact and the importance of accident settings for the well-being of so many individuals motivates this study. The most basic function of tort law, at least seen from an economist’s perspective, is the internalization of external effects (e.g., Endres 1991: 2). For instance, if drivers

1

do not have to compensate harm done to cyclists in the event of an accident, they will not consider these losses when optimizing individual behavior. The lack of internalization in accident settings is due to the high transaction costs, preventing internalization via bargaining among parties (Coase 1960). Tort law is one way aiming at and eventually achieving that the driver takes account of a broader set of consequences of his activity. The central interest throughout this study is with individual incentives to take precautionary measures. The most natural conception is that precaution affects the probability of an accident and/ or the magnitude of harm suffered in the event of an accident. For example, the driver of a car can decide whether to use the indicator in all circumstances or to adhere to speed limits. Decisions such as these are likely to influence the probability and/ or the magnitude of the harm in the event of an accident with a pedestrian or a fellow driver. The next section will present a brief sketch of results we offer with respect to individual incentives for precaution.

1.2

Contribution

The book is composed of two parts. The first part comprises a review of the law-andeconomics literature. In that section, we present the stock of knowledge which we took advantage of. The second part encompasses our own analyses. Stated alternatively, we present literature on selected themes in the survey and later on provide analyses that relate to these themes and contributions. The survey comprises not only literature which is intimately close to our analyses but instead allows for a wider focus. A more restricted presentation of literature with a more direct link to the respective analysis at hand will be given before the respective model and analysis are presented in the chapters following the survey. In the following, to set the stage, we will present a sketch of the building blocks of the second part of this book, i.e., our own projects. The outline of these projects will be somewhat crude at this point as its purpose is to give a general idea of what makes up this book. Further details will follow in due course. The basic accident setting involves an injurer and a victim. Both individuals can be 2

characterized by different aspects. One important dimension is the level of harm suffered by a victim in the case of an accident. We pick up the heterogeneity of victims with respect to the magnitude of harm consequent to an accident in ’On the Incentive Effects of Damage Averaging in Tort Law’, ’On the Superiority of Damage Averaging in the Case of Strict Liability’, and ’Screening Accident Victims’. In the first of these, we investigate the efficiency consequences of using an average measure built across different victim types. In practice, harm is assessed in every occasion. This undoubtedly causes administrative costs of significant magnitude. For instance, courts may call on experts to testify on the harm estimate in the case at hand. It has been established that the average can be utilized without negatively affecting efficiency in the case that the injurer is the only party able to affect expected harm and cannot anticipate the type of the victim (Kaplow and Shavell 1996). Our analysis delves into circumstances in which both parties to the accident can affect expected harm and inquires whether the previous conclusion holds for this setting. We find that it is no longer unambiguous which measure gives the average in such a setting. Most importantly, it is established that there is an average measure which indeed induces efficient behavior under the considered circumstances. The section ’On the Superiority of Damage Averaging in the Case of Strict Liability’, in contrast, starts from the basic model in which only the injurer can choose precaution to affect expected harm. However, an additional optimization is introduced, being the victim optimizing over the value of the good put at risk. The possibility of an accident is real and needs to be incorporated when considering the utility that an object of a given value conveys. We show that victims do not follow this efficiency prescription if they are fully compensated for any harm suffered. In contrast, if injurers are obliged to compensate average harm, social interests are better reflected in the individual optimization of affected parties so that efficient precaution and investment choices are induced. The averaging of damages is no longer of interest in ’Screening Accident Victims’, where we inquire into the potential of designing damage awards in a way that makes type-adequate behavior optimal for victims instead. Above, we have already alluded to

3

the fact that victims usually differ with respect to harm and that the accurate assessment of this heterogeneity in court often absorbs considerable resources. Victims are often better informed on the magnitude of harm suffered. It then becomes a question whether this informational advantage can be used by courts. Arguing against the unquestioned use of information provided by victims is the fear that victims might misrepresent the magnitude of harm to increase the compensation. Consequently, it is of interest to search for circumstances in which victims do not have an incentive to misstate the level of harm suffered. It is shown that this desirable property does not require any manipulation of damages away from compensatory damages in a number of cases, i.e., compensating precisely the harm suffered suffices for the self-selection of victim types. For the case in which the circumstances prove this to be insufficient, we specify a simple adaptation of the compensation levels of respective victim types which once again ensures that victims behave according to their type. Speaking of victims which vary in the harm suffered consequent to an accident, the injury an accident inflicts upon a victim is very often of considerable magnitude. As a consequence, the assets of injurers available to compensate the harm done often fall short of the harm itself. This has important effects on the behavior which liability law can instill. In fact, under the standard set of assumptions, injurers who are potentially judgmentproof take less care than individuals who can dispose over sufficient assets. Individuals who are potentially judgment-proof externalize the part by which harm exceeds assets. In ’A Note on Judgment-Proofness and Risk Aversion’, we establish that this may no longer hold true if risk aversion of actors is taken into account. The paper shows that potentially judgment-proof injurers who are assumed to be risk averse may very well take more care than affluent individuals. Thinking of real-world accidents, one comes across occasions not only in which the injurer is unable to compensate the harm due to limited assets but also those in which both parties suffer harm. The standard accident model assumes that one party, the injurer, inflicts harm upon another party, the victim. In many accident contexts, it is more descriptive of reality to acknowledge that both parties to the accident suffer harm. For

4

instance, every car accident usually entails that both cars are damaged. The literature has argued that such situations can be disaggregated into two different lawsuits. Incentives thus created are similar to incentives in the simple setting. In ’On the Similarity of Bilateral-Harm and Unilateral Harm with Role-Type Uncertainty’, we provide a characterization of the similarity taking account of role-type uncertainty. i.e., that individuals may be uncertain as to their role in an accident. In fact, a recent contribution to the literature has identified a circumstance in which the incentives of the bilateral-harm setting are markedly different from the unilateral-harm framework (Dharmapala and Hoffmann 2005). We point out that this contrast can disappear if the unilateral-harm framework is enriched by role-type uncertainty. Besides assuming that only one party suffers harm in the event of an accident, another simplifying assumption of the standard framework is that given victims have a righteous claim they will always sue and be satisfied by a court judgment. This does not need to hold for a multitude of reasons. For instance, the injurer might want to invest resources into reducing the probability that the judge will eventually decide in favor of the plaintiff. This can be achieved, for instance, by hiring a lawyer who makes up a story casting doubt on the identity of the actual injurer. In ’On Avoidance Activities After Accidents’, we allow for such investments and investigate into the effects of this change in assumption on various aspects commonly considered in the economics of tort law.

References Calabresi, G. (1970). The Costs of Accidents: A Legal and Economic Analysis. New Haven: Yale University Press. Coase, R. (1960). The Problem of Social Costs. Journal of Law and Economics 3: 1-44. Dewees, D., Duff, D. and M. Trebilcock (1996). Exploring the Domain of Accident Law. Oxford: Oxford University Press. Dharmapala, D. and S.A. Hoffmann (2005). Bilateral Accidents with Intrinsically Interdependent Costs of Precaution. Journal of Legal Studies 34: 239-272.

5

¨ Endres, A. (1991). Okonomische Grundlagen des Haftungsrechts. Heidelberg: Physica Verlag. Kaplow, L. and S. Shavell (1996). Accuracy in the Assessment of Damages. Journal of Law and Economics 39: 191-210.

6

Chapter 2 The Economics of Tort Law: Basics and Selected Core Themes 1

Introduction

Life is pervaded by risks of suffering harm to property or personal well-being. An obvious example in everyday life is commuting to work; be it by car, train, or bicycle. Often, discretion to reduce the probability of harm is outside the personal realm; instead it lies in the hands of others. For instance, the probability of suffering harm as a cyclist in an accident is very much affected by the way fellow citizens drive their cars. Tort law deals with situations such as this one. Specifically, it addresses relations between people that are not regulated by private agreement due to high transaction costs broadly conceived (Cooter and Ulen 2004: 310).1 In contexts of high transaction costs, the allocation of legal entitlements is critical for efficiency (Coase 1960). However, whether tort law purports to aim at efficiency is open to question. Tort law is frequently said to serve two purposes; the compensation of victims and the inducement of deterrence (e.g., Shapiro 1991). Between these, the focus of economic analysis is more on deterrence, whereas legal commentators, especially from Europe, tend to give more weight to compensation (see Adams 2002: 140 and Sch¨ afer and Ott 2005: 125). Deterrence in this context comprises the allocation of resources to reduce the expected harm of accidents. Thus, we will mainly concentrate on the effects that different legal rules have on the choice of precaution, or on the primary costs of accidents in Calabresi’s terminology.2 Calabresi (1970) introduced the distinction between primary, secondary, 1 Modern tort law also encompasses product liability and medical malpractice, and thus relationships in which parties have agreed on a contract. Still, the cost of negotiating with every customer (patient) about risk-reducing precautions is presumably prohibitive in many areas. Furthermore, it is often argued that the imperfect information of customers (patients) on risk is a factor lending importance to liability in these areas (see, e.g., Sch¨afer and Sch¨ onenberger 2000). 2 That tort law indeed influences care choices has been confirmed empirically. See, e.g., Sloan et al. (1994), K¨ otz and Sch¨ afer (1993).

7

and tertiary costs, all of which are affected by the tort law regime. Primary costs comprise the sum of accident avoidance measures and harm due to the accident. Secondary costs arise due to risk-bearing of risk-averse parties, whereas tertiary costs are due to the administration of the liability system. Except for one contribution, we will focus on risk-neutral individuals, leaving the sum of primary and tertiary costs as an objective function, the minimization of which can be interpreted as wealth maximization from a planner’s point of view. Deterrence and compensation may also be achieved using other policy means such as regulation or insurance. However, we focus on a delineation of the economics of liability law, and will not enter into a comparison of policy instruments (for this see, e.g., Shavell 1993, 2007a, Innes 2004).3 The rest of this survey is structured as follows. In Section 2, the very fundamentals of the economic analysis of tort law will be laid out. The framework on which this analysis builds is simple and delivers strong results. Furthermore, the model of precaution presented is sufficiently general to also be applicable in matters dealing with contract or property problems (Cooter 1985, 1991). Given the numerous assumptions of the basic setting, much attention of the ensuing literature was on the robustness of earlier results under relaxed assumptions. We will discuss several avenues of these extensions and variations in Section 3. The selection of topics is justified by our own contributions in the background. Thus, this discussion comprises an elaboration on the effects of (i) heterogeneity, (ii) uncertainty, (iii) administrative costs, (iv) risk aversion, (v) the limited ability to pay for harm caused, and (vi) bilateral harm. As said, each strand of the literature we cover bears importance for the analyses to follow and is thus presented in some detail. Within each of these sections we aspire to at least refer to the analyses most important for the respective theme. However, due to the richness of the literature on tort law, our coverage is admittedly selective and also shows in our neglect of the empirical contributions to the literature. The presentation in Section 3, however, comprises references to our own analyses and thereby helps to locate our work within the field. 3 Consequently, our exposition also does not include an analysis of the appropriate domain of tort law, i.e., whether it should be more or less embracing (see, e.g., Sch¨ afer 2000).

8

2

Basic Economics of Tort Law

In this section, we first present the simple framework and then continue to discuss variants, which can likewise be considered canonical. To structure the subsequent exposition, the sequence of which is similar to that in, e.g., Shavell (2007a), we briefly glance at the different classes to be discussed (see Table 1). In general, the model comprises activity and care levels of two individuals, being the potential injurer and the potential victim in an accident. The most basic version of the model treats only injurer care as the choice variable. This will be dealt with first. We then go on and introduce variable victim care. At this point of the discussion all standard liability rules, which are the tools of liability law, will have been introduced. Finally, the model is enriched by endogenizing activity levels. There are different ways to set up the framework once activity as well as care are variable. For instance, there are unilateral care and activity frameworks, in which only the injurer actively chooses behavior, whereas the most extensive model is that in which injurer and victim optimize concerning both variables. Classification of Model Unilateral Care Bilateral Care Care and Activity

Injurer Care Variable Variable Variable

Victim Care Fixed Variable Variable/ Fixed

Injurer Activity Fixed Fixed Variable/ Fixed

Victim Activity Fixed Fixed Variable/ Fixed

Table 1: Basic Models

The results to be presented build on the early findings of Calabresi (1961, 1970) and Posner (1972). Early contributions utilizing formal reasoning are Brown (1973) and Diamond (1974). Shavell (1987) is the first to present the fundamentals of the economic analysis of tort law comprehensively. The most basic model involves two risk-neutral individuals: the injurer and the victim who undertake some activity of fixed extent. The victim could potentially suffer harm in an accident. It is further assumed that the harm to be suffered with some probability is equally a private as well as a social harm.4 The probability of the accident can be lowered 4 It is often argued that if harm is purely economic, e.g., manifested in a loss of earnings, then there is no social harm and thus no need for compensation. See Dari Mattiacci and Sch¨ afer (2007) for a recent discussion.

9

by injurer precaution. A further assumption of the basic model is that there is no factor other than the behavior of the respective parties which might provoke harm to occur. The precaution of the injurer, which lowers expected harm, is costly for the injurer. In this regard, it is generally assumed that the function representing the precaution costs (accident probability) is (strictly) convex. The social objective in this context is the minimization of the sum of precaution costs and expected harm. The problem is specified in such a manner as to yield a unique costminimizing care level, leveling the cost of a little more precaution (marginal cost) with the resulting reduction in the expected harm (marginal benefit).5 Without any intervention, the injurer takes no care. The injurer knows about the cost of precaution, whereas the benefit manifested in the reduction of expected harm is external to her optimization. Tort law relies on private initiative to achieve its end. Given there is harm, victims must file a lawsuit against the tortfeasor in order to obtain compensation.6 In the basic framework, there are no administrative costs of the liability system. One obvious and important consequence of this assumption is that victims will not be deterred from filing a suit by litigation costs. The court will then test whether indeed a harm has been caused and whether the defendant is legally responsible.7 Legal responsibility may either comprise causation only or causation combined with the breach of a duty; which of these applies depends on the liability rule. A liability rule allocates the burden of harm in a way that may be contingent on the behavior of parties. It is assumed that damages paid by injurers equal the harm suffered by victims and that injurers command over sufficient assets to be able to actually compensate that harm. The behavior usually taken into consideration when allocating the burden of harm is the level of precaution. If the rule is no liability, the harm lies where it falls, i.e., the injurer is never made to compensate the 5 Cooter and Porat (2000) point out that courts often count only the reduction in the expected harm to others as marginal benefit and thereby leave out what may be a large chunk of the total marginal benefit, the reduction in self-risk. Leong (1989) is an earlier contribution along these lines. 6 Compensation usually requires that harm has occurred. However, the debate also concerns whether the exposure to the risk of harm should not suffice in some instances for a claim (see, e.g., Miceli and Segerson 2005). 7 Compensation usually requires that harm has occurred. However, the debate also concerns whether the exposure to the risk of harm should not suffice in some instances for a claim (see, e.g., Miceli and Segerson 2005).

10

victim. Application of that rule obviously does not help to internalize the social benefits of care. Thus, the injurer chooses no care. The mirror image of the rule no liability, i.e., the injurer compensates the victim irrespective of care taken, is denominated strict liability (SL). In this case, the objective function of the injurer coincides with that of society. As a consequence, the care chosen by the injurer is socially optimal. Finally, simple negligence (SN) is a liability rule that requires full victim compensation as long as injurer care falls short of a defined standard, the duty referred to above, whereas care equal to or higher than the standard relieves the injurer from the burden of harm. If the care standard is equal to the socially optimal level, the injurer will adhere to due care because injurer costs are minimized by this choice. This result obviously presupposes not only that the court has the information required to calculate socially optimal care and observe care taken but also that the injurer is aware of the precise expected harm and standard of care. There is an alternative conception of the negligence rule which states that only the share of harm actually caused by the negligence is to be compensated if injurer care falls short of the standard, which also induces socially optimal care, given the assumptions of the basic framework.8 Many if not most accident contexts are more appropriately described by allowing both parties to the accident, injurer and victim, to influence expected harm via care.9 It is possible that either both parties are in a position to take care and only precaution is desired by one party, or that both parties should actually take precaution. Regarding the first type, that of alternative precaution, it is usually argued that the burden of the expected harm should be imposed on the ’least cost avoider’, the individual who can prevent the accident with the least cost of care (Calabresi 1970).10 For instance, Cooter and Ulen (2004: 330), who call this setting one of redundant precaution, provide the following 8 Grady (1983) and Kahan (1989) consider this version of negligence. However, Shavell (2007a), for instance, is very hesitant concerning this conception as he doubts that information on harm given due care is readily available. Cooter (1991) similarly states that courts usually do not undertake the discount by the expected harm at due care. Van Wijck and Winters (2001) extend the alternative conception to variable activity and consider its social welfare implications, whereas Singh (2004) allows for bilateral care. 9 Coase (1960) already alludes to the fact that settings such as those of interest here are of a reciprocal nature, implying the the exclusive focus on the injurer is often misleading. 10 However, see Dari Mattiacci and Garoupa (forthcoming) for a discussion of the disadvantages of this approach in the presence of imperfect information.

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example: ”the manufacturer and the homebuilder can check electrical wire for defects, but the manufacturer can check at less costs than the homebuilder”. For the most part, analyses use the second way to reflect that both parties have an effect on expected harm, i.e., the bilateral-care model. This can be incorporated into the model straightforwardly. The victim care causes costs and affects expected harm in the same way as outlined for injurer care. Concerning the interaction of injurer and victim care, it is often assumed that they are substitutes, that is, the presence of more of one type of care reduces the effectiveness of the other type of care with respect to the reduction in expected harm at the margin. Importantly, the strategic interaction between individuals is usually captured by a non-cooperative game in which choices are made simultaneously.11 Furthermore, the game between injurer and victim is played only once.12 The minimization of social costs yields an optimal level of care for both types. In this bilateral-care setting, neither no liability nor strict liability can induce the efficient outcome since there is always one party without concern for the expected harm who accordingly chooses no care to minimize individual costs. Yet, if a defense of contributory negligence is added to the rule of strict liability (SLCN), requiring compensation of the harm only if the victim complies with due 11 Although few in number, there are also studies which consider respective care levels being chosen in sequence. Shavell (1983) provides a discussion of the case of a sequential structure to the choice of care. It is assumed that the party moving second can observe the care choice of the other party. For the case when the victim moves first, it is established that negligence is inferior to rules utilizing the defense of contributory negligence. The reason is that simple negligence makes taking care dominant for the injurer, who moves last, whereas taking care is optimal under strict liability (negligence) with a defense of contributory negligence only if the victim has taken care. If the injurer moves first and the victim moves second, only negligence guarantees the efficient outcome, whereas the other rules might fail to do so. Winter (1994), instead, assumes that the party moving second, the injurer, cannot observe the care choice of the party moving first, the victim. Due to the fact that the second party is called on to move only if a dangerous condition has occurred, this model is different from the simultaneous-move framework. In this setting, a ranking according to social welfare of the two liability rules with a defense of contributory emerges, with strict liability with a defense of contributory negligence coming out ahead, whereas the comparison with simple negligence is ambiguous. In another direction of inquiry, Endres (1992), assuming strict liability with a defense of contributory negligence, considers the case in which injurers move first and the choice of care is perfectly observable, while care standards are not equal to the respective first-best care level. He, for instance, shows that the sequential structure may yield results superior to the simultaneous-move setting. 12 Winter (1997) is a notable exception in this regard. He considers the consequences of a repetition of the game where individuals learn at the beginning of each period whether they are injurer or victim. One of his findings is that strict liability (no liability) may attain the efficient outcome in this setting although it is deficient in the one-period framework because, given the uncertainty on the role one plays in an eventual accident, it pays for sufficiently patient individuals to take care even as a victim (injurer) if the probability of being an injurer (victim) is high enough. The last requirement is intuitive since the benefit of victim care, being the reduction in the accident probability, is only enjoyed in the shoes of the injurer, given strict liability, for example.

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care, the efficient outcome results as unique equilibrium as long as due care equals the efficient level.13 Similarly, negligence and negligence with a defense of contributory negligence (NCN) attain that outcome, given the same prerequisite. These last two rules differ only in the event of substandard care by the injurer and the victim. In that circumstance, the injurer will be required to provide compensation only under simple negligence.14 Another negligence rule that attains efficient care in equilibrium if the standards are equal to the socially optimal level is comparative negligence (CN).15 This rule is different from those mentioned. It apportions liability according to relative fault if injurer and victim care fall behind due care, whereas the others allocate in an all-or-nothing manner.16 We summarize the different standard liability rules in Table 2. Liability Rule SL SN SN NCN NCN NCN NCN SLCN SLCN CN CN CN CN

Injurer Care Not decisive At fault Faultless At fault Faultless Faultless At fault Not decisive Not decisive At fault Faultless Faultless At fault

Victim Care Not decisive Not decisive Not decisive Faultless At fault Faultless At fault Faultless At fault Faultless At fault Faultless At fault

Liability Allocation Full injurer liability Full injurer liability Injurer not liable Full injurer liability Injurer not liable Injurer not liable Injurer not liable Full injurer liability Injurer not liable Full injurer liability Injurer not liable Injurer not liable In proportion to negligence

Table 2: Standard Liability Rules

All of these rules attain the efficient outcome by utilizing (at least) one threshold regarding care which allows the party at which it is directed to be free of the expected harm burden (Miceli 2004: 48). Or, to be more precise, they share the following properties: 13 Endres and Querner (1995) establish that there might only exist a mixed-strategy equilibrium if the standard deviates from the efficient level. 14 This is seen as a potential advantage of negligence with a defense of contributory negligence because this decreases the number of suits, which is desirable as soon as we allow for administration costs (Dari Mattiacci forthcoming). 15 See, for example, Curran (1992) on the empirical importance of comparative negligence nowadays. Rea (1987) analyzes comparative negligence in the case of simultaneous and sequential care, and when some individuals are not receptive to incentives. 16 The argumentation for the results outlined above is more elaborate in, e.g., Sch¨ afer and Sch¨ onenberger (2000).

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(i) given negligence by one party, the non-negligent party is completely free from liability, and (ii) when both parties are non-negligent, the whole accident burden falls on one party for all combinations of care. It can be shown that these two properties are sufficient for attaining the efficient care equilibrium under the usual assumptions (e.g., Kim 2004).17 Recently, it has been of interest whether liability should be shared given that tortfeasor and victim have complied with any existent due care standards.18 However, we will keep to the standard liability rules, which assign the liability wholly under that circumstance. Another important variant of the basic model allows for variable activity. The model now allows both parties to choose care as well as activity level. The activity level is generally conceptualized and modeled as the number of times one engages in a given activity (Shavell 1980). In an illustrative example, the attentiveness while driving is precaution taken and the number of miles driven is the activity level chosen. Activity may, however, be interpreted more broadly as all those variables which are at the discretion of individuals but are not controlled by a due care standard, where the lack of incorporation into the vector of controlled activities may be due to administrative costs (see, e.g., Dari Mattiacci 2005). In this framework, the socially optimal care and activity levels maximize the benefits due to the activity less the precaution and expected harm, i.e., net benefits, where it is usually assumed that respective benefits due to the activity can be represented by a strictly concave function and that the costs of precaution and expected harm are proportional to the level of activity. The central result in this regard is that there is no liability rule which ensures the efficient outcome (Shavell 1980). To attain efficiency, double responsibility at the margin is necessary (Cooter 1985). However, due to the threshold characteristic, no liability rule holds this property. Only the residual bearer internalizes all marginal benefits and marginal costs resulting from her choices. Take, for instance, simple negligence. As long as the injurer complies with the standard of care, he is free from any liability and will thus not take into account the effect her activity bears on expected harm. Excessive injurer activity is a consequence. Therefore, in order 17 Indeed, under more restrictive assumptions, it can be shown that property (i) is necessary and sufficient for efficiency. See Jain and Singh (2002) and Kim (2004). 18 Calabresi and Cooper (1996) ignited the discussion, to which Parisi and Fon (2004) and Singh (2006) contributed importantly.

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to attain the efficient outcome in this generalized setting, more than the imposition of a liability rule is needed.19 The general framework comprising the activity and care level of injurer and victim as choice variables can be modified so that some variables are taken as exogenous. For instance, Shavell (2007b) shows in a model in which only the injurer chooses care and activity that negligence will never discourage socially desirable activity even if standards are excessive, whereas regulation can do this. In that setting, it has also been established that strict liability is usually superior to negligence with respect to activity, since the latter does not impose expected harm on the injurer, which provokes excessive activity (e.g., Polinsky 1980). Recently, Kim (2006) established that this ranking of strict liability and negligence in terms of social desirability may be reversed if the activity of the injurer entails positive externalities. Since the injurer internalizes only the private benefits under strict liability, suboptimal activity may be the consequence. Under negligence, the attractiveness can be steered by designing the due care standards in an appropriate fashion, which, however, is detrimental to the efficiency of care choices.

3

Extensions to the Basic Model

Further on, we will elaborate on extensions of the basic model that bear importance to our own analyses. In the first section, models that allow for heterogeneity will be dealt with. The heterogeneity considered applies to the types of injurers or victims. For instance, victims are allowed to differ from one another when it comes to the magnitude of harm. This is a heterogeneity which bears particular importance in our own analyses. If individuals cannot foresee with which type the accident may occur, heterogeneity also implies uncertainty. However, the section actually entitled ’Uncertainty’ is the one consecutive to the one on heterogeneity. This can be explained by the fact that we speak of uncertainty different from that arising from heterogeneity in Section 3.2. The aspects we turn to in that section are not characteristics of the parties to the accident such as costs of care or 19

For an approach including fines, see, e.g., Finsinger and Pauly (1990), Goerke (2003).

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the like. Instead, it is considered (i) that injurers do not know the risk associated with a given activity, (ii) that injurers cannot pinpoint one care level as the applying standard of care, (iii) that the causation of the accident is uncertain since there are factors other than the activity of the injurer which contributed to the occurrence of the accident, and (iv) that the enrolment may be indeterminate, i.e., that a party to the accident may have a positive probability for being the victim and a positive probability for being the injurer in an accident. Factors such as heterogeneity of victims or uncertainty about the causation complicate suits. The assumption that administrative costs do not exist thereby becomes even less appealing than in the absence of such factors. As stated above, these costs are abstracted from in the basic model. It is, however, of obvious importance to consider how results are affected by administrative costs. Consequently, we broaden the setting and acknowledge the importance of administrative costs already mentioned in the triad of Calabresi’s goals for the liability system. After the discussion of tertiary costs in Section 3.3, we turn to secondary costs, i.e., costs associated with risk-bearing. In analyses on tort law, the focus is on risk-prone activities. Consequently, in the absence of insurance, at least one of the parties to the accident will be exposed to an income level contingent on the state of the world, i.e., whether the accident occurs or not. In the basic setting as well as most analyses till this day, individuals are assumed to be risk-neutral. Consequences of relaxing this assumption will be delineated in Section 3.4. Whereas individuals being risk-averse introduces a new dimension along which to compare different allocations, riskbearing costs, the consideration of insufficient assets to compensate harm questions the attainability of efficient care in the simpler setting. Obviously, individuals often do not have funds sufficient to fully compensate harm done; interesting and practically relevant complications arise therefrom. We lay out the consequences for efficiency that have been established in the literature in Section 3.5. Finally, we briefly elaborate on the case in which the accident setting is one in which both parties to the accident, instead of one party, suffer harm in the accident contingency.

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3.1

Heterogeneity

The basic setting assumes homogeneous injurers and victims. However, heterogeneity may be considered with respect to several aspects of the basic model. Heterogeneity frequently complicates matters. The first-best outcome will often call for type specificity of care taken, for example, which is often difficult to implement due to personal characteristics being either private information or not precisely foreseeable ex ante. In this section, we will discuss contributions that analyze the relative performance of liability rules given heterogeneity of actors and studies which consider ways in which the rules are or need to be adapted to deal with heterogeneity in an optimal fashion. The dimensions along which we will allow for heterogeneity are costs of care, levels of wealth, harm estimates, and heterogeneity in the actual harm level. Consequently, our discussion comprises the obvious factors with a bearing on the analysis.20 The aspects will come up in our elaboration in the order introduced before. Individuals are likely to differ in their costs of taking a certain amount of care.21 For instance, being an attentive pedestrian is likely to be less expensive for middle-aged individuals than it is for children or the elderly. In the unilateral-care model, it holds that the optimal level of injurer care is a decreasing function of the marginal costs of care. Consequently, injurers with different costs of care should be held to different due care levels from the stance of efficiency. To be able to do that, courts need to be able to observe the injurer type, i.e., care costs, which often proves prohibitively costly to do. This is one of the reasons why law usually does not try to make care standards type-specific.22 Instead, one level labeled the reasonable-person standard is applied to all injurers.23 As a consequence, care taken no longer displays the variation optimal for deterrence. Instead, all injurers with low and moderate costs of care exert standard care, whereas injurers with 20 Another heterogeneity that comes to mind is in the effectiveness of care. However, effects are comparable to the case in which the costs of care are allowed to vary (Shavell 1987: 73). 21 The capture of total care costs usually is constant marginal costs times the level of care. Consequently, differences in the level of costs for a given level of care are in such a setting synonymous with higher marginal costs of care. 22 Another reason might be the equal protection clause (e.g., Emons 1990a). 23 On this, see, for instance, Diamond (1974). Miceli (1997: 25-27) provides a brief and precise textbook discussion.

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high costs of care take first-best care, as taking standard care proves too costly for this group. Yet, there is an economic rationale underlying the legal practice.24 The preceding discussion on the reasonable-care standard related to the standard conception of negligence. Singh (2004) highlights that the alternative conception of negligence, in which a negligent injurer compensates the victim only to the position in which she would be, given due care by the injurer, i.e., bearing positive expected harm, can attain the efficient outcome in the unilateral-care framework. This can be reasoned as follows. As Kahan (1989) emphasized, the restated negligence rule induces efficient care in the standard framework even if the standard of care is excessive. The injurer needs to compensate the difference in expected harm which is due to her deviation from due care. In that case, the marginal effects of the individual problem are equal to those of the social problem. The reason is that the reduction in the expected liability payments, which amounts to expected harm at due care, is a fixed component of the injurer’s objective function. Hence, injurer costs are minimized at efficient care. The application to the case of varying costs of care is an intuitive transfer, where the standard set is that of the individual with the lowest costs of care, implying its excessiveness for all other types. Returning to the standard conception of negligence, Miceli (2006) revisits the question of varying costs of care and negligence. Using the unilateral-care model with discrete injurer types who differ in costs of care, he tries to attain the first-best with a variant of simple negligence. The liability rule which attains this goal can be specified as follows: The standard of care which frees from liability is the efficient care level of the injurer with the lowest cost of care. Besides, there is an array of care levels which are all optimal for some injurer type and imply a different level of liability. Smaller care levels imply some liability, which increases gradually as care taken falls and eventually attains the full level of harm. Taking due care ought to be optimal for the injurer with lowest cost. The gradation of liability borne by other types results from the incentive compatibility constraints of respective types. For instance, consider the type with the second lowest 24 Landes and Posner (1987: 126): ”The allocative costs of forgoing individual standards of care are undeniable but must be compared with the costs of ascertaining each individual’s due care level, an information cost. Information costs are real costs and it may be efficient to give up the allocative gains of an individualized standard in exchange for a reduction in them.”

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costs of precaution. If the liability contingent on taking the level of care which is socially optimal for this type, i.e., care smaller than due care, is too high, this injurer type might very well increase care to the level appropriate only for the type with the lowest care costs. This can be optimal for the individual because the additional costs of care caused by the increase in the care level is less than the expected liability avoided by the increase in care. Consequently, the liability which makes this type choose optimal care for the given type might have to be lower than full harm. The argumentation for other types follows analogously, always preventing that injurer types behave as intended for others by the appropriate utilization of liability levels. Turning from the unilateral-care model to the one, in which both parties to the accident take care, while maintaining the assumption that individuals are different with respect to costs of care, the analysis becomes more complicated. Green (1976) analyzes heterogeneity in costs of care in a bilateral-care framework. In Section 2, we have delineated that liability rules which can induce the efficient outcome in the basic bilateral-care setting direct a due care standard at at least one party to the accident. One of the recommendations by Green given heterogeneity in costs of care-taking is to direct the uniform care standard at the side of the accident which displays less differences in costs, supposing these are not too great. Since the other side bears expected harm and chooses optimal care given the choice of the other side, this may approximate the first-best outcome reasonably well. Stated alternatively, the side to the accident which is vastly heterogeneous is induced to act according to individual characteristics, whereas the other side to the accident will not necessarily translate individual differences into differences in behavior, but the latter is not too drastic for social costs as long as heterogeneity on that side to the accident is moderate. In contrast, Rubinfeld (1987), allowing for heterogeneity in care costs and a bilateral care choice, emphasizes another issue. He argues that this setting can provide a reason for the relative superiority of comparative negligence over other standard liability rules. The intuition behind this assessment lies in the fact that comparative negligence can smooth the discontinuity of expected damages present in the case of simple negligence, given that injurer and victim standards are set rather high, and thereby provide more

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incentives to deviate from due care. Individuals with very low care costs meet even the exaggerated standard. In contrast, most individuals prefer being comparatively negligent to spending the amount required to meet the exaggerated standard. The gradation in liability relative to comparative fault, which is characteristic for comparative negligence, effectuates that there is a price for a given reduction in liability payable in units of care. Individuals with higher care costs ’buy’ less reduction so that, in sum, a schedule of care choices results.25 Bar-Gill and Ben-Shahar (2003), however, argue that this feature is not restricted to comparative negligence, but other negligence rules might as well display this characteristic. Approaching the theme from a more general mechanism-design perspective while considering endogenous choices by injurer and victim, Emons (1990a, 1990b) and Emons and Sobel (1991) find a liability rule which only depends on actions taken, i.e., which is not type-specific, and allows for the attainment of the first-best even though individuals are not identical. This rule, however, also entails punitive damages which makes some individuals expect to be better off after an accident has occurred. Such punitive damages result for the following reason. For instance, if two types who are not supposed to bear any expected harm, due to their choice of the respective endogenous variable, meet in an accident, one party still has to bear the harm caused by the accident. An expected harm of zero for this party is nevertheless attained as it will be abundantly compensated if involved in accidents with other types. This party will thus obtain punitive damages in these latter instances. In acknowledgment of this shortcoming, the analysis continues to show that there still is another liability rule superior to that of applying a reasonable-person standard to all non-identical individuals. For a brief rationale for why the negligence rule with the uniform standard can be improved upon, suppose the following. There are two injurer types, denoted A and B, non-identical victims, and the reasonable-person standard s is too high (low) relative to the socially optimal level of A (B). Now, if the rule is changed so that the care level which guarantees no liability is increased somewhat to s + , and a range of care levels [s, s + ) is allowed for which 25

This logic runs in principle parallel to that in Miceli (2006).

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entails only some liability before full liability takes effect, then injurer type A continues to choose s, whereas type B increases her care to s +  to avoid any liability.26 For a very small , there are no noteworthy repercussions of this proceeding on the victims’ choices so that it only counts that the care level of injurer type B has been corrected somewhat for welfare. Both of the liability rules outlined, the one which instills the first best and the other one which improves upon the reasonable-person standard, attain their outcome by utilizing loss sharing between parties. The preceding exposition, in large part, examined the appropriate level of care and the possibility to induce it, given heterogeneity of care costs. Another dimension along which individuals realistically differ is their level of wealth. At this point in the discussion we are, however, not yet concerned with cases in which the individual level of wealth does not allow for making the victim whole. This aspect is discussed on its own in Section 3.5. What is immediately of interest is the fact that varying wealth levels might imply varying marginal utility of income, once we deviate from the risk-neutrality assumption. As a consequence, heterogeneity along the wealth dimension might justify a discussion on the appropriate level of care for a given level of wealth.27 Indeed, Arlen (1992a) argues that efficient care increases with the wealth of the injurer despite the availability of actuarially fair insurance. In response, Miceli and Segerson (1995) show that it often holds that efficient care does not dependent on the level of wealth even though individuals are not risk-neutral. The independence holds for the case in which there is a way to redistribute wealth among individuals irrespective of the liability system. If such a scheme is missing, it might be that the choice of care also plays a redistributional role, in which case efficient care, in fact, depends on wealth.28 A discussion on the role of legal rules regarding the redistribution of income is provided by, e.g., Kaplow and Shavell (1994, 2000). Before we turn to the case where harm levels are heterogeneous, which is an assumption that plays a prominent role in many of our own analyses and is of unquestionable 26

Note that this procedure again runs in analogy to Miceli (2006). Undoubtedly, if we were to take wealth maximization (cost minimization) as the objective given riskneutral individuals, as we mostly do, we do not need to consider this variation in wealth when deciding on efficient care. 28 However, note that there are circumstances in which even though the space of policy instruments is restricted, efficient care results as independent from wealth (Miceli and Segerson 1995). 27

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importance when turning to the application of the accident model to real-world contexts, we discuss a paper that deals with a related question. Polinsky (1987) considers the case in which there is actually only one level of loss, known to the court, but where injurers err in estimating this magnitude, having a point estimate from a range of loss estimate values. Hence, the case considered is one of heterogeneity in the harm estimate, given homogeneous victims. In the model, injurers distinguish themselves in that they derive different gains from the undertaking of the activity. For the setting, it is shown that some adjustment to compensatory damages is usually optimal to balance underdeterrence and overdeterrence, where it cannot be determined generally whether an upward or downward adjustment is asked for. There is underdeterrence (overdeterrence) since some individuals who underestimate (overestimate) the loss engage in (refrain from) the activity, although this is not justified given the true loss. Furthermore, it is argued that strict liability is preferable to negligence under these circumstances since it allows for both adjustments to have an effect on injurer behavior, whereas behavior under negligence is not affected by an upwards adjustment of damages from the level of compensatory damages. The reason for this lies in the fact that the upward change in compensation does not affect behavior given the same negligence standard. Further on, we will consider analyses that focus on heterogeneous harm levels among individuals. The actual magnitude of harm is often not foreseeable for the injurer or may even be best considered as private information. In the basic model, there is one level of harm. However, in reality, the consequences of a given accident greatly depend on the individual. Take for instance a broken leg due to a car accident. The harm due to the temporary disability caused by the accident if the victim is a professional tennis player will presumably be a multiple of the harm if the victim is an office worker. Take the unilateral-care model and that the injurer can anticipate the harm level. Then, strict liability will only induce efficient care only if damages transferred are equal to the harm suffered. The efficient care level very intuitively increases with harm. This follows in the basic model from marginal costs independent of the harm level and a marginal benefit which increases in the level of harm. In the

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case of negligence with due care set at the efficient level, injurers will take optimal care even for anticipated harm levels that are not equal to the actual one. The injurer will certainly abide by due care if she expects the harm to be greater than it is, and will also fulfill the standard of care as long as she does not expect harm to be drastically below the actual level.29 If the injurer cannot anticipate the precise level of harm, but knows about possible harm magnitudes and their respective probability, efficiency does not depend on whether the compensation to be paid by the injurer in the event of fault under negligence or irrespective of care under strict liability equals average harm or precise harm in every occasion or not. The choice between the two compensation measures is irrelevant to the injurer optimization. Kaplow and Shavell (1996) argue, based on this finding, that the costly assessment of harm should often not be undertaken in court.30 The additional costs entail no social benefit as the care decision is unaffected and the fact that accurate assessment allows for making victims whole is not considered as a value in itself. Certainly, this assessment needs to be qualified once injurers are in a position to learn about the accident risk. Given compensation according to average harm, there is no incentive to acquire information. In Chapter 3, we work ’On the Incentive Effects of Damage Averaging in Tort Law’ by introducing bilateral-care into the setting considered by Kaplow and Shavell (1996). The consequences of averages as damage measure are critically dependent on the weighing of respective harm levels. This implies that some measures which may be conceived of as damage average distort incentives. Importantly, we establish that there is an average measure which allows for the attainment of efficient care in the bilateral-care framework. This measure is particular in that it incorporates the effect of care on the accident probability at the margin. The analysis concentrates on incentives to take care and therefore does not incorporate the information acquisition incentives. Whereas most studies as ours 29 Endres (1989) analyzes the case of misinformation concerning expected harm and its consequences under negligence and strict liability with a defense of contributory negligence. On this matter, see also Singh (2003). Note that in the context of errors in the assessment of damages, the way negligence is specified plays a critical role (Kahan 1989). 30 Interestingly, compensation of expected harm can be observed in practice. For instance, §287 of the German Zivilprozessordnung gives discretion to the judge if the evidence of the harm is unreliable (Sch¨ afer 2005).

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in Chapter 3 try to work out whether damage averaging is worse than accuracy from an efficiency perspective, the analysis in Chapter 4 establishes that there are circumstances in which accuracy is clearly inferior to damage averaging. The paper ’On the Superiority of Damage Averaging in the Case of Strict Liability’ endogenizes a new behavioral dimension. As said, for the case of strict liability and unilateral care, the literature argues that if injurers cannot anticipate the precise level of damages courts might use the damage average without distorting care incentives. Our contribution shows that the use of the damage average is in fact preferable if one allows victims to choose the object at risk. If the court were instead to insist on the accurate compensation of the harm suffered, this would induce victims to choose an inefficiently high object value. In contrast, restricting the compensatory transfer to the damage average allows to instill the first-best outcome. Besides information acquisition of a given injurer ex ante as considered in Kaplow and Shavell (1996), there are other arguments for assessing the harm accurately in every case. The fact that courts indeed usually strive to make the victim whole may entail positive effects in an intertemporal perspective. Hua and Spier (2005) argue that litigation is an instrument for enabling other injurers to learn about a given victim, since the victim may suffer similar harm in future accidents involving different injurers. Consequently, upon the updating of beliefs, injurers can fine-tune their care choices. Such fine-tuning of care may also be achieved via the due care standard. Feess and Wohlschlegel (2006), using a unilateral-care model, consider a scenario in which the accident probability is determined by care and a factor independent of care, about which only some injurers are perfectly informed. The court and other injurers are only familiar with the distribution of this determinant, larger realizations of which decrease the accident probability. This setting therefore entails uncertainty concerning expected harm. Furthermore, the court cannot accurately observe care taken but receives a uniformly distributed signal. The signal is an element of an interval built around actual care taken by adding and subtracting a constant. In this setting, simple negligence can help to transmit the information that some injurers have on the expected harm to those with imperfect information. The reasoning is as follows. Whereas uninformed injurers optimize against the distribution

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of the factor which codetermines the accident probability besides care, informed injurers optimize knowing the precise level of that factor, and, since optimal care of the informed is a monotonically decreasing function of the determinant, the court can adapt the due care standard after a sufficient number of observations. This change in the standard transmits information to the hitherto uninformed injurers if the standard of care installed to achieve optimal care is likewise a monotonically decreasing function of the determinant. Above, it has been argued that potential difficulties involving heterogeneity in harm levels comprises not only the problem of foreseeability but possibly also the problem of private information. Consequently, mechanisms which achieve a separation of different victim types are potentially valuable. Polinsky and Rubinfeld (2007) consider victims which vary in harm suffered, where harm is difficult to measure, and thereby provide a rationale for coupon remedies. It often holds in consumer lawsuits that the court award consists of coupons allowing for the purchase of the defendant’s product at a discounted price. In the model, the distribution of harm on its support is dependent upon the firm type. Courts can observe neither harm suffered in a specific case nor the firm type. Hence, if the court awards a cash remedy, it has to determine it only knowing the distribution of firms. This makes all firms, which are strictly liable, take the same care level. The alternative to a pure-cash remedy is the coupon-cash remedy, under which either a specified amount of cash or a number of coupons can be opted for by the plaintiff. A critical assumption of the analysis is that the consumer’s valuation of a coupon increases in the harm suffered. Then, consumers will self-select into the groups accepting coupons and the cash remedy, respectively. By making coupons somewhat more expensive for the firm than the cash remedy, one obtains that expected costs increase in the level of harm as the share of victims who opt for the coupons increases. As a consequence, it is achieved that care is at least somewhat tailored to firm type, giving a schedule of care increasing in the expected harmfulness. In Chapter 5, we also deal with the separation of different victim types. Victims have private information on individual harm which can only be remedied by spending on verification. The necessity of these, in the real-world often sizable, expenditures on

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harm level assessment for the attainment of an outcome which minimizes social costs is questioned in this contribution. The analysis shows that, given strict liability with a defense of contributory negligence, victims can be induced to behave type-adequately in the face of a schedule of due care levels and compensation levels. The trade-off between additional care costs and higher compensation is under some circumstances solved even without any changes to the liability rule, i.e., if victim compensation is full. The rationale is that the value of additional care differs between different victim types so that the additional care required in order to obtain a higher compensation may weigh too heavily and thereby ensures that types self-select adequately. In this regard, our analysis parallels the one by Miceli (2006) since he similarly considers the incentive compatibility constraints of different types, who for efficiency reasons should exert different amounts of care. In our contribution, we consider the potential role of court awards in separating victim types. Likewise considering heterogeneous harm levels and asymmetric information on harm, Rasmusen (1995) considers the consequences for court awards when there is no possibility to separate types via different means of compensation. Given that harm is not certain, courts might err in their assessment. This error has repercussions on plaintiff incentives to bring suit in the first place. The central finding is that this should affect the damage award irrespective of whether the error has been predictable. There is a high probability that the damage level measured has been produced by error if this magnitude is extreme. A curtailment to the mean is the appropriate response. If the error is predictable and positive (unpredictable), the award should be adjusted downwards (upwards) due to plaintiff self-selection (signaling). The finding on the unpredictable error is best explained by the following fact. An unpredictable error implies that courts may observe apparently weak suits, and the upward correction is optimal since the plaintiff’s willingness to bring suit is a credible signal that his true damages are higher than the court’s assessment. After the preceding discussion on heterogeneity of individual characteristics, we now turn to aspects of uncertainty.

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3.2

Uncertainty

The variability which causes the uncertainty discussed in this section concerns aspects other than individual characteristics. Indeed, we will turn our attention to uncertainty concerning the risk involved in a given activity, the standard of care to which care taken is compared, the causation of an accident, and the role allocation in an accident. As a first step, one might assume that there is uncertainty concerning the risk attached to an activity and whether incentives to overcome this uncertainty by the procurement of information are optimal. Shavell (1992) analyzes this question in a unilateral-care model and finds that private incentives are aligned with social benefits and costs in the case of strict liability and in the case of negligence if care and information are somehow regulated by the liability rule. The latter can imply that a wrong decision on information acquisition is deemed negligent, or that the standard of care applied presupposes the optimal decision on the acquisition of information.31 If, instead, negligence uses standards given what the injurer knows (presuming the acquisition of information), inadequate (excessive) information acquisition incentives result. The distortion resides in the fact that the injurer is freed from expected harm upon exertion of the correct care level, whereas society only enjoys a decrease in expected harm.32 The presentation of the basic model used the assumption that individuals are aware of the expected harm and, therefore, did not require any discussion of information acquisition incentives. Another fundamental assumption of the basic model is that injurers know the standard of due care if liability rules apply, which utilize due care standards. This together with the assumption that due care is set equal to the socially optimal level of care ensured that simple negligence induces incentives for efficient care. However, it is clear that the assumption on the perfect knowledge of the due care level is rather strong, in particular given the fact that matters such as reasonable care are often expressed in rather vague legal terms. Consequently, the study of uncertainty with respect to due care is of obvious interest. 31 Note that these two conceptions differ in that the court does not have to establish the knowledge of the injurer under the latter. 32 This topic might be likened to the discussion on legal advice. See, e.g., Kaplow and Shavell (1992).

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Calfee and Craswell (1984) and Craswell and Calfee (1986) consider uncertainty of the injurer concerning the legal standard of care.33 The due care level is drawn from a closed interval so that increasing the care taken reduces the likelihood that it will be judged insufficient in comparison to the realization of the care standard. In such a circumstance, the injurer considers three effects of increasing care: (i) Precaution has positive marginal costs, (ii) additional care lowers the expected harm as usual, the difference being that this decrease is weighted by the probability that one is judged negligent given the care exerted, and (iii) additional care decreases the probability that one is indeed judged negligent. Consequently, in comparison to the social problem, two contradictory effects make the private problem different. The common marginal benefit is discounted by the probability of being judged negligent, whereas the decrease of this probability adds a marginal benefit to the private problem foreign to the social problem. Consequently, uncertainty concerning due care may result in care chosen being less than, equal to, or greater than optimal care contingent on the specification of the problem. Assuming a normal distribution combined with a small standard deviation provokes care above the social optimum. Cooter and Ulen (1986) find an advantage of comparative negligence over other forms of negligence precisely in this tendency to overcomply given evidentiary uncertainty.34 The rationale being that a wrong assessment of care does not automatically imply full responsibility, but a sharing of liability if the other party is likewise judged negligent. However, Edlin (1994) shows that contributory negligence and comparative negligence both yield efficient care, when the policy maker takes account of the distortions due to evidentiary uncertainty when deciding on standards of care. The resulting efficient due care standards for injurer and victim are higher for comparative negligence to enlarge the number of outcomes in which some responsibility for harm is borne, which is required for efficient care incentives. We noted above that uncertainty concerning due care may result in care chosen being 33

The analysis of the case in which the standard of care is known but actual care is observed with a random error runs analogously (e.g., Shavell 1987: 97). Endres (1989) considers cases in which standards are not equal to the efficient levels but certain and Lando (2007) analyzes the case in which one party to the accident is imperfectly informed, i.e., over- or underestimates the standard. 34 Evidentiary uncertainty comprises situations in which the care taken is assessed with a random error. As noted earlier, this is similar to the case in which the individual is uncertain about the standard of care.

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less than, equal to, or greater than optimal care. Bartsch (1997a) considers the comparative statics of the individual minimization problem and shows that even if the two counteracting forces, expected harm being discounted by the probability of being judged negligent and the additional benefit due to the decrease in this probability at the margin, exactly balance out so that privately optimal care equals socially optimal care, this no longer holds in the event of a change in one of the exogenous parameters such as marginal costs of care. This follows because the private reaction to the change is different from the social one.35 In the basic model, care is considered as unidimensional. However, there are many circumstances in which a host of precautionary activities help to affect expected harm. These different activities may differ in characteristics such as observability. Bartsch (1997b) models uncertainty concerning care taken in a unilateral-care setting in which negligence is the liability rule and care is two-dimensional, but only one dimension can be observed by courts, albeit imperfectly. The interesting finding is that uncertainty may improve care incentives. As an extreme case, consider that, initially, care can be observed without error. In that case, injurers will take none of the unobservable precaution measure. This no longer follows if uncertainty is introduced. The deviation of individually optimal care from the social optimum depends on the direct effect of the uncertainty and the interdependence of respective care types. In the analysis by Bartsch (1997b), it no longer holds that uncertainty necessarily increases social costs. In Chapter 8, entitled ’On Avoidance Activities After Accidents’, we provide an extension of the standard framework along the following lines. The basic model allows for precaution to reduce the expected liability, for instance, in a unilateralcare model with strict liability. We allow for avoidance activities, which only concern the probability with which the injurer is made to compensate the victim but not the probability of the accident occurring and may therefore be classified as unproductive from a social point of view. This framework is applied to analyze different aspects of 35 In that article, starting from the imperfect observability of care, a principal-agent framework is presented. Given that the injurer is assumed to be risk averse and the policy maker risk neutral, the result known from contract theory, agents have to bear some risk in order to incentivize desired behavior, can be established in this setting as well.

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importance to the economic analysis of tort law. For instance, we find that uncertainty on due care is no longer necessarily detrimental to the minimization of social costs because the uncertainty reduces the incentives to spend on avoidance activities. Another important contribution of Chapter 8 is the establishment of the fact that punitive damages, i.e., damages above harm, can be welfare-reducing if injurers can choose avoidance. Bartsch (1997b) complicates the setting not only by evidentiary uncertainty but also by considering multidimensional care and highlights consequences of the interaction. The introduction of certain other aspects may similarly allow for interesting conclusions. Young et al. (2006) claim that evidentiary uncertainty and its implied tendency to take excessive care may be countered by the defense of the absence of causation. Causation and uncertainty over causation is a topic different from uncertainty on due care and the theme we turn to now. We stated above that in order to to have a promising claim against the defendant victims usually must present evidence for the defendant being the cause-in-fact and the proximate cause of the harm suffered by the victim. Realistically, cases will often be characterized by uncertainty over causation. For instance, your lung cancer may very well be due to your neighbor, the nuclear power plant, which, however, might be able to establish sufficient doubt due to a multitude of other factors, which might have caused the disease. Shavell (1985) provides the groundbreaking analysis of causation. He considers uncertainty over causation, distinguishing between nature and another individual as alternative causes of the harm. There may be cases in which the accident can be clearly attributed to a cause and cases where the harm is of ambiguous origin. It is established that a threshold criterion, which applies full liability if the probability of causation, i.e., the probability that the considered actor has caused the harm given that it is of ambiguous origin, is greater than a critical threshold and no liability otherwise is inferior to proportional liability, i.e., liability proportional to the probability of causation. The reason lies in the fact that the threshold rule inappropriately reflects social costs. To illustrate this, take the simple case in which an individual with a given benefit from an activity has to decide whether or not to engage in it. The participation in the activity is desirable as long as the

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benefit is greater than the expected harm caused by it. If we deem the individual liable not only in accidents caused by the activity but also in cases in which the activity does not have an impact, this can deter desirable activity. Proportional liability is free from such distortions. However, it is argued that using a probability threshold is considerably less expensive to administer. It can be argued that the possibilities considered in Shavell (1985) are restricted. In Schweizer (2006), it may be that (i) harm results only if the injurer is negligent, (ii) harm results only if the injurer is not negligent, (iii) harm follows given any care choice, or (iv) harm never ensues, irrespective of the care chosen. The outcome is a function of the choice of the injurer as well as a move by nature, introducing the uncertainty over causation. If the move by nature were observable, full liability should result only in case (i) since in that case harm is actually caused by the injurer’s negligence. Interestingly, the case where negligence prevented the occurrence of an accident emerges in this more encompassing analysis, case (ii), which is of importance for the appropriate damage award. Exactly this case is ruled out by Shavell (1985). In reality, the move by nature is rarely observable. This leaves the application of the award on average, contingent on the observed outcome. An observable outcome can be that harm has been suffered and that the injurer has been negligent. This event cannot be partitioned further into case (i) or (iii) since information required for this assessment is unavailable. The analysis by Schweizer contains the proportionality rule of Shavell (1985) as a special case. The central difference is that the multiplier applied to the level of harm to reach damages also contains the probability that negligence prevents the accident, which is set equal to zero by Shavell. Young et al. (2004) also provide a discussion of uncertainty over causation and highlight a distinction between causality and causation. The probability that an act caused some outcome in the sense of causality (causation) is the probability for the outcome occurring given that the act was undertaken (the probability by which the probability for the outcome is reduced if the act is not undertaken). These generally differ. Assume, for instance, that two different causal chains can cause the accident and that the act can effect the accident only when compounded by a certain move of nature in the first causal chain.

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A difference between the respective probabilities arises in this framework, for example, since the concept of causality would include into the probability the case in which the outcome resulted from the second causal chain and could not have been provoked by the act of the individual. The latter being due to nature choosing to move differently from the act which would have completed the first causal chain to the accident. This inclusion does not follow in the case of causation. As intuition suggests, the use of the probability of causation and not of causality will induce efficiency when used in proportional liability. Our discussion of contributions in the law-and-economics literature dealing with uncertainty closes with an elaboration on a very fundamental uncertainty, which is the uncertainty concerning the role in an eventual accident. This is of interest since there are several contexts in which the enrolment is not fixed in any way. A prime example is the case of driving. Individuals who drive on the highway are very likely unable to exclude a positive probability of hurting someone or of being hurt by someone. Kim and Feldman (2006) show that this variation of the standard framework changes the result of a unique equilibrium in efficient care only if the subjective victim probabilities are not consistent. The equilibrium in efficient care always exists under standard liability rules, but may no longer be unique. For example, if parties underestimate the victim probabilities, there may also be an equilibrium in suboptimal care under negligence since parties’ perception of expected liability is inadequate. Aside from Kim and Feldman (2006), the possibility of individuals being uncertain whether they are the injurer or the victim in an accident is treated by Hasen (1995) and Winter (1997). Hasen (1995) thereby complements the literature on the duty to rescue a stranger, in which it has been assumed that roles, rescuer and rescuee, are certain. Whereas the efficiency of a duty to rescue has been in dispute before, this precisely results in Hasen’s setting if the probability of becoming the rescuer or the rescuee are similar. Winter (1997) has been discussed in footnote 12. The preceding discussion comprises, for example, analyses on uncertainty over causation. Issues such as this uncertainty often imply that the resolution of any given case by the court may be very costly. Since one of our own analyses explicitly accounts for the reality that the use of the liability system imposes real costs, we provide a brief overview

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of the most important contributions allowing for administrative costs in the next section.

3.3

Administrative Costs

The administrative costs of the liability system comprise expenses in time and money of litigants and the state. These costs are of considerable magnitude. For the USA, it is said that, on average, one dollar received as harm compensation creates a dollar or more in administrative costs (Shavell 2007a). Positive administrative costs raise a multitude of aspects. First, the relative desirability of liability, compared to other policy measures that are able to attain the internalization of external effects, is questioned. Second, agents have to make decisions not yet brought into light, such as whether to bring a suit, when to settle, or how much to spend on pretrial or trial expenditures.36 Third, our central concern, the choice of precaution, is affected by administrative costs and their incidence. Since we do not consider administrative costs in our later analyses, except for one, and the field opened by that topic is wide, we restrict ourselves to present some key insights with respect to precaution. For a more extensive discussion of litigation see, e.g. Spier (2007), and Katz (2000) for a survey on the effects of different cost-shifting rules. Our central focus in this contribution lies on incentives to take care. Hylton (1990) provides a treatment of the deterrence effects of litigation costs under strict liability and under negligence, disregarding the possibility of settlement. His model comprises a binary care choice, as well as random harm levels and costs of taking care. Both injurer and victim bear litigation costs, i.e., he assumes the American rule for the allocation of legal costs. He finds that litigation costs cause too little care under strict liability for two reasons: (i) some victims do not sue since their harm, i.e., the compensation if a judgment is obtained, is less than their costs of trial and these victims are therefore ignored in the calculus of the injurer, and (ii) litigation costs of victims who sue are not considered by the injurer. Thus, there are cases in which the costs of care are less than the social benefit, thus taking 36 For instance, the private versus socially correct incentives to bring suit are discussed in Shavell (1982a, 1997a, 1999), Menell (1983), Kaplow (1986), and Rose-Ackerman and Geistfeld (1987).

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care is socially desirable, but taking care is not in the private interest of the injurer. This is due to the fact that the individual benefit is less than the social benefit as a consequence of the effects referred to above. This can be remedied by a tax which introduces these elements of the social problem into the private optimization. Under negligence, if there is an equilibrium, it implies that some do not take care although they should from a social perspective. The reasoning is as follows. The victim does not know the injurer costs of care. Victims will recover damages only if the injurer abstained from care-taking although the injurer’s cost of care deemed it socially desirable. Victims will only sue if there is a positive probability of winning, given an accident has occurred, which requires that some injurer types indeed fail to take efficient care. The social objective in the presence of administrative costs is no longer assumed to be the minimization of primary costs. Consequently, it is worthwhile to consider policy responses, given administrative costs, to minimize the sum of primary and tertiary costs. Polinsky and Rubinfeld (1988a) consider the optimal level of compensation with the goal of minimizing total social costs, which also comprise litigation costs. Consequently, society would like to optimally balance the potential tension between first-best care and litigation costs. It is assumed that exertion of care lowers the magnitude of harm in their framework and litigation costs remain where they fall. As a consequence, a critical care level is implied under strict liability at which the amount of compensation the injurer owes the victim makes the latter indifferent between suing and bearing the loss but saving on litigation costs. They find that usually some adjustment to compensatory damages is optimal. The adjustment may be positive or negative, depending on the marginal productivity of care, and can induce the first-best outcome if the adjustment is type-specific. Care is exerted under simple negligence only given the threat of suit. The litigation costs of the victim erect a range of loss levels for which no suit occurs, i.e., causes too little care incentives without adjustment to compensation. A positive adjustment is intended to correct this. A negative adjustment intends to achieve the opposite, that is, deter suits if care has a low productivity since given the threat of suits and high litigation costs, care will be exerted irrespective of its productivity. This latter scenario can be precluded under negligence by

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the appropriate choice of due care. However, it is also possible under this liability rule that the care level which forestalls suits due to positive victim litigation costs is less than efficient care, arguing for an upward adjustment of compensation. Hylton (2002) uses the framework put forth by Polinsky and Rubinfeld (1988a) and extends their analysis along several dimensions for the case of strict liability. First, he evaluates welfare effects of increases in litigation costs. For example, if the injurer care does not affect the magnitude of harm but only the accident probability, then increases in victim litigation costs unambiguously lower welfare. This follows since the increase in victim litigation costs decreases the number of suing victims, which, in turn, decreases incentives for care. This does not generally follow if care affects the harm level, since, in that case, care has an additional marginal benefit in changing the marginal victim which is indifferent between suing and bearing the harm while saving litigation costs. Higher litigation costs then imply a higher marginal benefit due to this effect, an effect on care incentives which may dominate the lessening of care incentives due to the ’number of victims’ effect explained before. Second, the optimal adjustment to compensatory damages is shown to be capped from above by the victim’s litigation costs. Hylton (2002) also considers settlement along the lines of Polinsky and Rubinfeld (1988b), where results are dependent on the shares of the settlement surplus that injurer and victim can appropriate, respectively. The preceding contributions already allow for changes to the liability system as we observe it in reality to reduce overall social costs. For going further down that road, it is noteworthy to recognize the following. The liability of the injurer for harm done induces injurer care, whereas the claim for compensation provides incentives to bring suit for victims. To affect litigation costs and the costs due to the prevention and occurrence of accidents in an optimal fashion, Polinsky and Che (1991) propose the decoupling of liability, i.e., introducing a divergence between what the injurer pays and the victim receives.37 The optimality is established in a unilateral-care setting given strict liability. The basic 37 The original idea of decoupling liability was proposed by Schwartz (1980) and Salop and White (1986) in the context of antitrust. In Chapter 5, we will advocate a potential ’decoupling’ of injurer payment and victim receipt to disentangle victim types.

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intuition is as follows: Proceeding from the assumption that liability equals harm and harm equals compensation, we can conclude that an increase in liability increases injurer care. This can be exactly offset by lowering the compensation received by the victim, since the incentives to bring suit are lowered which, in turn, decreases care incentives. Victims vary in their litigation costs and suit will be brought only if the award at trial is marginally larger than trial costs. Thus, decreasing the compensation transferred to victims decreases the probability for suit ex ante. In consequence, after the change in liability and compensatory payments, care incentives are still optimal but expected litigation costs have fallen. This rationalizes that the procedure is continued until the liability payment of the injurer cannot be increased further.38 Choi and Sanchirico (2004) revisit the optimality of decoupling, altering the framework of Polinsky and Che (1991) by allowing for endogenous litigation effort.39 An increase (decrease) in liability (compensation) effects an increase (decrease) in injurer (victim) incentives to spend on litigation effort. It is shown that the effects via changes in how a given suit proceeds in terms of litigation expenditures can dominate the desirable effects of decoupling, arguing against the general desirability of decoupling liability. Choi and Sanchirico (2004) change the framework by adding a policy variable, an additional filing fee, by which the number of suits can be controlled but which does not affect litigation efforts, whereas the level of compensation is the means to achieve the number of suits in Polinsky and Che (1991). A change in injurer liability and victim recovery have an effect on the number of suits and litigation efforts. As a consequence, liability and recovery should be set in order to reduce the costs per suit brought, while maintaining the desired level of deterrence. Chu and Chien (2007) also extend Polinsky and Che (1991). They introduce a settlement stage into the game with asymmetric information on the probability with which the victim prevails. The victim makes a take-it-or-leave it settlement offer, being less informed than the injurer. The focus lies on the effects that the magnitude of injurer payment and victim recovery have on the bargaining positions in pretrial negotiation. For instance, lowering awards to the 38 Note, however, that this does not necessarily imply that liability is larger than compensation. Liability is bounded from above by injurer wealth. It may be, in cases of very high harm, that additional care incentives are desired which may require setting recovery above liability. 39 Kahan and Tuckman (1995) is an extension along similar lines.

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victim makes the threat to sue less credible. We assert above that the liability for harm done increases care incentives of the injurer. This similarly works with other payments imposed on the injurer in the contingency of being required to compensate the victim. Consequently, shifting the litigation costs of the victim to the injurer likewise has this effect on care incentives. Kaplow (1993) compares shifting the plaintiff’s fees with increasing damage awards and thereby provides an analysis very similar in spirit to Polinsky and Che (1991). Shifting the litigation costs of victims to injurers increases care incentives by increasing the probability of suit and the cost of a given suit lost by the injurer. Similarly, increasing the award makes it profitable for more victim types to sue, again, increasing the incentive to take care. In comparison, it is proven that an increase in the damage multiplier combined with a decrease in the share of litigation costs borne by the injurer, in such a way as to keep deterrence constant, decreases the number of victim types for which suit is profitable. To obtain an intuition, consider increasing the damage multiplier while decreasing the share of the plaintiff’s litigation cost borne by the injurer, keeping the marginal victim indifferent. Suing victims have lower litigation costs than the marginal victim. Consequently, relevant expected costs of the injurer for these victim types, expected payment plus expected share of victim litigation costs, rise. A leveling of costs would require a further decrease in the share of victim litigation costs borne by the injurer, where the decrease becomes larger the smaller the victim litigation costs are. Hence, keeping deterrence constant allows for a decrease in the marginal victim type. This decreases total expected litigation costs. Abbreviating the above, we can say that the increase in the damage multiplier is an inducement of equal value to all victim types, whereas the sharing of legal fees has special appeal to types with high costs. There are further methods of reducing expected litigation costs without affecting deterrence. In a framework characterized by heterogeneity with respect to the probability of prevailing at court and homogeneous litigation costs, Polinsky and Rubinfeld (1996) show that it is desirable to penalize plaintiffs who lose while increasing the reward for those who win. This has beneficial effects, since the impact of the stated changes are

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succinctly different for low-probability and high-probability plaintiffs, implying an overall decrease in expected litigation costs. Note that Polinsky and Rubinfeld (1996), as well as Polinsky and Che (1991) and Kaplow (1993), make use of the prescription of Becker (1968), namely that of lower enforcement due to its high cost, while maintaining deterrence via increasing the payment in the event of enforcement. This prescription critically relies on the assumption of risk-neutrality. The next section considers the assumption of risk aversion and its effects on the basic conclusions of the economic analysis of tort law. However, before we come to that, we present a final analysis dealing with administrative costs. Miceli (2000) considers litigation costs in his analysis on a totally different matter, the statute of limitations for tort suits. The statute is usually justified on grounds of error costs since evidence is presumed to deteriorate over time, increasing the likelihood of error. Miceli argues that such a statute can also be rationalized by taking litigation costs into account. The search for the optimal length of time involves trading off deterrence against litigation costs since shorter time spans permit fewer suits.40 One finding of the framework is that the optimal statute length appears to be longer under negligence than under strict liability and can be explained as follows. If the time span is increased, expected litigation costs also increase. The extent of this increase is lower under negligence because the probability that the injurer was indeed negligent declines with a longer time horizon and this lowers victim incentives to sue.

3.4

Risk Aversion

The basic model and most models to date assume risk-neutral individuals. Indeed, risk aversion is of importance in only one of our analyses.41 Risk neutrality is, without doubt, a heroic assumption in many contexts. In this section, we will discuss analyses on different aspects involving risk aversion. Certainly, care incentives, as in all sections throughout, will be of interest. Furthermore, the possibility to insure against liability, punitive dam40 The model does not contain any reason for the postponement of suit. Instead, the length of time after an accident that elapses until a victim files suit is treated as a random variable. 41 From this follows that most aspects which will be discussed have no bearing on our studies. Yet, risk aversion is an important consideration and therefore deserves at least a brief digression.

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ages, and the consequences of harm being nonmonetary will be dealt with. Shavell (1982b) is a central contribution incorporating risk aversion. He, first, establishes that the first-best is characterized by care so that the marginal costs of care are equal to the absolute value of the reduction in expected harm. This feature thus parallels the outcome under risk neutrality and follows from the second result. Individuals do not have to tolerate any variation in income in the first-best outcome. Next, it is of importance to establish what can be attained with the available means liability and insurance. In the absence of insurance, the optimal outcome attainable differs from the case in which injurers and victims are risk-neutral. If strict liability is utilized, it proves to be advantageous to lower damages below the level of harm in order to decrease the burden imposed on a risk-averse injurer. Consequently, the victim is not fully compensated and shares the risk. The outcome is not first-best. If negligence is applied as a liability rule, the standard of care differs from the optimal care level in the case of risk neutrality, since the victim is fully exposed to the variation in income. The variation in victim income is due to the fact that injurers comply by the standard and implies that a further reduction in the likelihood of accidents is often desired. The reduction in likelihood is attained via the elevated standard of care. Again, the outcome is not first-best. In the presence of a competitive insurance industry, which offers insurance at actuarially fair rates, the firstbest can be achieved. However, under strict liability, this only holds if liability insurers can observe precaution taken. Without the ability to condition insurance policy terms on behavior, injurers will usually purchase only partial insurance under strict liability. Partial insurance induces care-taking but implies risk-bearing costs. In contrast, negligence allows the attainment of the first-best even without insurers’ being able to observe care. The reason is that injurers adhere to the standard of care and victims do not bear risk because of their purchasing first-party insurance. A central conclusion is that liability insurance is socially desirable. Even in the case of strict liability and non-observability of precaution, liability insurance does not harm the victim who will be fully compensated but increases the expected utility of the injurer. Let us stick to the topic of liability insurance for a brief moment. Feess and Hege

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(1999) develop a role for insurance in the accident framework even if individuals are risk neutral. In a setting in which there are multiple tortfeasors, tortfeasors’ actions are unobservable, and damages have to be nonpunitive, traditional liability rules fail to induce efficient incentives. Feess and Hege (1999) introduce insurance-based liability, where the court conditions the liabiity of a given tortfeasor on having signed an incentive-compatible insurance contract. In that sense, the problem of inducing optimal behavioral incentives is delegated to insurance companies. The insurance contract that instills first-best incentives uses a bonus paid out in the no-harm state of the world in order to harmonize individual and social marginal benefits, i.e., to circumvent the team-production problem. After this digression on insurance if individuals are risk neutral, we now return to the more traditional setting in which insurance is offered in the presence of risk-averse individuals. Nell and Richter (1996, 2003), using a framework with constant absolute risk aversion preferences and possibly more than one victim, consider insurance which is imperfect in the sense that there is a proportional loading to expected harm. The possibility of having more than one victim is of interest only if individuals are risk averse, since, in that case, it has more effects than upscaling expected losses. Nell and Richter (1996) approximate the certainty equivalent by a mean variance approach and derive that no liability rule with full liability of one party can attain the first-best or even the second-best since these outcomes require a strict division of losses.42 The imperfectness of insurance makes coinsurance optimal, which, in turn, makes some risk sharing among injurer and victims optimal. Only in the absence of loading will the injurer’s share be equal to one, implying strict liability. Interestingly, whereas the injurer’s liability share tends toward zero if the number of victims goes to infinity in the case without insurance, there is a lower bound to the injurer’s share if insurance is available. The former holds since the risk costs of the injurer increase in a non-linear fashion with the number of victims, whereas the latter is true under reasonable circumstances, as the coverage will be full for a critical number of victims. Consequently, negligence can approximate the optimal outcome if insurance 42 In the first-best, care and the share of losses are decision variables, whereas in the second-best, the sharing of losses is set while trading-off optimal risk allocation and the inducement of appropriate precaution.

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is not available or only has limited coverage. Nell and Richter (2003) present a very similar analysis but add a section on the activity level, in which it is shown that strict liability (negligence) induces an activity level lower (higher) than in the social optimum. Another recent study dealing with liability insurance is Winter (2006). He analyzes the case of joint tortfeasors with limited wealth, whose insurance demand may be strategically interdependent. The interest is on the insurance decision, whereas the decision on care is not modeled.43 Limited wealth makes it advantageous not to purchase insurance if liability is higher than a critical level. If interdependence is introduced via the liability rule of joint and several liability, this may lead to Pareto dominated outcomes. Whereas injurers fully insure themselves if individual liability is oriented only at the wealth and insurance level of the individual itself, an equilibrium of no insurance is possible over the same range of liability levels if the wealth and insurance of other injurers may be utilized to make up for shortcomings of the injurer considered. An important observation is that an increase in the liability can deter insurance incentives. This implies less compensation for victims and incentives for precaution.44 Courts sometimes impose punitive damages, i.e., damages in addition to compensatory damages awarded against a defendant.45 Analyzing a setting with risk-averse actors, Chu and Huang (2004) provide a rationale for punitive damages and, specifically, the characteristics of their granting, being the fact that punitive damages are often only granted for outrageous conduct and are frequently related to the defendant’s wealth. Given risk aversion, being required to compensate a given amount of damages is less threatening to wealthy injurers than to poor ones. This translates into care incentives being lower for wealthy injurers if the utility function is separable and marginal costs of care do not decrease too rapidly with wealth. As a consequence, it might turn out that only poor injurers abide by the negligence standard, whereas wealthy injurers put up with being liable. In this case, courts can use punitive damages to fine-tune the deterrence 43

It is stated that the care level will be in step with the compensation of the victim. Another very recent and interesting contribution on liability rules and risk aversion is provided by Zivin et al. (2005). However, since they allow for negotiations and side payments, their study is very much outside the realm of our survey. 45 This practice is touched upon in Chapter 8, ’On Avoidance Activities After Accidents’. 44

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of poor and wealthy injurers. The standard, which wealthy injurers do not adhere to, is tailored to poor injurers and their victims. Wealthy injurers are confronted with the following options: abide by due care and be free from liability, take lower than due care but within a tolerated range and compensate the harm in the event of an accident, or take even lower care and face the threat of punitive damages. The tolerance level and punitive damages will then be set to incentivize desired precaution by wealthy injurers. In effect, injurers self-select the more appropriate care level for their level of wealth without the court being able to observe wealth. The basic model relies on harm being compensable. There are cases in which the appropriate estimate on the amount needed to make the victim whole is not readily available. For instance, accidents often result in physical injury. This can decrease or even nullify the earning potential of individuals. It is then of interest what amount of money would fully compensate the victim. It is often assumed that full compensation can be attained by granting the present value of expected future lost earnings. However, judges are often somewhat restrictive in relation to this measure. Skogh and Tibiletti (1999) argue that judges are thereby in line with full compensation since individuals would bear uncertainty in the absence of the accident, which implies that the expected value needs to be corrected for the risk premium. The appropriate compensation ought to make the individual full in expected utility terms. For that purpose, Skogh and Tibiletti supply the immediate certainty equivalent, which is a function of the expected losses, the level of risk aversion, and the correlation of expected losses, inter alia. Compensation may imply further difficulties other than measurement. As said, it often holds that accidents cause losses that are nonpecuniary. In the case of risk-neutral individuals, nonpecuniary losses do not change the analysis. Under these circumstances, the major problem is often measuring the nonpecuniary part of the harm consequent to an accident. If individuals are risk averse, however, other matters arise. It is usually assumed that the suffering of nonpecuniary losses does not affect the marginal utility of wealth. For instance, losing a child might not affect the marginal value of wealth. This implies, from an insurance perspective, that victims in their pursuit of equating marginal utility of wealth across the

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accident and no-accident state would not want to insure themselves against nonpecuniary losses. Consequently, injurers should internalize full losses, i.e., the sum of pecuniary and nonpecuniary losses, whereas victims ought to be compensated only for pecuniary losses (e.g., Shavell 2007a). Nonpecuniary losses furthermore raise the issue of state-dependent utility. Consequently, the marginal utility of wealth differs for given wealth across states of the world, where it is not obvious a priori whether the marginal utility in the accident state is higher or lower than in the no-accident state, which further complicates the analysis (e.g., Friedman 1982, Arlen 1992b, Frech 1994, Fraser 1996). For example, think of being paralyzed consequent to an accident. This will presumably affect the value assigned to additional income, but it is not undisputed in which way. Before Winter (2006), who allows for insurance and limited assets, Shavell (1986) contributed greatly by analyzing the effect of limited wealth on the incentives to take care and the interaction with liability insurance. This will be of concern in the following subsection.

3.5

Judgment-Proofness

Injurers lacking sufficient funds to cover the harm caused by an accident is a situation which is often descriptive of reality. The repercussions of this scenario for the incentives to take precaution are therefore of utmost interest.46 The contributions differ in the way in which they model the effect of care on expected harm and the nature of care. Given limited wealth, it can be hypothesized that since injurers do not internalize the complete social harm, externalizing the part that exceeds personal funds, they will exert only insufficient precaution. Indeed, Shavell (1986), using a framework with nonmonetary care which reduces the accident probability, gave formal support to this intuition by showing that all injurers with (i) asset levels below the level of harm in the case of strict liability and (ii) asset levels lower than a critical value (which is below the level of 46 The deviation from the standard model due to limited liability may of course be coupled with other deviations from standard assumptions. For instance, the case of several injurers is interesting if judgmentproofness is an issue, which we, however, do not cover here (see, e.g., Kornhauser and Revesz 1990, Watts 1998).

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harm) in the case of negligence, will take less care than individuals with sufficient funds.47 Under negligence, injurers adhere to due care as soon as the expected individual costs of failing to do so, comprising precaution costs and expected liability, are higher than care costs of standard care, given the wealth level of the individual. This transition therefore occurs at a wealth level lower than harm. The fact that the jump in individual costs at due care under the negligence rule can be utilized to trigger more care than is individually optimal according to the minimization of the sum of precaution costs and expected liability is developed further by Ganuza and Gomez (2004) for the setting employed by Shavell (1986). It is argued that optimal due care is the minimum of first-best care and a level of care dependent on the individual’s wealth level. The latter care level might be labeled second-best care and is derived as follows. Given assets less than harm, individuals compare the costs of standard care to the costs of bearing precaution costs and expected liability. If assets of the injurer are sufficiently lower than harm, failing to take due care implies lower individual costs. Second-best care is defined as the care level which causes precaution costs equal to this level of individual costs. The negligence rule, utilizing this optimal due care level, evokes first-best care by all injurers who choose due care under the standard negligence rule and higher care by all potentially judgment-proof injurers who do not abide by the prescribed care level under the standard negligence rule. Given that the minimization of the sum of precaution costs and expected harm is assumed to maximize welfare, increasing the suboptimal care somewhat in the direction of first-best care is desirable. Boyd and Ingberman (1994) augment the analysis by considering the possibility that precaution, which is non-monetary, might impact on the loss magnitude instead of on the accident probability, or that it might lower both. They consider the case of strict liability and find that, given their general description of risk reduction, damages that are noncompensatory or punitive can improve precaution incentives depending on the specification. For instance, if losses can take the form of two magnitudes the higher 47 Similar conclusions are reached by Summers (1983), who groups judgment proofness with the case in which not all victims bring suit. However, Dari Mattiacci and Mangan (forthcoming) show that these instances of ’tort law failure’ instill different incentives in a variety of contexts, e.g., if precaution has an effect on the magnitude instead of on the probability of losses.

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value of which bankrupts the injurer and care lowers the probability of both events, increasing damages above the compensatory level in the lower loss event can increase care incentives. In contrast, if the probability of the accident is fixed and only the magnitude can be affected by precaution, noncompensatory damages may increase care from the level it would be with compensatory damages. Again, consider two loss magnitudes of which the higher bankrupts the injurer. If the amount of damages in the lower loss event is decreased, this increases the marginal benefit of the discrete precautionary measure and might therefore motivate it. Consequently, the optimal damage level in the event of potential insolvency will be idiosyncratic to the technology. Dari Mattiacci and De Geest (2005) follow this lead and add the separate-probability-magnitude model in which two different precautions can be taken. Care is of the non-monetary type. It is established that negligence is the superior liability rule for all the different precaution technologies considered.48 The discussion above introduced an alternative assumption about the way that care affects the expected harm. We now turn to an alternative assumption about the nature of care. In line with Shavell (1986), Beard (1990) presumes that care lowers the accident probability but, in contrast to Shavell, assumes that care expenditures reduce the assets available for compensation, i.e., monetary care. He shows that, in the case of strict liability, potentially judgment-proof injurers may take more care than injurers who are not bankrupted by the compensation of victims.49 Miceli and Segerson (2003) detail Beard’s results for strict liability and negligence in a simplified framework. If care is monetary and strict liability is the applicable rule, injurers with asset levels in the range of harm exert more care than injurers whose wealth constraint does not bind in the case of an accident. The fact that care reduces the assets available for compensation de facto 48 Actually, if care only reduces the accident probability, negligence should utilize average compensation, whereas compensatory damages are used in all the other cases. In the probability model considered, the injurer cannot anticipate the precise level of harm and basing compensation on the average can then ameliorate the truncation present otherwise due to limited wealth. 49 Posey (1993) likewise assumes monetary care in a framework of strict liability. She shows that if the benefit due to the activity can be lost to compensation requests, limited liability not necessarily implies excessive activity. Although the expected costs of additional activity are less given limited wealth, the benefit of the activity is only realized in the no-accident state, which lowers the marginal benefit of activity.

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reduces the care costs because these only arise as costs in the case where no accident occurs. The possibility that care of potentially judgment-proof individuals exceeds the care level taken by injurers with sufficient assets thus follows if, in comparison to the cost minimization problem of injurers with sufficient assets, the reduction in marginal costs of care overcompensates the reduction in the marginal benefit of care, where the latter reduction is due to limited liability. Surprisingly, the fact that some potentially judgment-proof injurers may take excessive precaution can rationalize the introduction of a liability cap if care is monetary. This is established by Dari Mattiacci (2006). There is a range of asset levels for which individually optimal care exceeds first-best care. If the liability is capped to a level less than harm, care levels such that the injurer is not bankrupted result from the individual cost minimization at smaller asset levels. The divergence of the resulting care level from the first-best level will depend on the difference between the harm and the liability cap. However, there are considerable cost to the introduction of a liability cap since all injurers without binding asset constraint will also choose care other than the first-best, which unambiguously increases social costs. The contribution by Dari Mattiacci and De Geest (2006) contains the results of Miceli and Segerson (2003) for the framework in which care affects the accident probability, but adds results for the case in which the magnitude of harm is affected by precaution, when the accident probability is exogenous.50 The finding for the model in which care affects the magnitude of the harm is that it does not matter whether care is monetary or nonmonetary. Precaution is optimally equal to zero from the individual’s stance as long as assets are below a critical threshold, while it is equal to socially optimal care for asset levels larger than that. The rationale is that harm as a function of care is only relevant if precaution places the injurer in the solvent zone. Before that holds, care entails only costs since expected liability consists only of the exogenous probability and wealth level. MacMinn (2002), while being more traditional regarding the precaution technology by sticking to the case in which care affects the accident probability, contributes due to 50 Endres and L¨ udecke (1998) show for this setting that there may only be a mixed-strategy equilibrium if the liability limit falls within a certain range.

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the consideration of risk-averse individuals under both strict liability and negligence. He shows that injurers who turn out judgment proof in the case of an accident exert more (less) care under negligence than under strict liability if care is non-monetary (monetary). In the case of non-monetary care, precaution under strict liability eventually becomes greater than care under negligence since individuals do not exert more than standard care under the latter liability rule. Notably, MacMinn considers the negligence rule making injurers liable only for the harm caused by their negligence (see Kahan 1989). In Chapter 6, we like MacMinn (2002) assume that injurers are risk averse and revisit the central conclusion of the literature heretofore, i.e., that care taken by potentially judgment-proof injurers falls short of first-best care under strict liability if care is nonmonetary. In ’A Note on Judgment Proofness and Risk Aversion’, we establish that the reverse may hold if individuals are risk averse, i.e., some potentially judgment-proof injurers spend more on care than some injurers with assets greater than the harm. Individuals with sufficient funds may decrease care in response to increases in asset value because this change decreases both the marginal benefit and the marginal costs of care. The lowering of the marginal benefit of care does not result for judgment-proof individuals since the asset value change has no effect on the utility of the accident state. Due to this irrelevance, increases in assets actually increase the marginal benefit of care for individuals who are judgment proof in the accident contingency. The above analyses including ours establish the individually optimal precaution as a function of injurer assets by treating the latter as exogenous. Boyd and Ingberman (1999) allow wealth to vary with the legal rule. The potential injurer, a profit-maximizing firm, who is assumed to be strictly liable, can reduce expected liability by either increasing precaution or decreasing capital exposed to liability. The factor price of capital is higher if it can be lost to tort claimants, which can imply an inadequate use of capital and an inappropriate level of safety. The central finding is that an increase in damages can sometimes effect a decrease in safety since the exposed wealth has been reduced as well. This argues against punitive damages, for which it is usually assumed that they improve care incentives. There indeed exists empirical support in favor of the thesis that firms try

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to shield from liability payments by divestiture, i.e., by contracting out activities prone to large liability requests to small and potentially judgment-proof firms (Ringleb and Wiggnis 1990, Merolla 1998).51 Besides firms who are set up with consciously few assets, employees who determine safety in the production of goods often have little to loose in the event of an accident. Usually, the most that can be lost if the employee causes an accident to some third party is the job itself. This individually perceived loss will often be less than harm caused. To incentivize high care, firms may then decide to pay above-market wages. Shavell (1997b) analyzes such a context in which the firm is strictly liable and the employee chooses care. He establishes that firm incentives to pay supernormal wages usually deviate from social optimality for principally two reasons. First of all, whereas an increase in wages is a costless transfer from a social standpoint, it reflects additional costs for the firm. Secondly, these costs will be reflected in prices which distorts the quantity of goods purchased. Polinsky and Shavell (1993) similarly start from the limited ability of firms to make employees internalize the full magnitude of harm and deduce that it may therefore be optimal to impose fines and imprisonment on the employee.52 In addition to the consequences of limited assets with regard to our primary interest, incentives to take care, limited liability also entails effects on insurance demand. Shavell (1986) establishes that potential judgment proofness diminishes incentives to purchase insurance. This may imply either that no insurance at all or less coverage than without the limit in wealth is purchased for certain asset levels and dependent on the observability of precaution.53 The observability of precaution also determines the level of care taken if liability insurance is available. However, even if care taken may not correspond with the first-best level, this does not question the social desirability of liability insurance (Shavell 2000). 51 Brooks (2002) argues that there is a counterargument to this calculated use of small and potentially insolvent firms in the possibility that the larger corporation will be held vicariously liable. The firm with limited assets takes too little care which can imply very high liability costs for the large corporation even if the probability of being held vicariously liable is small. 52 It is argued that the state can usually extract more financial means from the employee than the firm, for instance, by using the threat of imprisonment. 53 A related contribution by Winter (2006) was discussed at the end of the previous section.

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There are several policy options that can be thought of to ameliorate problems resulting from judgment proofness. Since these have little bearing on the analyses which follow, we will only briefly enumerate the most important policy responses. The imposition of liability on a party who has a relationship with the potentially judgment-proof party, i.e., vicarious liability, is one such option (see, e.g., Sykes 1984, Chu and Qian 1995, Pitchford 1995, Boyd and Ingberman 1997, Demougin and Fluet 1999, Feess and Hege 2003, Hay and Spier 2005). An obvious possibility is the imposition of minimum asset or liability insurance requirements (Jost 1996, Polborn 1998, Shavell 2005, Dari Mattiacci and Parisi 2003). Furthermore, the direct regulation of the activity is a way to proceed with areas in which liability is severely hampered by limited wealth (e.g., Shavell 1984, Kolstad et al. 1990, Schmitz 2000, De Geest and Dari Mattiacci 2007). It has also been shown that decoupling penalties imposed on the injurer from losses due to the accident, along the lines of Boyd and Ingberman (1994), can characterize optimal liability in a setting with potentially judgment-proof individuals (Lewis and Sappington 1999, Innes 1999).54

3.6

Bilateral Harm

The standard model of tort law assumes that consequent to an accident, only one party to the accident suffers harm. In many accident contexts it is more sensible to assume that both parties actually suffer harm. Take for instance the often cited and empirically important car accident setting. In the event of a collision, both cars are usually damaged. However, we may disregard this simplification of the standard model as long as it has no effects on the conclusions derived. This irrelevance is questioned by Leong (1989). In his model, expected losses of the injurer and the victim depend negatively on both injurer and victim care.55 One of his findings is that victims do not take optimal care under negligence. He reasons as follows. If injurers take due care to avoid being liable, then victims choose care without considering 54 Indeed, in a setting of stochastic harms, optimal non-care-contingent liability might take a threshold form so that in the event of high harm realizations injurers pay all assets, whereas no compensation is required in the event of low harm realizations. 55 Reference to the case of bilateral harm has been made before the contribution of Leong, however, without comprehensive study (see Rea 1987).

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the effect on injurer’s losses. Complications in attaining efficient care may also arise under strict liability with a defense of contributory negligence. Due to the effect of precaution on the expected losses of both the injurer and the victim, the victim due care level may be so high that the victim rather chooses care lower than the standard since that level minimizes the sum of precaution costs and expected victim losses. It turns out that Leong (1989) uses an assumption which is critical for the results but does not necessarily accord with reality. He assumes that victims cannot be held liable for injurer’s losses.56 This is highlighted by Arlen (1990), who asserts that standard liability rules induce efficient care as long as it is accounted for that individuals engaged in bilateral-harm contexts are potential tortfeasors and potential victims at the same time. Consequently, each may sue the other.57 Returning to the finding of Leong (1989) spelled out above, negligence erects a standard of care directed at both parties, which is why the particular result of Leong (1989) will not occur in equilibrium. Consequently, the simplification of treating harm as unilateral is often without bearing on the conclusions attained. However, there are circumstances in which incentives are dependent on whether only one or both parties eventually suffer harm. Dharmapala and Hoffman (2005) establish an example by showing that efficient care incentives may result in a model characterized by interdependent precaution costs in the bilateral-harm case, whereas the efficient care equilibrium is not attained in the unilateral-harm case. The outcome concerning the unilateral-harm framework is attributed to a lack of available causes of action. Assume simple negligence as the liability rule and that injurers adhere to the due care level, which is set equal to efficient care. The victim bears total expected harm and chooses care, taking into account the effect on her costs and expected harm. Hence, she does not internalize the effect of her care on injurer costs of care. If care by one party lowers precaution costs of the other party, injurer costs of care turn out to be higher than optimal as a consequence of the incomplete internalization by the victim. The injurer has no cause of action for compensation of the damage in the form of higher 56

This is already acknowledged in Leong’s concluding remarks. ”With rare exceptions, the law allows the injured parties in an accident to sue each other ... Such a suit can be factored into two parts and analyzed as if it were two separate accidents.” Cooter and Ulen (2004), 331. 57

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precaution costs. Thus, the equilibrium entails an inefficient care choice by at least one party. In contrast, in the bilateral-harm context, standards are directed at both parties and there may be incentives to adhere to the standard for both parties, given due care by the other party. This follows since taking standard care removes the burden of the expected harm of the other party. The very interesting analysis by Dharmapala and Hoffmann (2005) with the contrasting result for unilateral harm and bilateral harm is the igniting spark for the analysis presented in Chapter 7. In ’On the Similarity of Bilateral Harm and Unilateral Harm with Role-Type Uncertainty’, we show that their result depends on role-type certainty. The introduction of role-type uncertainty transforms the ex ante status into one of bilateral harm in expectation terms. Since both parties can turn out to be the victim, both parties bear some expected harm. This can ensure that both parties take efficient care, as in the bilateral-harm framework. Arlen (1992b) considers a bilateral-harm framework in which individuals are risk averse, can obtain actuarially fair insurance, and suffer physical harm. Physical harm results in pecuniary and non-pecuniary losses. Suffering from non-pecuniary losses is captured by a change in the utility function, implying for utility of given wealth a different absolute magnitude and derivative. This, in turn, implies that optimal risk spreading cannot be described by full insurance coverage. In such a setting, the standard model with unilateral harm allows a conclusion to be drawn that rules based on strict liability cannot induce efficiency because they attain full insurance of the victim. For the bilateral-harm setting, Arlen (1992b) shows that strict liability with a defense of contributory negligence can obtain the efficient outcome. This requires that both parties to the accident face the same liability, as a consequence of which payment and receipt of compensation in the event of an accident cancel out. This allows for optimal risk-spreading, and that liability is sufficient to make the efficient equilibrium incentive compatible and unique.58 A key difference between the two models is that, in the unilateral-harm framework, the damage measure simultaneously controls care-taking and risk-spreading, whereas, in the bilateral58 It is possible to have levels of liability which are different for respective parties if further conditions are met.

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harm framework, the magnitude of damages determines care incentives and the difference between respective liability is important for risk-spreading.

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Chapter 3

On the Incentive Effects of Damage Averaging in Tort Law∗ Abstract It has been generally accepted for unilateral-care models that care incentives are not affected by the use of either accurate damages or average damages if injurers lack knowledge of the precise damage level they might cause. This paper shows that in bilateral-care models with heterogeneous victims, consequences of averages as damage measure are critically dependent on the weighing of respective harm levels. Importantly, we establish that there is an average measure which allows the attainment of efficient care in the bilateral-care framework.

Keywords: accuracy, average harm, expected damages, care interdependency

JEL-Classification: K 13, K 42, H 23

∗ This chapter is an edited version of: Friehe, T. (2007). On the Incentive Effects of Damage Averaging in Tort Law. Economics Bulletin 11: 1-7. This work was presented at the 2005 Annual Meeting of the Austrian Economic Association in Innsbruck.

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1

Introduction

The optimal functioning of tort liability critically depends on efficient damage awards (Arlen 2000). However, courts cannot always calculate victim damages correctly and without cost. Court errors in estimating damages and administrative costs for the accurate determination of damages are important factors that may plague tort liability. Dari Mattiacci (2005) and Singh (2003), for instance, analyze the effects of biased court errors in assessing victim damages under different liability rules. Regarding administrative costs, Kaplow and Shavell (1996), using a unilateral-care model, find that if injurers do not know harm ex ante, care incentives are unaffected by the court’s choice between accurate and expected damages. Then, administrative costs necessary for the accurate assessment can be saved because accuracy does not influence injurer behavior and, therefore, is costly without associated gain. We approach this same question by inquiring whether using the harm average as the damage measure also leaves injurer incentives unaffected in the accident model with bilateral care. This is of interest as the assumption of bilateral care applies to more real-life contexts than that of unilateral care. Furthermore, the use of the average is not only a matter of theoretical inquiry since damage averaging is actually applied in class action suits.1 The fact that both parties can take care complicates the analysis. It is no longer unambiguous how the average to be used as damage measure ought to be constructed and no longer clear whether damage averaging continues to be irrelevant for the outcome. This paper establishes that there are different ways of weighing individual harm levels which correspond to different outcomes. Very importantly, we find that there is an average measure available which allows for the efficient outcome. In Section 2, we first lay out the model used. Next, we provide the analysis of different average measures. Subsequently, we discuss assumptions used in that analysis, namely standard care by victims and courts observing victim type. Section 3 concludes. 1 Given different individual damage levels, this practice - inter alia - affects who intitiates a class action suit (Marceau and Mongrain 2003) and the composition with respect to damage levels of the class action (Che 1996).

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2

The Model and Analysis

2.1

The Model

The model is based on the standard tort model (see, e.g., Shavell 1987). All parties are risk-neutral. Injurers are identical and decide on continuous care x with unit costs of one. We consider two victim types. Victim type j, j = L, H, can take continuous care yj with unit costs of one. Victims are identical except for their density function on the ¯ The high-damage victim has more cumulative density on support of damages, d ∈ [d, d]. higher damage magnitudes, causing her expected value of damages to be higher than the expected value of the low-damage type, E[d, H] = DH > DL = E[d, L].2 Neither injurers nor victims can anticipate the precise level of damages that follows from an accident. Care reduces the accident probability p(x, yj ) at a decreasing rate, i.e., pyj (x, yj ), px (x, yj ) < 0 < pyj yj (x, yj ), pxx (x, yj ) holds. We assume that injurer and victim care are substitutes with respect to the accident probability, pxyj (x, yj ) > 0.3 The liability rule in place is strict liability with a defense of contributory negligence.4 Parties are completely informed about their expected payoffs and the applicable legal standard. Care levels are observable and verifiable in court in the case of litigation. Shares of different victim types, α as low-damage and (1 − α) as high-damage victim fraction, are common knowledge. The social goal is the minimization of total social costs, consisting of precaution costs and expected harm.

2.2

Analysis

Following Kaplow and Shavell (1996), we assume that injurers cannot anticipate which victim type they might harm. This, along with our assumption of stochastic damages, 2

The use of stochastic damages separates the type identification from the damage level assessment. This assumption eases our analysis, especially with respect to the discussion of victim incentives, and is quite usual in the literature (e.g., Miceli 1997: 18). 4 This liability rule entails compensation made in equilibrium, which is important for our interest. Negligence allows for efficient care. If injurers adhere to the standard, respective victim types optimize given their full damage and standard injurer care without consequence of the compensatory measure. Similarly, negligence induces efficient care more often than strict liability in a context of errors in measuring damages (e.g., Miceli 1997: 35). 3

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prevents the attainment of the first-best, in which care choices are tailored to the damage level (see also Ganuza and Gomez 2005). Accordingly, social costs are defined as follows

SC(x, yL , yH ) = x + αyL + (1 − α)yH + αp(x, yL)DL + (1 − α)p(x, yH )DH

(1)

∗ solve the first-order conditions of the associated Socially optimal care levels x∗ , yL∗ and yH ∗ minimization problem.5 It holds that yL∗ < yH since

dy ∗ dD

py = − pyy > 0. D

Individuals minimize the sum of private care costs and expected liability. Due care is assumed to equal the efficient level for the respective victim types yj∗. The victim type is observable in court, whereas the accurate damage magnitude is not.6 If victims are accurately compensated by injurers, established reasoning shows that victims take ∗ ) may result due care and injurers choose x∗ . However, the efficient outcome (x∗ , yL∗ , yH

in this setting even without spending on the accurate assessment of damages. If typespecific expected harm is used instead of accurate damages and yj = yj∗, the efficient-care equilibrium is attained. This results because the first-order condition of the injurer is the same as in the case of accurate damage assessment. Hence, the result of Kaplow and Shavell (1996) can be carried forward to the bilateral frame, with due qualification. We now turn to damage averaging, as average harm is usually referred to in the literature.7 Above, we argued that type-specific expected harm attains the efficient outcome. To realize the difference between an average measure which is applied across the board and using type-specific expected harm, imagine that the number of victim types approaches infinity. Then, in the latter case, courts would use an infinite number of compensation measures, whereas there is only one uniform measure with damage averaging. In the following, we assume that victims stick to their type-specific due care. This assumption is discussed explicitly in the next section. 5

The specification of p(x, yj ) suffices for the second-order conditions. Consider the following example. Highly educated individuals are more likely to suffer higher losses in earnings than poorly educated persons. The educational status is easily observed as part of the trial proceedings, whereas the assessment of accurate damages requires reliance on costly experts and the like. 7 For instance, Arlen (2000: 717) states ”Injurers can be optimally deterred by basing damages on the average harm caused.”, while Dari Mattiacci (forthcoming) notes ”accuracy is needed if parties can foresee the magnitude of the harm; if parties face average expectations, average compensation is cheaper than, and as effective as, accurate compensation.” 6

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It is important to note that there are different ways to average harm over victim types. First of all, there has to be an incorporation of victim-type shares. This leads to

D0 = αDL + (1 − α)DH

(2)

Secondly, we may wish to account for the fact that different victim types might take different levels of care. Different levels of victim care result in different accident probabil which accounts for respective ities. This has its consequences for the observed average D, accident probabilities, and is given by x, y¯H )DH x, y¯L )DL + (1 − α)p(¯  = αp(¯ D αp(¯ x, y¯L) + (1 − α)p(¯ x, y¯H )

(3)

 are the same as where x¯ and y¯j represent equilibrium care levels. Note that D0 and D long as different victim types take the same level of precaution.8 The efficiency characteristics of average measures are at the heart of our study and summarized in the following proposition. Proposition 1 Suppose injurers cannot anticipate victim type, victim types are observable in court, and victims choose yj = yj∗, j = L, H, then (a) damage averaging distorts injurer care incentives, unless the average measure equals D∗ =

∗ )D +(1−α)p (x∗ ,y ∗ )D αpx (x∗ ,yL x L H H ; ∗ )+(1−α)p (x∗ ,y ∗ ) αpx (x∗ ,yL x H

(b) injurers exert more care than the socially optimal level x∗ , if (i) D0 is used as average measure, and (ii)

∂px (x,yj ) ∂yj

 is used as average measure (with px > 0 holds in the case D

as elasticity of the accident probability with respect to injurer care); (c) the individual marginal benefit of injurer care is greater for the average measure D0  than for D. Proof. (a) Given marginal costs of injurer care of 1, injurers choose the same care level with a damage average as with accurate damages if the marginal benefit of injurer care 8 Shavell (1987: 152) states for the unilateral case ”If ...the magnitude of liability for an accident equals ... expected losses conditional on the occurrence of an accident, then liable injurers ... will be led to act  we set liability equal to the expected loss conditional on the optimally under liability rules.” By using D, occurrence of an accident.

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is larger than (equal to/ less than) 1 for x < (= / >)x∗ . We equate the marginal benefit of injurer care given accurate damages with that given the damage measure D ∗ ∗ ∗ −αpx (x∗ , yL∗ )DL − (1 − α)px (x∗ , yH )DH = −αpx (x∗ , yL∗ )D ∗ − (1 − α)px (x∗ , yH )D ∗

(4)

and solve for D ∗ , which has required properties. Note that D ∗ = D0 if pxyj (x, yj ) = 0 holds. (b) (i) We evaluate the first-order conditions for injurer care at x  which solves the condition if D0 is used. ∗ ∗ −αpx ( x, yL∗ )DL − (1 − α)px ( x, yH )DH = −αpx ( x, yL∗ )D0 − (1 − α)px ( x, y H )D0 (= 1) (5)

∗ x, yL∗ ) = px ( x, y H ). The We insert D0 = αDL + (1 − α)DH and simplify to reach px (

marginal effect of injurer care on the accident probability is absolutely larger for lower victim care levels if injurer and victim care are substitutes. Thus, injurers exert more care if D0 is used as a compensation measure. (ii) Proceeding similarly to (i), with x solving  is used, the condition we find after simplification the first-order condition for x if D is px (x, yL∗ ) =

x ∗ ∗ ) px (x, yL ) p(x,yL

=

x ∗ ∗ ) px (x, yH ) p(x,yH

∗ = px (x, yH ). Larger care if the average

 is used, identified by |px (x, y ∗ )|> |px (x, y ∗ )|, can only result for substitutional measure D L H care. (c) We proceed as above, that is we relate the marginal benefit of injurer care when  at x = x. After simplification, we get α(−1)px (x, y ∗ ) > using D0 and when using D L ∗ (1 − α)px (x, yH ).

Comments: (a) If victims can be held to type-specific standards, it is not efficient to use average expectations concerning damages unless the marginal damage average D∗ is available and  in that employed. The marginal damage average differs from the observed average D the first derivative of the accident probability function with respect to injurer care is used instead of the absolute level. This ensures that care incentives at the margin are not distorted. Substitutional care is usually assumed in the literature (e.g., Ganuza and 69

Gomez 2005) and is sufficient (necessary) for supraoptimal injurer care in response to  Note that the marginal effect of injurer care on the damage averaging with D0 (D). accident probability function being dependent on the level of victim care is critical for our results. If, for instance, p(x, yj ) = q(x) + r(yj ) holds, then the simplest measure D0 , which only allows for respective victim fractions, suffices for efficiency. (b) The divergence in care incentives originates from the fact that the marginal reaction of expected liability to injurer care is no longer equal to the response of expected damages to care. The injurer needs to incorporate the fact that victims with higher (lower) damage potential take more (less) care, as she does under full compensation. The different levels of victim care imply different levels of productivity of injurer care with respect to its application to the accident probability function of low-damage and high-damage victims.  is preferable to D0 as the arising distortion in injurer care (c) With substitutional care, D is smaller.

2.3

Discussion

In this subsection, we provide results for the case in which two assumptions used above do not necessarily hold. First, we scrutinize victim care incentives instead of assuming that victims take standard care, and, second, we consider the case in which the type observation for the court is costly. 2.3.1

Victim Incentives to Comply in the Case of Damage Averaging

Both victim types know that if they take due care, they will be compensated by the  D∗ }, whereas they will be punished ¯ D ¯ ∈ {D0 , D, amount of the average measure used D, for suboptimal care by no compensation. Regarding victim care, the use of average damages as a compensation measure, on the one hand, questions the optimality of due care for the high-damage victim because the usual discontinuity in victim costs at efficient care is weakened by the fact that compensation does not cover full harm. On the other hand, we need to acknowledge that low-damage victims not only recover full harm but actually gain something. Indeed, the size of the respective differences that the victims 70

¯ > 0 and DL − D ¯ < 0. bear if they take at least standard care is DH − D The victim cost function of type j is ⎧ ⎪ ⎨ yj + p(x, yj )(Dj − D) ¯ yj ≥ yj∗ V Cj (x, y) = ⎪ ⎩ yj + p(x, yj )Dj yj < yj∗

(6)

The injurer cost function depends on the care choices of both victim types in the following way ⎧ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x + αp(x, yL)D ¯ IC(x, y) = ⎪ ¯ ⎪ x + (1 − α)p(x, yH )D ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x + αp(x, y )D ¯ + (1 − α)p(x, yH )D ¯ L

∗ y H < yH and yL < yL∗ ∗ y H < yH and yL ≥ yL∗ ∗ yH ≥ yH and yL < yL∗

(7)

∗ yH ≥ yH and yL ≥ yL∗

The injurer minimizes individual costs with respect to care, given victim care. No ∗ and yL < yL∗ hold. Suppose only one victim type adheres to care is chosen if yH < yH

the standard, for instance, the low-damage type. The injurer chooses x = arg min{x + ¯ with x smaller than x∗ . The same argument applies if only the high-damage αp(x, yL∗ )D} ∗ victim adheres to the behavioral standard. Finally, consider the case yH ≥ yH and

yL ≥ yL∗ . We can conceive of x being greater than, equal to or less than x∗ . We reason in three steps what victim decisions can be expected. (1) It is never optimal for victims to respond with yj < yj∗ to x ≤ x∗ . The minimization of the respective victim cost function yields that it is preferable to choose at least standard care because yˆj = arg min{yj + p(x, yj )Dj } is greater than or equal to yj∗ if x ≤ x∗ . In addition, adherence to standard care yields at least some compensation. We stated above that if only one victim type exerts at least standard care, the injurer takes care less than x∗ . Thus, there is no equilibrium in which at least one victim is negligent and the injurer takes care less than x∗ . (2) It is possible that victims respond with substandard care to injurer care being substantially larger than x∗ . However, the injurer responds individually optimal to victims being negligent by lowering care below x∗ . According to (1), victims respond to this by 71

increasing their care to at least standard levels. Consequently, there may exist only a mixed strategy equilibrium if the injurer responds to victim standard care by very large x > x∗ which is reciprocated by a drop in victim care.9 To exclude this possibility requires ¯ for any assuming the following: It holds that [p(x, yˆj ) − p(x, yj∗)]Dj > yj∗ − yˆj − p(x, yj∗ )D ¯ This states the following. The care choice x > x∗ lowers the incentive for x > x∗ and D. victims to exert care. It is true that for x > x∗ , it holds that −

∂p(x,yj∗ ) Dj ∂yj

< 1. Then, the

¯ is ignored. increase in victim care from yˆj to yj∗ is not worthwhile as long as the effect of D Relating this to the assumption, the difference in probabilities weighted with the respective damage Dj is lower than the difference in care levels yj∗ − yˆj . The assumption declares ¯ > DL into account makes the investment in victim care that taking the compensation D cost-justified. Consequently, there is no mixed strategy equilibrium if we exclude cases in ¯ as which the difference in care levels is not sufficiently weakened by the compensation D to make standard care preferable. This will not be restrictive except for rather extreme contexts. Consolidating the arguments made in (1) and (2), we conclude that, by and large, there is no equilibrium in which victims do not exert at least standard care. ∗ and yL ≥ yL∗ . The low-damage victim (3) It may be that x < x∗ results for yH ≥ yH

¯ < 0. However, the highcertainly does not take more than standard care since DL − D damage victim might prefer to take care beyond the standard. The first-order condition H) ∗ ¯ = 1. There , is given by − ∂p(x,y (DH − D) of the high-damage victim, given yH ≥ yH ∂yH

are two effects: First, due to low injurer care, x < x∗ , the absolute value of the partial derivative of the probability function is larger than that in the first-order condition of the social problem with x∗ . This indicates an increase in the incentive to take care for the ¯ < DH . high-damage victim. Second, this increase is at least partially offset by DH − D So, if care chosen by the injurer does not deviate too strongly from x∗ , which contains ¯ is sufficiently different from DH , which enforces the second effect, the first effect, or if D ∗ . the high-damage victim will stick to standard care yH

9

See Endres and Querner (1995) for a closer description of such a case of circularity in tort law.

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2.3.2

Information on Type and Magnitude Is Costly

Assume that courts cannot observe the type without cost. In consequence, courts most likely use one care standard.10 The social cost function which reflects that injurers cannot anticipate victim type and that the due care level for victims will apply irrespective of the type reads

SC(x, y) = x + y + αp(x, y)DL + (1 − α)p(x, y)DH

(8)

Let (x , y  ) minimize (8) and note that victim care being undifferentiated is costly with respect to SC. Within this context, does the use of practical damage averages distort injurer care incentives? Proposition 2 Suppose injurers cannot anticipate victim type, victim types are not observed in court and victims choose yj = y  , j = L, H, then injurers will respond in accor is used as a compensation dance with the social interest to uniform victim care if D0 or D measure. ¯ = Proof. The first-order condition of the injurer given standard victim care is 1+px (x, y  )D ¯ = D0 . Likewise, incentives are unaf0. Injurer care incentives are socially optimal if D ¯ = D,  as D  = D0 = D ∗ in this case. fected if D

3

Conclusion

Victims frequently differ in their damage levels. In the unilateral-care framework, it does not warp care incentives if the court applies average damages instead of accurate damages in the case in which injurers cannot anticipate the victim type. We consider the accident model with bilateral care. This has an ample effect on the averaging of damages. We establish that two measures which put themselves forward due to their simplicity distort care incentives. However, we also establish that there is an average measure which induces 10 A uniform care standard in face of agent heterogeneity, labeled reasonable man standard, has been studied in the context of heterogeneous precaution costs. A recent contribution is Miceli (2006).

73

efficient injurer care. The special feature of this measure is that it not only accounts for different levels of victim care but also for their consequences at the margin.

References Arlen, J. (2000). Tort Damages. In: Boudewijn Bouckaert and Gerrit De Geest (eds.), Encyclopedia of Law and Economics, Vol. II, Cheltenham: Edward Elgar: 682-734. Che, Y.-K. (1996). Equilibrium Formation of Class Action Suits. Journal of Public Economics 62: 339-361. Dari Mattiacci, G. (2005). Errors and the Functioning of Tort Liability. Supreme Court Economic Review 13: 165-187. Dari Mattiacci, G. (forthcoming). Tort Law and Economics. In: Hatzis, A. (ed.), Economic Analysis of Law: A European Perspective, Cheltenham: Edward Elgar. Endres, A. and I. Querner (1995). On the Existence of Care Equilibria Under Tort Law. Journal of Institutional and Theoretical Economics 151: 348-357. Ganuza, J. J. and F. Gomez (2005). Caution, Children Crossing: Heterogeneity of Victim’s Cost of Care and the Negligence Rule. Review of Law and Economics 1: Article 3. Kaplow, L. and S. Shavell (1996). Accuracy in the Assessment of Damages. Journal of Law and Economics 39: 191-209. Marceau, N. and S. Mongrain (2003). Damage Averaging and the Formation of Class Action Suits. International Review of Law and Economics 23: 63-74. Miceli, T. J. (1997). Economics of the Law. Torts, Contracts, Property, Litigation. Oxford: Oxford University Press. Miceli, T. J. (2006). On Negligence Rules and Self-Selection. Review of Law and Economics 2: Article 1. Shavell, S. (1987). Economic Analysis of Accident Law. Cambridge, MA: Harvard University Press. Singh, R. (2003). Efficiency of ’Simple’ Liability Rules When Courts Make Erroneous Estimation of the Damage. European Journal of Law and Economics 16: 39-58.

74

Chapter 4

On the Superiority of Damage Averaging in the Case of Strict Liability∗ Abstract For the case of strict liability and unilateral care, the literature argues that, if injurers cannot anticipate the precise level of damages, courts might use the damage average without distorting care incentives. This paper shows that the use of the damage average is in fact preferable if one allows victims to choose the object at risk. If the court insists on accurate compensation of the harm suffered, this induces victims to choose an inefficiently high object value. In contrast, restricting the compensatory transfer to the damage average instills the first-best outcome.

Keywords: strict liability, incentives, care, endogenous harm

JEL-Classification: K 13, D 62

∗

This chapter is joint work with Florian Baumann. Both authors contributed equally to the project.

75

1

Introduction

The optimal functioning of tort liability is critically dependent on efficient damage awards (Arlen 2000). Referring to the unilateral-care model and strict liability, we briefly sketch the reasoning on the optimal compensation magnitude to highlight where our central point is made. If the magnitude of harm is known and liability equals the harm done, incentives to take care are optimal. If instead, liability exceeds (falls below) the actual level of harm, the injurer’s incentives will be superoptimal (suboptimal). Assuming that the setting is changed solely due to the magnitude of harm being stochastic, the injurer will minimize social costs if the compensation is equal to either the precise harm level for each accident or average harm. Both types of compensation induce the optimal care choice because they do not differ from the injurer’s ex ante perspective (see, e.g., Arlen 2000, and Dari Mattiacci forthcoming). This paper argues that the choice between accurate and average compensation might be irrelevant for obtaining efficient care incentives but not for overall efficiency. To show this, we sensibly assume that victims adapt their choices according to the legal setting or in more precise words, that the victim’s choice of the value of the object exposed to the accident risk depends on the liability regime. For instance, individuals tend to derive higher utility from driving valuable vehicles than from driving others due to performance or other aspects. Furthermore, individuals are likely to differ in their preferences on the value of objects, meaning that the efficient value will vary accordingly. The application of the liability rule determines whether the expected harm from an accident is incorporated into the calculus of the individual when determining the object value. Full compensation of the victim excludes expected damages from the victim’s optimization, whereas, as we will show, the use of average harm allows for correct marginal incentives. Considering the endogeneity of the harm in this fashion is to our knowledge nouveau to the literature.1 There is little literature closely related to our pursuit. Another paper that points 1

In some frameworks, the magnitude of harm is variable and may be affected by precaution in a selfinsurance sense (see, e.g., Dari Mattiacci and de Geest 2005). Other variants of the economic model of accidents allow for a variable activity level (see, e.g., Shavell 1980), which might be likened to our model. However, both aspects are different as regards content and formal structure. See, for instance, Arlen (2000) for a survey and discussion of the activities that have been considered in the tort law realm.

76

to the advantage of using average harm as a compensation measure is by Kaplow and Shavell (1996). They use a unilateral-care model with heterogeneous victim harm levels and assume that the assessment of the harm is costly. In reference to the fact that care incentives are unaffected by the court’s choice between accurate and expected damages if injurers do not know precise harm ex ante, Kaplow and Shavell argue that courts should use the average to save on administrative costs. We do not consider assessment costs but show that the use of the average as a compensation measure is advantageous to ensure efficient incentives for all affected individuals. Thus, we promote the use of compensation of the average harm to ensure overall efficiency. Damage averaging is also referred to in Dari Mattiacci and de Geest (2005), who touch upon compensation of average harm as a means to improve incentives in the context of the judgment-proofness problem. The use of an average as compensation measure can actually be found in class action suits. Given different individual harm levels, this practice - inter alia - influences who initiates a class action suit (see Marceau and Mongrain 2003) and the composition with respect to damage levels of the class action (see Che 1996). In the next section, we lay out the model. In the analysis, we compare the outcomes under accurate compensation and compensation of the average harm to the social optimum, and derive our central result. Section 4 concludes.

2

The Model

The model involves injurers and victims, whose respective populations are of equal size. In an accident, one injurer harms one victim. The accident probability p(x) decreases at a diminishing rate in the care the injurer takes, x,

dp(x) dx

< 0 and

d2 p(x) dx2

> 0. We assume

strict liability as liability rule so that the injurer is legally liable irrespective of her care level. The level of liability is denoted D(l) and might depend on the level of losses actually suffered by the victim, l.2 The victim suffers harm in the event of an accident. She has discretion, however, 2

If the compensation is a function of the losses, we assume that conditions.

77

d2 D dl2

≤ 0 to ease the second-order

regarding the value placed at risk. We distinguish different victim types θ ∈ [θ, θ], where the type indicates the individual’s esteem for a given level of value. The victim type is private information. We take F (θ) as the commonly known distribution function over types. The value placed at risk is synonymous to the level of losses incurred in the case of an accident, l. The utility function of victim type θ is U V (l, θ) = u(l, θ) − l − p(x)[l − D(l)]

with

∂u(l,θ) ∂u(l,θ) ∂ 2 u(l,θ) , ∂θ , ∂l∂θ ∂l

> 0 and

∂ 2 u(l,θ) ∂l2

(1)

< 0. Consequently, we assume that the victim

in all contingencies enjoys the value placed at risk as it is replaced in the accident contingency, causing additional costs of l − D(l) for the victim. The individually optimal value level for type θ, l(θ), follows from the first-order condition ∂u(l, θ) dD(l) ∂U V (l, θ) = − 1 − p(x)[1 − ]=0 ∂l ∂l dl and is increasing in the type since

dl dθ

(2)

∂ 2 u(l,θ)

= − ∂ 2 u(l,θ)∂l∂θ

2D dl2

+p(x) d

∂l2

> 0. The injurer minimizes her

expected total costs, which are the sum of expected liability and precaution costs. She cannot generally anticipate the precise level of damages she will be required to compensate in the event of an accident and, thus, optimizes subject to the distribution of compensation levels for all victim types. The objective function may be stated as 

θ

U I (x) = −x − p(x)

D(l(θ))dF (θ)

(3)

θ

and therefore the injurer behaves to satisfy dU I (x) dp(x) = −1 − dx dx



θ

D(l(θ))dF (θ) = 0.

(4)

θ

The above specifies the injurer and victim behavior given strict liability using the yet unspecified transfer D(l).

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3

The Analysis

In this section, we derive the social optimum, and compare it to the outcome under two alternative specifications for the level of compensation, accurate and average compensation. Social Optimum We assume that the benevolent social planner seeks to maximize the sum of individual utilities of society’s constituents. Individuals replace the value lost in the event of an an accident. Consequently, we consider this use of resources as opportunity costs in the welfare function. Moreover, the planner acknowledges that injurers, like herself, optimize given only knowledge on the distribution of victim types. This implies the maximization of  SW (l(θ), x) =

θ

[u(l(θ), θ) − (1 + p(x))l(θ)] dF (θ) − x.

(5)

θ

The according first-order condition with respect to the level of losses for each type θ, given socially optimal care x∗ , is ∂u(l∗ (θ), θ) − 1 − p(x∗ ) = 0. ∂l(θ)

(6)

The socially optimal care level, x∗ , fulfills the first-order condition −

dp(x∗ ) dx



θ

l∗ (θ)dF (θ) − 1 = 0.

(7)

θ

The optimal value level for type θ, l∗ (θ), requires a positive net benefit in order to account for the possibility that the value is lost in the event of an accident. The optimal level of care ensures the equalization of the marginal benefit, the reduction of the probability of an accident given the distribution of socially optimal values exposed to the risk, and the marginal costs, amounting to one. The decentralization of the choices via a liability rule needs to consider that victim types are private information. This hinders a direct regulation of the value of victim type

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θ.3 Accurate Compensation In many jurisdictions, courts strive to make victims whole in the sense that the compensation transfered by the injurer should put the victim in a position as if the accident had not occurred (see, e.g., Shavell 2004: 237). Consequently, the case treated here can be regarded as common court practice. This practice implies D(l(θ)) = l(θ). First of all, we check the performance of this legal setting in our framework with respect to injurer incentives to take care. The injurer’s first-order condition (4) can now be stated as dp(x) dU I (x) = −1 − dx dx Using

dD(l) dl



θ

l(θ)dF (θ) = 0.

(8)

θ

= 1 in (2), we obtain the victim’s first-order condition with respect to the

magnitude of the value ∂U V (l, θ) ∂u(l, θ) = − 1 = 0. ∂l ∂l

(9)

This first-order condition describes the private optimum of the victim, which does not coincide with the social optimum. The reason is that the victim does not take into account the possibility that the value perishes due to an accident. The victim is in fact fully insured by the application of the liability rule. The condition regarding the precaution of the injurer replicates the condition known from the social optimization and therefore corresponds to the optimal response given the choice of l(θ). Consequently, the victim chooses an inefficiently high value of the object at risk and the injurer responds to this with a care level higher than in the social optimum. Compensation of the Average Harm If the compensation consists of the average harm, then most victims are not made whole. Instead, some receive more than their harm, whereas others remain undercompensated. However, our interest resides in the incentive effects concerning the choice of precaution and the choice of object value. Assuming θ compensation of the average harm implies D(l(θ)) = l = θ l(θ)dF (θ). 3 It is undisputed that it is very difficult to ascertain individual benefits. This also reasons why care standards can be regulated via standards, whereas activity levels practically cannot (see, e.g., Shavell 2004).

80

Checking (4) yields that injurers have optimal care incentives in this setting as well. Turning to victims, using

dD(l) dl

= 0 in (2), we obtain

∂U V (l, θ) ∂u(l, θ) = − 1 − p(x) = 0 ∂l ∂l The choice of victim type θ does not affect the average loss,

(10)

θ θ

l(θ)dF (θ). This ensures

that the compensation obtained does not change with the choice of l by the individual victim. Note that this is not peculiar to our reference of average harm but is indeed a necessary prerequisite for the attainment of the efficient level of losses, l∗ (θ). The compensation term enters the objective function as a constant with regard to the variable chosen by the victim. Consequently, the victim takes the full effect of a higher value on the expected harm into account at the margin and decides in accordance with the social optimum. Thus, the resulting outcome is x = x∗ and l = l∗ (θ) for all θ.

4

Conclusion

In many jurisdictions, full compensation of the victim is the starting principle that courts try to adhere to. The literature on the economic analysis of tort law has pointed out that it is without incentive effects if the court uses the average harm instead of the accurate harm in cases where injurers cannot tell the precise magnitude of harm they might cause. Our contribution lies in showing that the use of the average harm is preferable to the accurate harm if the victim can choose how much to expose to the accident risk. Full compensation creates an externality since victims do not consider the expected harm in their calculus as they are fully insured by the liability regime. In contrast, compensation of average harm instills optimal incentives at the margin for both, the injurer and the victim.

81

References Arlen, J. (2000). Tort Damages. In: Bouckaert, B., de Geest, G. (Eds.), Encyclopedia of Law and Economics, Vol. II. Cheltenham: Edward Elgar: pp. 682-734. Che, Y.-K. (1996). Equilibrium Formation of Class Action Suits. Journal of Public Economics 62: 339-361. Dari Mattiacci, G. (forthcoming). Tort Law and Economics, in Hatzis, A. (Ed.), Economic Analysis of Law: A European Perspective. Cheltenham: Edward Elgar. Dari Mattiacci, G. and G. De Geest (2005). Judgement Proofness Under Four Different Precaution Technologies. Journal of Institutional and Theoretical Economics 161: 38-56. Kaplow, L. and S. Shavell (1996). Accuracy in the Assessment of Damages. Journal of Law and Economics 39: 191-209. Marceau, N. and S. Mongrain (2003). Damage Averaging and the Formation of Class Action Suits. International Review of Law and Economics 23: 63-74. Shavell, S. (1980). Strict Liability Versus Negligence. Journal of Legal Studies 9: 1-25. Shavell, S. (2004). Foundations of Economic Analysis of Law. Cambridge, MA: Harvard University Press.

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Chapter 5

Screening Accident Victims∗ Abstract This paper considers victim heterogeneity in harm levels in a bilateral-care model, where harm is private information. In practice, resources are expended on the verification of damages suffered. We establish a sufficient condition for the possibility to accurately deduce the harm level from the observable care choice without spending on verification. For cases in which this condition does not hold, this paper sets out a simple screening mechanism that induces victims to reveal their type truthfully and induces optimal care in equilibrium without verification costs.

Keywords: bilateral care, incentives, heterogeneous victims, information asymmetry

JEL-Classification: K 13, K 41, D 62

∗ This work was presented at the 2005 French-German Talks in Law and Economics in Saarbr¨ ucken and the 2006 Conference of the Verein f¨ ur Socialpolitik in Bayreuth.

83

1

Introduction

1.1

Motivation and Main Results

Administrative costs of the tort system are a considerable burden on society. Shavell (2004: 281) concludes for the United States that, on average, every dollar received by a victim creates a dollar or more in administrative costs. The literature on administrative costs has hitherto been concerned with, e.g., the divergence of the private and social motive to bring suit or the relative desirability of cost-shifting rules (see Spier forthcoming). We turn to administrative costs that arise due to the assessment of damages and their efficiency justification. For instance, Shavell (2004: 237) asserts that compensation of the actual level of harm is the starting principle in most legal systems. Consequently, courts and/or litigants usually put forth considerable resources to establish the magnitude of actual damages.1 This paper is concerned with heterogeneous harm levels, bilateral care, and administrative costs associated with the verification of harm levels in court.2 It is often the case that victims have a pretty good picture of the harm suffered after an accident has occurred, whereas courts and injurers are notably less well informed. It might then be expected that victims use this information asymmetry to their advantage by misrepresenting the harm magnitude. For instance, Kaplow and Shavell (1996) state that the primary objective of the plaintiff is usually to collect as much as possible in litigation. Hence, resources spent on the accurate verification appear justified to correct for this tendency. The framework we consider distinguishes victim types by their harm magnitude and assumes that the court cannot tell the victim type without spending on administrative 1 In analogy to court proceedings, the credible establishment of private information usually causes costs in pretrial negotiations as well. Shavell (1989) analyzes voluntary disclosure and discovery in a framework in which plaintiffs have private information and do not incur any costs to credibly reveal this information to defendants. 2 Our focus is on verification per se, not on the fact that it occurs in court. A parallel in pretrial negotiations can, for instance, be seen in the discovery process which might be invoked despite its costliness (see Farmer and Pecorino 2005). If parties can observe care, our elaborations below can be transferred to expenditures associated with pretrial settlement bargaining. With regard to discovery, Hay (1994) concatenates care-taking behavior and discovery expenses, and concludes that in the face of high costs, accuracy may be sacrificed to lower social costs.

84

costs.3 The importance of uncertainty over harm is, for instance, evidenced by the ability of judges to bifurcate trials into one on liability and one on damages (see, e.g., Cooter and Ulen 2004: 428). Our informational structure brings to the fore the role played by administrative costs in determining the harm magnitude. Take the instance of damages for loss of earnings, which is an essential element of tort liability. In establishing the present value of lost future income, a court will typically undertake the following steps: ”(1) wages are forecasted based upon an assumed growth rate; (2) if the injured victim is still employable, wages from an alternative occupation should be subtracted, but if the victim is deceased, amounts attributable to foregone personal consumption should be subtracted; (3) income taxes are subtracted; (4) each year’s remaining amount is then multiplied by the probability that the victim would still have been working if the accident had not occurred; (5) and finally, the residual is discounted to present value at a risk-free rate of interest” (Krauss and Levy 1996: 328). It is obvious that small changes in the assumptions on respective aspects have a tremendous impact on the resulting damages aggregate and may be highly disputed. Furthermore, note that several of these aspects are highly type-specific. For instance, future wages are dependent on the skills the plaintiff possesses as well as his ability to create new capabilities. Likewise, the probability that the plaintiff would still be working is some function of - inter alia - the past employment experience. Consequently, courts usually rely on the testimony of experts in these matters. This creates huge administrative costs, the circumvention of which is the focus of this study. Similar examples amount. For instance, to find a clear example of the importance of administrative costs resulting from the necessity to assess damages, we do not have to stray far afield since many car accident cases similarly involve the testimony of experts, which alone can cause a high relation of administrative costs to accurate harm done. Actually, in the German Code of Civil Procedure, one of the aims of §287 is to allow judges not to hear all evidence offered by the parties when it comes to the magnitude of 3 Imagine that cases are sorted on a dimension according to the ease with which harm suffered can be deduced. On one end of the spectrum, the harm magnitude can be ascertained without further question or non-negligible verification effort, whereas on the other end, the harm level is highly disputable and its assessment is associated with verification costs of a significant magnitude. Our concern is more towards the latter end of the continuum.

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the harm to save on administrative costs. In this regard, our analysis can contribute an economic rationale for this rule since judges only need to observe care to be appropriately equipped to decide on the damage level in our setting. We consider cases in which the expenditure on the accurate assessment can be made redundant because victim behavior reliably reveals the private damage level information. The analysis shows that, in many circumstances, victims do not have an incentive to lie about their true harm level. Furthermore, in cases where there is an incentive to claim the occurrence of a harm level different from the one actually incurred, a simple incentive scheme can effectively screen victims and thereby ensure truthfulness while implementing optimal precaution. The reasoning is as follows. The court cannot observe the harm level but tries to separate privately informed victims with available means. Victims with different harm potential are held to different care standards under strict liability with a defense of contributory negligence.4 The compensation obtained by victims can be made contingent on the care exerted. These variables, care standard and compensation level, can therefore be utilized by courts to distinguish different victim types. To that extent, our approach parallels adverse selection models such as monopolistic price discrimination, for instance, where payment and quantity may be the two variables used to achieve sorting. We find that, in a number of cases, the resulting outcome concerning optimal respective victim care standards already prevents incentives to masquerade as the other victim type. For cases in which this does not hold, we show that reducing the compensation of some victims appropriately can ensure that respective incentive-compatibility constraints hold, which means that lying about the harm magnitude does not occur.5 If it can be guaranteed that victims are telling the truth in the sense that they are choosing type-adequate care levels, there is no longer a need to spend resources on the verification of harm. In an extension, the model is widened to allow for variable activity choices to 4 The analysis focuses on this liability rule because it entails compensatory transfers in equilibrium, which does not hold for negligence, for instance, unless further frictions such as uncertainty on due care are introduced. 5 Despite the fact that full compensation is legal practice, recommendations for a deviation from this principle can be found in the literature. For instance, Calfee and Craswell (1984) argue that in cases of uncertainty, a reduction in the compensation can curb incentives to overcomply with legal standards. Furthermore, Polinsky and Rubinfeld (1988), using a unilateral-care model, find that it is sometimes optimal to compensate victims less than fully.

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reflect participation constraints of individuals. Starting from the outcome that results with free verification, we find that the incentive scheme can improve upon this reference by changing the behavior toward that of the social optimum.

1.2

Relation to the Literature

Kaplow and Shavell (1996) also deal with the accurate assessment of damages. Using a unilateral-care model with heterogeneous victim harm levels, they find that, if injurers do not know precise harm ex ante, care incentives are unaffected by the court’s choice between accurate and expected harm for compensation, and that private incentives to establish damages are often socially excessive. Our model, in contrast, incorporates bilateral care and inquires into the beneficiality of harm verification in that more realistic yet more complicated setting. The idea concerning the incentive scheme is close in spirit to the study of Miceli (2006), in which he defines requirements for an efficient self-selection equilibrium in a model with unilateral care and negligence as liability rule, given the problem of varying injurer precaution costs. The literature on settlements also often employs the assumption of private damage level information of victims but rarely considers care choices (see Spier forthcoming). In a notable exception, Spier (1994) concatenates care-taking and settlement behavior in a model with two victim types and strict liability. Investigating which damage award structure minimizes social costs consisting of unilateral precaution costs, expected harm, and litigation costs, she finds that optimal damage awards are flat and equal to expected damages if litigation costs are high. Remember that we, instead, try to implement efficient care while lowering litigation costs. A recent study with some relation to ours is Emons and Fluet (forthcoming). In that analysis, the accurate magnitude of a real number is in dispute, which may represent damages, and it is acknowledged that the fabrication of evidence is costly but leads to fewer court errors. In the presence of large evidence submission costs, the judge should decide without hearing either of the parties, i.e., put up with the relatively lower error costs.6 6 Literature in the realm of our study but with little bearing on it includes Cooter and Emons (2003), who find a mechanism that ensures truth telling of witnesses in trial. In their model, the true state is observed after the testimony which allows sanctions for false signals. In our framework, there is no

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In the next section, we lay out the model. We start the analysis with free harm level verification for courts and set out the benchmark levels for social costs and care. We proceed to the case in which courts can obtain information on the harm level only at strictly positive verification costs. In Section 3, we discuss the option of screening victims without and with an incentive scheme to implement optimal care. In the fourth section, we extend the framework to allow for variable activity choices of injurers and victims. Concluding comments are given in Section 5.

2

The Model, Benchmark, and Common Court Practice

2.1

The Model

The model builds on that by Kaplow and Shavell (1996). An important departure is our consideration of bilateral care. Due to this extension, we add a defense of contributory negligence to their strict liability rule. Injurers are risk-neutral and identical, and decide on continuous care x ∈ R = [0, ∞) with unit costs of one. Risk-neutral victims of type j = L (H) make up α ∈ (0, 1) (1 − α) of all victims and take continuous care yL (yH ) ∈ R with unit costs of one.7 Injurer and victim care reduce the accident probability, represented by the strictly convex function p(x, yj ), at a diminishing rate, px , pyj < 0, pxx , pyj yj > 0.8 As is standard, we assume pxy > 0 (see, e.g., Ganuza and Gomez 2005). This allows a more straightforward reasoning but is not critical for the result. The harm level Dj suffered by victim type j in the event of an accident is private information, and 0 < DL < DH holds.9 Ex ante, injurers do not know the exact harm level they might ex post verification of the type. Hua and Spier (2005) consider the informational value of litigation for injurers. Thus, they consider litigation which uncovers the specific harm level of the accident and thereby reveals information about this victim. This can be used by other injurers to fine-tune care decisions. In our framework, we have a large population of injurers and victims so that this effect is not present. 7 The findings below cannot generally be transferred to the case of continuous victim types. However, we supply a generalization to the case of N victims where N may be very large. See Appendix A and D respectively 8 Subscripts of p denote partial derivatives. 9 The analysis would not be affected if we were to assume that the harm magnitude recruits itself from a ¯ on which respective types have different distribution functions and compensation is equal to support [d, d]

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cause, i.e., the victim type they might harm.10 To establish the actual harm level, courts have to spend k on verification.11 Parties are fully informed about their expected payoffs and the legal standard. Care levels are observable and verifiable in court in the case of litigation.12 Further, we follow Kaplow and Shavell (1996) in assuming that all accidents result in trials and in neglecting litigation costs other than verification costs.13 The social objective, given constant activity, is the minimization of total social costs, being the sum of precaution costs, expected harm, and verification costs. As we consider contexts with imperfect information, care levels that minimize social costs are not first best. The first-best solution is characterized by injurer and victim care tailored to the harm level. The ignorance of injurers concerning the precise harm prevents the attainment of the first-best solution (see Ganuza and Gomez 2005). We define social costs and call the minimizing care levels socially optimal or efficient, for terminological simplicity. Concerning the timing, we have simultaneous care decisions at the first stage. If there is an accident, the case proceeds to the tribunal which ascertains care levels chosen in stage one. Next, courts spend k if that is necessary to determine the according compensation. This last action is what we will focus on in this study.

Dj . The high-damage victim has more density on higher damage magnitudes, causing her expected value of damages to be higher than the expected value of the low-damage type, E[d, H] = DH > DL = E[d, L]. 10 Kaplow and Shavell (1996) also consider the possibility that injurers acquire the harm level information ex ante. We do not follow in that regard as this seems to exclude many accident contexts. 11 Alternatively, victims may arrange for the verification to be done since documentation of harm is a prerequisite in tort law. The verification costs k will in our analysis be borne by the defeated party. For endogenous litigation expenditures under alternative fee-shifting arrangements, see Farmer and Pecorino (1999). 12 Following the literature, we assume that courts can observe care levels without costs. If we were to allow for expenditure on care level verification, it would leave our analysis unaffected as the expenditure is necessary in every case discussed below and might therefore be interpreted as fixed costs due to the utilization of a liability rule that entails care standards for one side to the accident. If the verification of injurer and victim care causes disproportionate costs, this can be one basis for the decision for a specific liability rule (Dari Mattiacci 2005). 13 These assumptions do not impact on the qualitative nature of our conclusions attained. Concerning the former, we commented on possible analogies of the verification process in courts that can be found in the settlement realm in the introduction. Regarding our abstraction from other litigation costs, we find that we otherwise would deal with particularities already known, such as that some victims may not sue due to the fact that compensation would be lower than individual litigation costs or that injurers do not consider all the costs, harm plus total litigation costs, due to accidents under the American rule. If we were to assume the English rule of allocating legal expenses, the analysis would be largely the same.

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2.2

The Benchmark

In this subsection, we set out our benchmark by considering the case in which courts can verify harm without verification costs. Next, we introduce the costly verification of the harm level and depict how courts commonly deal with harm heterogeneity and positive verification costs. Given the information available ex ante and ex post in this subsection, i.e., injurers cannot anticipate the accurate harm magnitude ex ante and courts can condition care standards and compensation levels on victim type since types are observable ex post without cost, social costs SC are defined as follows SC(x, yL , yH ) = x + α[yL + p(x, yL )DL ] + (1 − α)[yH + p(x, yH )DH ]

(1)

∗ as socially optimal care levels.14 Optimal low-damage victim care is with x∗ , yL∗ , and yH ∗ , since less than optimal high-damage victim care, i.e., yL∗ < yH

dy ∗ dD

py = − pyy > 0. The D

∗ ∗ combination of care levels (x∗ , yL∗ , yH ) and social costs SC(x∗ , yL∗ , yH ) is the benchmark

as it describes the best possible outcome given that the ignorance of injurers cannot be alleviated. The benchmark is attainable by application of strict liability with a defense of contributory negligence. Individuals minimize the sum of private care costs and expected liability. Due care is assumed to equal the efficient level for the respective victim types yj∗ , j = L, H. Total individual costs of victim type j in the benchmark case (B), V CjB , are

⎧ ⎪ ⎨ yj if yj ≥ yj∗ V CjB (x, yj ) = ⎪ ⎩ yj + p(x, yj )Dj if yj < yj∗

(2)

Given x = x∗ , victims respond by taking due care. Given victim due care, injurers choose ∗ ) as an equilibrium. Irrespective of injurer care, victims will x∗ . This establishes (x∗ , yL∗ , yH

never take more than due care. It remains to show that there is no equilibrium in which victims choose care below the due care level. Note that injurers reduce their precaution 14 We assume interior solutions throughout for all care variables. The specification of p(x, yj ) suffices for the second-order condition.

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below x∗ if some victims take substandard care, because some of the expected liability drops from the injurer’s cost function. It is never optimal for victims to respond with yj < yj∗ to x ≤ x∗ . The minimization of the respective victim cost functions always yields that it is preferable to choose standard care because yˆj = arg min{yj + p(x, yj )Dj } ≥ yj∗ if x ≤ x∗ . Thus, there is no equilibrium in which at least one victim is negligent and the injurer takes no more care than x∗ . In sum, strict liability with a defense of contributory negligence implements the social optimum in the case of no verification costs.

2.3

The Common Court Practice

From now on, courts have to spend k to verify the precise harm level of the case at hand, i.e., the victim type is private information. The assessment expenditure can be imagined, for instance, as being caused by the salaries of experts testifying with regard to harm suffered or the critical study of voluminous documentation on harm by the court. Harm assessment is common practice in courts as making the victim whole is a fundamental principle in many legal codices. The social cost function in this case of type verification (TV) considers that administrative costs k arise in the case of an accident because the court verifies damages. SC T V (x, yL , yH ) = x + α[yL + p(x, yL )(DL + k)] + (1 − α)[yH + p(x, yH )(DH + k)] (3) ∗∗ . The minimizing care levels are denoted x∗∗ , yL∗∗ and yH

The implementation of this outcome may be achieved by the following variant of strict liability with a defense of contributory negligence. Victims are fully compensated if they adhere to due care prescribed for their type. In such a case, the injurer will pay the compensation and the verification costs k. In that way, injurers will internalize verification ∗∗ ). If costs and choose x∗∗ as a best response to due care by both victim types, (yL∗∗ , yH

victims are negligent, they are faced with the burden of their harm and verification costs k. In this regard, the modification of the rule is similar to the consideration of litigation

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costs and the English cost allocation rule (see, e.g., Katz 2000).15 Since suing when negligent is dominated, negligent victims will not sue so that costs of victim type j in the case of type verification (TV), V CjT V , are

V CjT V (x, yj ) =

⎧ ⎪ ⎨ yj

if yj ≥ yj∗∗

⎪ ⎩ yj + p(x, yj )Dj if yj < yj∗∗

(4)

Assume that victims adhere to due care. Injurers best response is x∗∗ . Victims will optimally respond to x∗∗ with standard-care taking if yj∗∗ ≤ yˆj + p(x∗∗ , yˆj )Dj

(5)

∗∗ The outcome (x∗∗ , yL∗∗ , yH ) is an equilibrium if this weak inequality holds.16 An alternative

equilibrium would have to entail yj < yj∗∗ since yj > yj∗∗ is dominated by yj∗∗. In such an equilibrium, injurers do not take care. Victims incur yˆj + p(0, yˆj )Dj and would thus find deviating to yj∗∗ always advantageous if (5) holds. If (5) does not hold, then there will be an equilibrium in substandard victim care if yˆj + p(0, yˆj )Dj < yj∗∗ or only a mixed-strategy equilibrium. Irrespective of the outcome, it holds that resultant costs are higher than those of the benchmark. Remark 1 The accurate assessment of harm levels in court, given verification costs ∗ ). k > 0, leads to higher social costs than SC(x∗ , yL∗ , yH ∗∗ ). We can apply the envelope theTake, for instance, the likely outcome (x∗∗ , yL∗∗ , yH

orem to the optimal value function SC T V (α, DL , DH , k) starting at k = 0 to see that implied social costs are higher than in the benchmark. In fact, total social costs contain ∗∗ )]k. Kaplow and Shavell (1996) propose information costs [αp(x∗∗ , yL∗∗ ) + (1 − α)p(x∗∗ , yH 15 The possible consequences of not shifting these expenditures include the familiar inefficient care choice by injurers, the fact that some victims may not sue (if DL < k), and that taking standard care yj∗∗ is no longer generally optimal for victims, if victims also have the option of avoiding litigation. However, note that our main point, namely that the outcome with costly verification is worse than the benchmark, would not be affected. 16 ∗∗ ∗∗ The standard reasoning does not ensure that the inequality holds due to the fact that (x∗∗ , yL , yH ) are distorted due to verification costs.

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the use of damage tables, a radical departure from practice, to save on verification costs without affecting precaution incentives too much.

3

Screening of Victims

The court, in this section, tries to avoid any expenditure on verification. If the court succeeds in this pursuit, the most desirable outcome is the benchmark. However, without verification, courts minimize social costs subject to the uncertainty concerning the type of victims who appear before court. The variables at the disposal of the court to sort victims comprise care standards and compensation levels. Successful screening is described by a menu of care standards and compensation levels that induces every victim to select the care standard intended for the type. The observation of care then allows the identification of the actual harm level of the victim, as there will be a one-to-one relationship of care and harm of each victim type. Courts can observe care taken free of costs but cannot freely verify the harm level of the case. The central question is: Can the court rely on what it can observe, namely the care behavior of the victim, as an indicator of the private information, namely the relevant harm level of the victim, which is verifiable only at a cost? Sole Use of Liability Rule The court could use the liability rule alone, if it were incentive compatible for victims to comply with the respective care standards defined for their type, given the associated compensation levels. Suppose that courts were indeed to base their judgment regarding the magnitude of damages solely on the victim care ∗ ) would obtain DL (DH ) in the case observed. In that case, victims who take yL∗ (yH

of an accident, whereas, for simplicity, victims with other care levels would be judged negligent and left uncompensated.17 The respective victim costs in the case of the sole 17 We can without qualitative consequence on our results use a more realistic liability rule entailing no ∗ ∗ ∗ ∗ compensation for care below yL , compensation of DL for care yL ≤ yj < yH , and DH for yj ≥ yH . See Appendix C on this. The way we treat it here, for one, allows a more straightforward exposition, and, for another, allows more easily to see the parallel to other screening problems.

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use of liability rules (S), V CjS , are ⎧ ⎪ ⎪ yL if yL = yL∗ ⎪ ⎪ ⎨ ∗ V CLS (x, yL) = yL + p(x, yL)(DL − DH ) if yL = yH ⎪ ⎪ ⎪ ⎪ ∗ ⎩ yL + p(x, yL)DL if yL =  yL∗ , yH

(6)

for the low damage victim and ⎧ ⎪ ∗ ⎪ yH if yH = yH ⎪ ⎪ ⎨ S (x, yH ) = V CH yH + p(x, yH )(DH − DL ) if yH = yL∗ ⎪ ⎪ ⎪ ⎪ ∗ ⎩ yH + p(x, yH )DH if yH = yL∗ , yH

(7)

for the high-damage victim. In Subsection 2.2, we have established that victims prefer to exert due care instead of bearing the full expected damages. Hence, victim type j will not ∗ but pick the standard care level, which lowers consider the option of taking yj = yL∗ , yH

her expected individual costs the most. It is possible that victims find it cost-minimizing to pretend to be of the other type by taking the care level prescribed for that type. Such an outcome would be inefficient and is to be prevented. On this matter, note that it is not in the interest of low-damage victims to take the care level intended for high-damage victims if ∗ ∗ + p(x∗ , yH )(DL − DH ) yL∗ ≤ yH

(8)

Likewise, high-damage victims will not take yL∗ if ∗ ≤ yL∗ + p(x∗ , yL∗ )(DH − DL ) yH

(9)

The ranking of three terms is of importance with regard to the above weak inequalities ∗ being fulfilled. These terms are ΔDL = p(x∗ , yL∗ )(DH − DL ), ΔDH = p(x∗ , yH )(DH − DL ) ∗ − yL∗ . There are several possibilities for the ranking of these terms: (i) and Δy = yH

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ΔDL > ΔDH > Δy, (ii) ΔDL ≥ Δy ≥ ΔDH , and (iii) Δy > ΔDL > ΔDH .18 Turning to incentive compatibility of choosing type-adequate care, i.e., inequalities ∗ if the ranking of case (i) (28) and (9), we note that the low-damage victim chooses yH

holds. This is true because low-damage victims trade additional precaution costs against ∗ and a higher expected compensation. In case (i), the high-damage victim chooses yH

therefore behaves as intended. In case (ii), no victim type chooses the care level of the other type, i.e., the care choice is perfectly revealing victim type. Hence, if this ranking occurs, no verification or other means are necessary to accurately deduce the harm level from the care taken. To the contrary, the ranking of case (iii) will give the high-damage victim incentives to act as low-damage victim, i.e., to exert only yL∗ . This is attractive for high-damage victims because the precaution cost saving is greater than the share of the expected liability borne due to the compensation of only DL while DH is suffered. The fact that cases (i)-(iii) are possible shows that we cannot generally exclude mimicking incentives for any type. However, if there are incentives to choose the other type’s optimal care, they are present for only one type. Our interest resides in circumstances in which types’s choices are truthful in the sense that they allow the deduction of the harm magnitude. We have stated that case (ii) sets out these circumstances. In the following, we delineate prerequisites for this case to ∗ ) allows us to derive conditions that are sufficient to occur. The social optimality of (yL∗ , yH

exclude case (i) or case (iii). Combining these, we obtain a sufficient condition to ensure that the ranking of case (ii) ensues. If that condition holds, courts can resolve the issue of harm by ascertaining victim care and deducing the according harm magnitude of the victim at hand. Turning to the exclusion of case (iii) first, we need to establish conditions which prevent ΔDL from falling beneath Δy, i.e., for which the high-damage victim would rather not choose yL∗ . The high-damage victim trades off the saving in precaution costs against the 18

∗ ∗ Note that p(x∗ , yL ) > p(x∗ , yH ) implies ΔDL > ΔDH and excludes some possible permutations.

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change in expected damages and compensation. Restating (9), we can say that if ∗ ∗ )]DH + p(x∗ , yH )DH − p(x∗ , yL∗ )DL Δy < [p(x∗ , yL∗ ) − p(x∗ , yH

(10)

∗ )]DH , and the change in exthe increase in the expected damages, [p(x∗ , yL∗ ) − p(x∗ , yH ∗ )DH − p(x∗ , yL∗ )DL , is higher than the difference in prepected compensation, p(x∗ , yH

caution costs, where the former are the costs of pretending to be the low-damage victim whereas the latter is the benefit. In this case, high-damage victims do not have an ∗ given x∗ incentive to mimic the low-damage victim type. The social optimality of yH ∗ implies that Δy < [p(x∗ , yL∗ ) − p(x∗ , yH )]DH holds. Hence, inequality (10) holds for sure ∗ if p(x∗ , yH )DH ≥ p(x∗ , yL∗ )DL .

Next, turning to case (i), we need to establish conditions that prevent Δy from falling ∗ beneath ΔDH , i.e., for which the low-damage victim would rather not choose yH . The ∗ is chosen but simullow-damage victim bears higher precaution costs in the case that yH

taneously obtains a change in expected damages and compensation. Consequently, we deduce from restating (28) that no mimicking incentives are present for the low-damage victim if ∗ ∗ )]DL + p(x∗ , yH )DH − p(x∗ , yL∗ )DL Δy > [p(x∗ , yL∗ ) − p(x∗ , yH

(11)

∗ )]DL holds. The social optimality of yL∗ given x∗ implies that Δy > [p(x∗ , yL∗ ) − p(x∗ , yH ∗ holds. Inequality (11), therefore, always holds if p(x∗ , yH )DH ≤ p(x∗ , yL∗ )DL . ∗ )DH and p(x∗ , yL∗ )DL is crucial for the exConsequently, the rank order of p(x∗ , yH

clusion of any of the cases. Indeed, both terms have to be equal to exclude mimicking incentives for both types. The rank order can be shown to depend on the accident probability function. For this, we restate the first-order conditions of the social problem with respect to victim care as py (x∗ , yL∗ ) p(x∗ , yL∗ )DL = 0 p(x∗ , yL∗ ) ∗ ) py (x∗ , yH ∗ )DH = 0 1+ p(x∗ , yH ∗ p(x∗ , yH ) 1+

96

(12) (13)

∗ Hence, for p(x∗ , yH )DH = p(x∗ , yL∗ )DL to hold,

∗) py (x∗ ,yL ∗) p(x∗ ,yL

To shed light on this condition, note it holds that

= ∂

∗ ) py (x∗ ,yH ∗ ) (< p(x∗ ,yH

py (x,y) p(x,y)

∂y

=

0) must be satisfied.

p(x,y)pyy (x,y)−py (x,y)2 . p(x,y)2

This

derivative sets out a condition concerning the curvature of the accident probability function.19 If the returns to additional care in terms of reducing the accident probability are sufficiently diminishing, ppyy > (py )2 , then, at optimal care levels, the level of expected harm due to high harm will be greater than the level of expected harm due to low harm, ∗ p(x∗ , yH )DH > p(x∗ , yL∗ )DL . This holds because care is not increased to totally counter

the existing motive for further care, DH − DL .20 We use the above to set out a condition regarding the accident probability function that is sufficient to ascertain that victim types choose type-adequate precaution, which allows courts to deduce the correct harm magnitude. Proposition 1 In a model with heterogeneous victims and strict liability with a defense of contributory negligence, courts can rely on observed care to deduce the harm level while implementing optimal care if the accident probability is such that ∗ )DH = p(x∗ , yL∗ )DL holds (which is true if Proof. If p(x∗ , yH

∂

py (x,y) p(x,y)

∂y ∂

py (x,y) p(x,y)

∂y

= 0 holds. = 0), then, as

was shown, case (i) and (iii) cannot occur. The ranking is indeed that of case (ii). In that case, there are no mimicking incentives, which make care taken a reliable indicator for harm suffered. We have established that simply setting up the schedule of optimal victim care levels suffices for truthful self-selection in some cases. Consequently, courts can deduce damages that victims have actually suffered from care taken. Therefore, no verification costs need to be incurred. The sufficient condition for case (ii) is fulfilled, for example, by an exponential accident probability function, p(x, y) = e−(x+y) , used by Rubinfeld (1987). Note that the given requirement on the probability function,

∂

py (x,y) p(x,y)

∂y

= 0, states that the difference in

the marginal rate of substitution between the compensation and the care level between p (x,y)

y 19 Note that p(x,y) is the elasticity of the accident probability divided by the victim care level. Accordingly, the sign of the considered derivative and that of the elasticity are not generally the same. 20 Take yH + p(x∗ , yH )DH and see that the difference in harm drives additional precaution in the ∗ ∗ ∗ in the derivative 1 + py (x∗ , yL )DL + py (x∗ , yL )(DH − DL ), the first two following way. With yH = yL terms are equal to zero and only the last term reasons for a further increase in care.

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d

different types is constant, irrespective of the level of care,

(

d dD dy dDj

dy

)



= 0.21

In other circumstances, however, incentives to mimic are present. Kim and Feldman (2004), for example, use p(x, y) =

1 . 1+x+xy+y

In this case, low-damage victims find it

advantageous to act as high-damage victims. For such cases, we need to search for an incentive scheme that induces every individual to take type-adequate optimal care. Liability Rule and Incentive Scheme The incentive scheme ideally installs optimal precautionary effort and does not deviate too much from the principle of full compensation since that is rooted in many legal systems. It is often only to a limited extent that the exposure to harm is a matter of choice. That is why we focus on incentive compatibility in the following and neglect individual-rationality constraints.22 This seems to be reasonable given that some activities that are accident-prone, such as commuting, are prohibitively costly to avoid. However, since the activity level is a way to adapt, we provide an extension capturing variable activity in the next section.23 We propose a scheme that entails a reduction in the compensation of some victim type to achieve the desired outcome. If the low-damage (high-damage) victim has an incentive to mimic, the compensation of high-damage (low-damage) victims can be lowered in order to change the ranking from that of case (i) (case iii) to that of case (ii) in which, due to the modified transfers of respective victim types, no more mimicking incentives are present. As the procedure is similar for the cases in which the ranking of case (i) or case (iii) is the valid description of the initial outcomes, we only elaborate on the case that low-damage victims have an incentive to pretend to be the high-damage type.24 The costs of victim type j in a context where a reduction t in the high-damage victim compensation level is supplemented to the liability rule for incentive purposes, i.e., when 21

See the related discussion on (18) in Appendix A. Individual-rationality constraints are sometimes not considered in standard adverse selection models as well, for example, in some variants of problems of optimal income taxation. The reason is that in the absence of the possibility to emigrate, the government can simply impose its tax scheme on individuals. 23 For instance, if one does not obtain any compensation in a case where one is victimized by a fellow driver, one might reduce the own mileage to a bare minimum not necessarily greater than zero. 24 The other case can be treated analogously in this as well as the next section. 22

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an incentive scheme (IS) is used, V CjIS , are ⎧ ⎪ ⎪ yL if yL = yL∗ ⎪ ⎪ ⎨ ∗ V CLIS (x, yL ) = yL + p(x, yL )(DL − (DH − t)) if yL = yH ⎪ ⎪ ⎪ ⎪ ∗ ⎩ yL + p(x, yL )DL if yL =  yL∗ , yH

(14)

for the low-damage victim and ⎧ ⎪ ⎪ yH + p(x, yH )(DH − DL ) if yH = yL∗ ⎪ ⎪ ⎨ IS ∗ (x, yH ) = V CH yH + p(x, yH )t if yH = yH ⎪ ⎪ ⎪ ⎪ ∗ ⎩ yH + p(x, yH )DH if yH =  yL∗ , yH

(15)

for the high-damage victim. Again, for reasons given above, low-damage victims consider ∗ ∗ , as yL = yL∗ , yH implies higher personal costs than yL∗ . Highonly the exertion of yL∗ or yH

damage victims no longer obtain full compensation because their compensatory transfer is reduced by t. This weakens the incentive to take any standard care. However, it holds for high-damage victims as well that they take one of the standards as long as t ≤ DH . If they were to take care different from the standards, injurers would lower their care below ∗ ∗ + p(x, yH )t would result. The high-damage victim x∗ and higher personal costs than yH ∗ if x < x∗ , compensates for lower injurer care, yˆH = arg min{yH + p(x, yH )DH } > yH ∗ ∗ ∗ without attaining costs as low as yH + p(x∗ , yH )DH . Thus, taking yH is preferred to ∗ . choosing any yH = yL∗ , which is smaller than yH

To avoid distortions in injurer care incentives, the reduction is not handed over to the injurer. To that extent, the incentive scheme entails some ’decoupling’.25 The reduction in the compensation of high-damage victims which is necessary to make truthful selfrevelation via the care level optimal for low-damage victims follows from the incentivecompatibility constraint that ought to hold in the optimum. Thus, from inserting such a 25 Polinsky and Che (1991) have established that a decoupling of the amount plaintiffs receive from the amount defendants pay can lower social costs because incentives to sue can be reduced without affecting precautionary incentives.

99

reduction in (28) to make it hold as intended follows the critical level t∗ = DH − DL −

∗ − yL∗ yH ∗ ∗ p(x , yH )

(16)

This critical level t∗ > 0 makes the low-damage victim indifferent between the exertion of ∗ for the transfer of DH − t∗ .26 This reduction, yL∗ for a compensation of DL and taking yH

on the one hand, allows for the difference in compensation levels forfeited by behaving as a low-damage type and on the other hand, corrects this measure for the difference in care costs that can be saved in this way. Naturally, the reduction also affects the incentivecompatibility constraint for high-damage victims. This reduction obviously should not tempt the high-damage victim to imitate the low-damage victim. From considering t in (9) follows another critical level for the reduction, an upper boundary, which combined with the reduction from (16) spans a range of reductions inducing incentive compatibility for both types (See Appendix A for a generalization of the reduction to the case with N victim types, where N > 2). Proposition 2 In a model with heterogeneous victims and strict liability with a defense of contributory negligence, courts can rely on observed care to deduce the harm level while implementing optimal care if the liability rule is changed in that high-damage victims are y ∗ −y ∗

p(x∗ ,y ∗ )

y ∗ −y ∗

compensated only by DH − t, t ∈ [DH − DL − p(xH∗ ,y∗L) , p(x∗ ,y∗L ) (DH − DL ) − p(xH∗ ,y∗L) ] if case H

H

H

(i) is the initial description of outcomes. Proof. Follows from the above. The afore-mentioned details an incentive mechanism that can be relied on to induce truthful self-selection of victim types. Given that self-selection is truthful, the court can deduce the harm level from the observed care level. Stating differently what we found above, we arrive at ∗ ) can Remark 2 Assume verification is costly. Social costs no higher than SC(x∗ , yL∗ , yH

be attained without further modification of the liability rule if the rank order of case (ii) is 26 ∗ The sign of t∗ is the same as the sign of p(x∗ , yH )t∗ = ΔDH − Δy. Note that we implement an incentive scheme because there are incentives for the low-damage victim to mimic the high-damage victim without it. This fact implies that ΔDH > Δy.

100

valid. Otherwise, social costs do not need to be higher since the simple incentive scheme ∗ ). outlined allows the implementation of optimal care (x∗ , yL∗ , yH

Relating the result of Proposition 2 to that of Proposition 1, we solve the range of feasible reductions if case (ii) is the description of the initial outcome and find that zero is an element so that, indeed, no incentive mechanism is needed in that circumstance.27 The desirability of the incentive scheme is dependent on not only the established efficiency properties concerning care but also what use the reduction can be put to. We assume that resources collected in this way are given back to victims via lump-sum payments. It is plausible to assume that individual actors are so small that their individual impact on the total resources transferred back is negligible. A constant γ reflecting this lump-sum transfer is, therefore, added to victim objective functions without affecting incentives. Hence, the budget is balanced as payments by injurers are matched by transfers received by victims. In the following, we provide a brief numerical example. This helps to obtain a better meaning of the likely magnitude of the reduction t and of the fact that t can be drawn from a set. Furthermore, the included graphical representation may be conducive to seeing the parallel of our analysis to the standard screening setting. Illustration To illustrate the relative importance of the reduction t, we consider a simple example. Suppose DL = 200 < 300 = DH , α = .5 and p(x, y) =

1 . 1+x+xy+y

∗ We obtain (x∗ , yL∗ , yH ) = (5.28, 4.64, 5.91) and the rank order ΔDL = 2.82 > ΔDH =

2.3 > Δy = 1.27 that indicates that the low-damage victim will mimic the high-damage victim. The reduction that prevents this from happening is t∗ = 44.95.28 The modified ΔDH equals Δy and the modified ΔDL is equal to 1.79 and, therefore, still higher than Δy, indicating that high-damage victims have no incentive to masquerade as low-damage victims. The lump-sum payment that is financed by the holdings of the court amounts to γ = .52. 27

∗ ∗ yH −yL ∗ ) < 0, we find that this requirement equals the true statement p(x∗ ,yH p(x∗ ,y ∗ ) ∗ ∗ ∗ ∗ yH − yL if p(x∗ , yH )DH = p(x∗ , yL )DL . Likewise coming from p(x∗ ,y∗L ) (DH − H

Starting from t∗ = DH − DL −

∗ ∗ ) − p(x∗ , yH )]DL < [p(x∗ , yL y∗ −y ∗

∗ ∗ ∗ ∗ ) − p(x∗ , yH )]DH > yH − yL DL ) − p(xH∗ ,y∗L) > 0, we can reduce the term to the true statement [p(x∗ , yL H ∗ ∗ if p(x∗ , yH )DH = p(x∗ , yL )DL . 28 Note that the reduction becomes smaller once we consider more than two types. See Appendix A.

101

To illustrate graphically, consider the two compensation options as being drawn from a continuum, which implies cost function y +p(x∗ , y)[Dj −D]. Figure 1 depicts two isocost 1 2 and ICH , and one of the low-damage victim, ICL. curves of the high-damage victim, ICH

ICL

D

1 ICH 2 ICH

DH − t∗ DL

yL∗

∗ yH

y

Figure 1: Respective Isocost Curves ∗ The combinations (yL∗ , DL ) and (yH , DH − t∗ ) are on the same isocost curve of the low-

damage victim. In contrast, acting as a high-damage victim is strictly preferred by the high-damage victim to masquerading as a low-damage victim, which is depicted by the 1 representing lower costs. This strong preference of the high-damage isocost curve ICH

victim reflects the leeway present with respect to high-damage victim incentives if the lower bound of the interval of reductions t is chosen.

4

Screening Victims with Variable Activity

We have shown that in some cases victims sort themselves correctly without any verification of damages or other added incentives. However, we have also shown that there are circumstances in which victims find it beneficial to pretend to be of another type. Our proposal in this regard is the reduction of transferred compensation to establish a ranking of outcomes, which prevents any incentives to mimic. In the derivation of the reduction making this feasible, we focused on incentive-compatibility constraints. Yet, it is also of importance to consider what consequences result regarding the participation in ∗ as care level might the activity. Reducing the compensation below DH while requiring yH

102

question the individual rationality of the activity for high-damage victims. That is why we will treat both injurer as well as victim activity as variable in this section. First, we set out the benchmark for later comparison with the case in which the scheme is actually applied. In the derivation of this benchmark, we consider verification to be free of costs to ease the comparison. For reasons similar to those outlined in Section 2, the case in which verification is costly is associated with lower welfare and would thus make living up to the reference easier. The framework we use in the following is rather standard (see, e.g., Miceli 2004). Injurers [victims] choose an activity level a [b] ∈ R which increases utility u(a) [v(b)] at a diminishing rate, i.e. we assume u (a), v  (b) > 0, u (0), v (0) = ∞, and u (a), v  (b) < 0. It is assumed that if, for instance, the injurer engages in the activity a times, she takes care x each of these times. The same applies to expected harm. The social objective is to maximize the utility that individuals obtain from the activity, net of precaution costs and expected harm.

SW (a, bL , bH , x, yL , yH ) = u(a) − ax + α[v(bL ) − yLbL − abL p(x, yL )DL ] + (1 − α)[v(bH ) − yH bH − abH p(x, yH )DH ]

(17)

¯, y¯L, y¯H ). Define the optimal levels that solve the system of first-order conditions as (¯ a, ¯bL , ¯bH , x Strict liability with a defense of contributory negligence defines standards for care but does not regulate activity by standard setting.29 The care standards equal y¯L and y¯H and we suppose that victims adhere to them. In that case, victims will choose their activity by the level that solves v  (bj ) = y¯j , which is socially excessive as it does not consider the expected damages inflicted on society by additional activity. Injurers respond socially optimal to given victim behavior.30 The inefficiency portrayed is not specific to the liability rule considered but a general result concerning all standard liability rules in models with bilateral variable care and activity (Shavell 1980). None of the liability rules 29 This is the by now standard way to capture the different types of behavior (see, e.g., Miceli 2004). Recently, Dari Mattiacci (2005) questioned this way of modeling and explicitly derived the activities included in the liability standards from a deterrence vs. administrative costs trade-off. 30 This is because injurers bear expected damages due to the standard obedience of victims and maximize u(a) − ax − αabL p(x, yL )DL − (1 − α)abH p(x, yH )DH with respect to x and a.

103

is able to establish the double responsibility at the margin necessary to attain the efficient outcome (Cooter 1985).31 The justification for the ability to enforce care standards and the inability to do so for activity, as laid out by Shavell (2004: 198), relies on both the generally missing knowledge of benefits that parties derive from their activities, and the high burden of assessing activity levels.32 For the following, we will take the outcome that uses y¯L and y¯H from above as victim standards as a benchmark (whereas, in Appendix B, we sketch an alternative avenue for the policy maker). Victims try to minimize their expected costs per activity unit. Their objective function without any liability rule is v(bj ) − bj [yj + ap(x, yj )Dj ]. The derivation of the reduction needed for successful screening is the analog of that in Section 3, albeit we need to take into account that the injurer activity is a factor determining expected damages. From this follows a critical threshold in the context of variable activity and care, t+ = DH − DL − y¯H −¯ yL , ap(x,¯ yH )

which is the lower endpoint of the interval of values t that prevent mimicking of

either low-damage or high-damage victims, t ∈ [DH − DL − y¯H −¯ yL 33 ]. ap(x,¯ yH )

y¯H −¯ yL p(x,¯ , yyHL)) (DH ap(x,¯ yH ) p(x,¯

− DL ) −

While everything else remains principally unchanged, the objective function of

the high-damage victim who takes due care is v(bH ) − y¯H bH − abH p(x, yH )t+ + γˆ , because the reduction in damage payments t is borne by high-damage victims. Consequently, the incentive scheme introduces an additional cost per activity unit into the optimization of the high-damage victim. The activity level will still be socially excessive since t+ < DH holds, but less so than in the benchmark. Proposition 3 In a model with heterogeneous victims, variable activity and care, strict liability with a defense of contributory negligence as well as the rank order of outcomes of case (i), courts can save verification costs and improve on the vector of care and activity 31 For instance, Goerke (2003) analyzes to what extent fines unrelated to the occurrence of accidents can help to achieve the efficient outcome in a model with bilateral care and activity choice. 32 Note that if we take these limitations seriously, it also restricts other options that might be used in conjunction with liability rules to correct the laid out defect in contexts with bilateral care and activity choice. So, if we assume that the major impediment to the inclusion of activity into the behavioral standards of the liability rule is the observability of activity levels, this restricts the practicability of the regulation of activity as well as the imposition of a tax per unit of activity. 33 The level of injurer activity and care which enters these values is that which optimizes u(a) − ax − αabL p(x, yL )DL − (1 − α)abH p(x, yH )DH , given that each victim takes type-adequate due care and individually optimal victim activity.

104

levels if the liability rule is changed in that high-damage victims are compensated only by DH − t+ , t+ ∈ [DH − DL −

y¯H −¯ yL p(x,¯ , yyHL )) (DH ap(x,¯ yH ) p(x,¯

− DL ) −

y¯H −¯ yL ]. ap(x,¯ yH )

Proof. Follows from the above. The reduction in high-damage victim compensation has several effects. First, it lowers the activity level of the high-damage victim as delineated above. Second, the care choice, x, can be less excessive due to the lowered bH in the first-order condition for injurer care. Third, the injurer can increase her activity level, a, since the expected marginal costs of one activity unit decrease due to the lower activity of high-damage victims (and the induced reduction in injurer care). These consequences move the vector of chosen activity and care levels closer to that of the social optimum. Note that the case for reductions would be even easier to make if we were to utilize the practical reference, namely the one with costly verification.

5

Conclusion

The liability system imposes huge costs on society. These are partly caused by proceedings dealing with the assessment of the exact magnitude of damages suffered in accidents. Consequently, considerations that may help to reduce this burden are of value. This paper shows for the case of strict liability with a defense of contributory negligence that the due care menu suffices to trigger truthful sorting in some cases. In other cases, reducing the compensation transfered to some victim types can induce truthful self-selection and, by making the costly verification of damages redundant, lowers social costs. This scheme does not need to distort care choices to effect the screening but installs optimal care levels. Our suggestion can lower social costs and does not necessarily entail a stark deviation from common practice. The latter is due to the fact that reductions employed to ensure incentive compatibility can be rather small in the case with many victim types. In an extension, we allow for adaptations to the modification of compensation via changes in the activity level. We find that, instead of arguing against the scheme, the utilization of the incentive mechanism changes the vector of optimal choices in the direc105

tion of the social optimum. As a consequence, the deployment of the incentive scheme may not only be of value due to the verification costs saving but also because the overall activity and care structure can be improved.

Appendices Appendix A: The Case of N ≥ 2-Types In this addendum, we derive the reduction that attains the desired screening with N different victim types. Let us order types according to harm magnitude, D1 < D2 < ... < N DN . Accordingly, y1N < y2N < ... < yN . Principally, this presents a vast problem as type j

could consider pretending to be any type. This then creates many incentive-compatibility constraints. However, the objective function of victim type j is y + p(x, y)[Dj − D] with D as the compensation obtained, and fulfills the single-crossing property as dD d dy j

dDj

=

pyj (x, yj ) 0, which occurs only if the high-damage victim care standard tempts the individual to deviate from the standard, the effect of the reduction is no longer unambiguous. There is a positive effect due to the restraining effect on activity, but there also is a negative effect owing to the restriction of implementable due care standards.

108

Appendix C: The Case of A Step-Liability Function In the text, we assume that victims bear full harm unless they take a level of care optimal for one of the victim types. Another way to capture the problem is to assume a stepliability function. This implies the following costs for victims ⎧ ⎪ ⎪ yL + p(x, yL )DL if yL < yL∗ ⎪ ⎪ ⎨ ∗ V CLS (x, yL ) = yL if yL∗ ≤ yL < yH ⎪ ⎪ ⎪ ⎪ ∗ ⎩ yL + p(x, yL )[DL − DH ] if yL ≥ yH ⎧ ⎪ ⎪ yH + p(x, yH )DH if yH < yL∗ ⎪ ⎪ ⎨ S ∗ (x, yL ) = V CH yH + p(x, yH )[DH − DL ] if yL∗ ≤ yH < yH ⎪ ⎪ ⎪ ⎪ ∗ ⎩ yH if yH ≥ yH

(26)

(27)

Given that victims rather take due care than bearing full expected harm (shown in Section 2.2), the respective first lines can be disregarded. ∗ . This holds In the next step, note that low-damage victims will either take yL∗ or yH ∗ ) true since there is no marginal benefit to increasing care from yL∗ to a level yL ∈ (yL∗ , yH

for low-damage victims. For the efficient choice to be individually optimal, as in the text, we require that ∗ ∗ yL∗ ≤ yH + p(x∗ , yH )(DL − DH )

(28)

holds. The irrelevance of which liability function is used, the pointwise function of the text or the stepwise function considered here, which is present for the low-damage victim, does not necessarily hold with respect to the high-damage victim type. The high-damage ∗ ∗ , now choose between yˇH and yH , where victim will, instead of choosing between yL∗ and yH

yˇH = arg minyL∗ ≤yH yL∗ . ˇ = p(x∗ , yˇH )(DH − DL ), ΔDH = Proceeding as we do in the text, we define ΔD ∗ ∗ ∗ )(DH − DL ), Δy = yH − yL∗ , and Δˇ y = yH − yˇH to discriminate different possible p(x∗ , yH ∗ ˇ > ΔDH , Δy > Δˇ cases. Note that the following holds because yH > yˇ > yL∗ : ΔD y.

In order to have both types behaving as intended for their type, we require with respect to low-damage victims that ΔDH ≤ Δy and with respect to high-damage victims that ˇ ≥ Δˇ ΔD y. The possible cases are detailed in the following table. Constellation ˇ > ΔDH > Δy > Δˇ ΔD y ˇ > Δy > ΔDH > Δˇ y ΔD ˇ > Δy > Δˇ ΔD y > ΔDH ˇ > Δˇ Δy > ΔD y > ΔDH ˇ > ΔDH > Δˇ y Δy > ΔD ˇ > ΔDH Δy > Δˇ y > ΔD

LDV: Take yL∗ ? No Yes Yes Yes Yes Yes

∗ ? HDV: Take yH Yes Yes Yes Yes Yes No

Table 1: Possible Rankings

From the table, we can deduce that low-damage victims (LDV) will choose to take ∗ yH

under the circumstances of the first line, whereas high-damage victims (HDV) pick yˇ

∗ given that the last line describes the ranking. instead of yH

A sufficient condition for making sure that low-damage victims behave as intended for the type is

∂

py (x,y) p(x,y)

∂y

= 0, a fact explained in the text. Here, we inquire whether high-

damage victims likewise will act type-adequately if this condition holds. The proceeding is similar to that in the text. ∗ ˇ ≥ Δˇ if ΔD y . This can also be stated as As said, high-damage victims pick yH

∗ ∗ )]DH + p(x∗ , yH )DH − p(x∗ , yˇH )DL Δˇ y ≤ [p(x∗ , yˇH ) − p(x∗ , yH

110

(30)

∗ ∗ The optimality of yH given x∗ implies that Δˇ y < [p(x∗ , yˇH ) − p(x∗ , yH )]DH holds. The ∗ )DH ≥ p(x∗ , yˇH )DL . above inequality, therefore, holds for sure if p(x∗ , yH

The individual optimization, which obtains yˇ as optimal choice, gives 1 + py (x∗ , yˇH )[DH − DL ] = 0

(31)

This can be reformulated as p(x∗ , yˇH )DL = −

p(x∗ , yˇH ) [−1 − py (x∗ , yˇH )DH ] py (x∗ , yˇH )

(32)

∗ ∗ )DH = 0 and pyy > 0 with yˇH < yH . where −1 − py (x∗ , yˇH )DH > 0 due to −1 − py (x∗ , yH

This compares to ∗ )DH = − p(x∗ , yH

∗ p(x∗ , yH ) ∗ ∗ py (x , yH )

(33)

which can be derived from the first-order conditions of the social minimization problem. Dividing (33) by (32), one obtains

1 −1−py (x∗ ,ˇ yH )DH

if

∂

py (x,y) p(x,y)

∂y

= 0 holds. The question is

whether this is smaller or larger than one. If it is larger than one, then high-damage victims behave as intended, given that the condition spelled out in Proposition 1 holds true. Now, note that 1 >1 −1 − py (x∗ , yˇH )DH

(34)

2 > −py (x∗ , yˇH )DH

(35)

1 − py (x∗ , yˇH )DL = −py (x∗ , yˇH )DH

(36)

requires

Since (31) holds, we know that

However, since yˇH > yL∗ , pyy > 0, and 1 + py (x∗ , yL∗ )DL = 0, −py (x∗ , yˇH )DH < 2 is true. As a consequence, if

∂

py (x,y) p(x,y)

∂y

∗ = 0 holds, it will hold true that p(x∗ , yH )DH > p(x∗ , yˇH )DL .

Summarizing, Proposition 1 is still valid if the liability function entails stepwise increases in the level of compensation. 111

Appendix D: The Case of Type Being Continuous In this annex, we inquire as to whether or not it is incentive compatible for victims to reveal their type when the optimal victim care and perfect compensation are to be implemented for the case in which type D, mirroring the harm magnitude, is continuous, ¯ instead of being confined to a discrete set. Our finding will be that this D ∈ [D, D], obtains only in remote circumstances. First of all, we delineate for this setting what is meant by optimal victim care and perfect compensation. The optimal victim care follows from 1 + py (x∗ , y)D = 0

(37)

py (x∗ , y) dy =− >0 dD pyy (x∗ , y)D

(38)

and develops according to

Consequently, with y = arg min{y + p(x∗ , y)D}, we can state optimal care for victim type D as



D

y(D) = y + D

−

py (x∗ , y(τ )) dτ pyy (x∗ , y(τ ))τ

(39)

Besides inducing optimal victim care, we are able to fully compensate victims under specified circumstances in the case of discrete victim types. In the continuous case, this demands that compensation C is equal to the harm suffered D, C(D) = D, and that dC(D) dD

= 1.

Relying on the taxation principle34 , we proceed in the manner of a direct revelation mechanism, i.e., a certain y and C is presented to the victim when hearing a type declaˆ The functions y and C will thus be functions of the announcement instead of ration D. the actual type. The cost function of the individual can be stated as ˆ ˆ ˆ D) = y(D) ˆ + p(x∗ , y(D))[D − C(D)] V (D,

(40)

ˆ which representing the expected costs of a victim of type D when announcing type D, 34

See, e.g., Salani´e, B. (1998). The Economics of Contracts: A Primer. Cambridge, MA: MIT Press.

112

gives the derivative  ˆ ˆ D) ˆ dy(D) ∂V (D, ˆ ˆ − p dC(D) 1 + py (x∗ , y(D))[D = − C(D)] ˆ ˆ ˆ ∂D dD dD

(41)

with respect to the type announced. For truthful revelation to be incentive compatible, ˆ = D, it is necessary that the derivative is equal to zero at D dy(D) dC(D) ∂V (D, D) = [1 + py (x∗ , y(D))[D − C(D)]] − p =0 ˆ ˆ ˆ ∂D dD dD

(42)

or using C(D) = D ∂V (D, D) dy(D) = −p=0 ˆ ˆ ∂D dD

(43)

∂ 2 V (D, D) dC(D) ∂ 2 V (D, D) + py (x∗ , y(D))(− )≥0 = ˆ2 ˆ ˆ ∂D ∂ D∂D dD

(44)

The second-order condition is

The statement in (43) gives one requirement for the proposed mechanism (y(·), C(·)) to be implementable. We can readily interpret this condition. Suppose that there is some type Dq , who might announce his type accurately or claim to be Dq −  where  → 0, for instance. In the latter case, the individual costs are y(Dq − ) + p(x∗ , y(Dq − )), whereas in the former case, individual costs turn out to be y(Dq ), with y(Dq ) > y(Dq − ) given that  > 0. Consequently, at the margin, if the increase in costs of care,

dy(D) ˆ dD

is equal to

the decrease in expected harm, −p, then the individual of type Dq will have no incentive to pretend to be of type Dq − . The requirement spelled out in (43) needs to hold not only for type Dq but for all ¯ Thus, it furthermore needs to hold that types D ∈ [D, D].

where

d2 y(D) dy(D) ∂ 2 V (D, D) − py (x∗ , y(D)) = =0 ˆ ˆ2 ˆ ∂ D∂D dD dD

(45)

  dy dy py pyy + D dD pyyy − Dp2yy dD d2 y(D) = 2 2 2 ˆ D pyy dD

(46)

113

For the expression in (45) to hold, it is necessary that the term in (46) is less than zero > 0. This is ascertained for pyyy ≥ 0 and may also hold for since −py (x∗ , y(D)), dy(D) ˆ dD negative pyyy , provided its absolute value is sufficiently small. ˆ = D being the optimum for type Note that the necessary second-order condition for D D, (44), is compatible with (45) being equal to zero. In summary, the requirements for optimal care and perfect compensation to be truthfully implementable in the case of continuous victim type comprise that (43) and (45) hold.

References Arlen, J. (2000). Tort Damages. In: B. Bouckaert and G. De Geest, eds., Encyclopedia of Law and Economics, Vol. II, Cheltenham: Edward Elgar: 682-734. Bolton, P. and M. Dewatripont (2005). Contract Theory. Cambridge, MA: MIT Press. Calfee, J.E. and R. Craswell (1984). Some Effects of Uncertainty on Compliance with Legal Standards. Virginia Law Review 70: 965-1003. Cooter, R. (1985). Unity in Tort, Contract, and Property: The Model of Precaution. California Law Review 73: 1-51. Cooter, R. and W. Emons (2003). Truth-Revealing Mechanisms for Courts. Journal of Institutional and Theoretical Economics 159: 259-279. Dari Mattiacci, G. (2005). On the Optimal Scope of Negligence. Review of Law and Economics 1: Article 2. Emons, Winand and Claude Fluet (forthcoming). Accuracy Versus Falsification Costs: The Optimal Amount of Evidence Under Different Procedures. Journal of Law, Economics, and Organization. Farmer, A. and P. Pecorino (1999). Legal Expenditure as a Rent-Seeking Game. Public Choice 100: 271-288. Farmer, A. and P. Pecorino (2005). Civil Litigation With Mandatory Discovery and Voluntary Transmission of Private Information. Journal of Legal Studies 34: 137-159.

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Ganuza, J.J. and F. Gomez (2005). Caution, Children Crossing: Heterogeneity of Victim’s Cost of Care and the Negligence Rule. Review of Law and Economics 1: Article 3. Goerke, L. (2003). Road Traffic and Efficient Fines. European Journal of Law and Economics 15: 65-84. Hay, B.L. (1994). Civil Discovery: Its Effects and Optimal Scope. Journal of Legal Studies 23: 481-515. Hua, X. and K.E. Spier (2005). Information and Externalities in Sequential Litigation. Journal of Institutional and Theoretical Economics 161: 215-232. Kaplow, L. and S. Shavell (1996). Accuracy in the Assessment of Damages. Journal of Law and Economics 39: 191-209. Katz, A.W. (2000). Indemnity of Legal Fees. In: B. Bouckaert and G. De Geest, eds., Encyclopedia of Law and Economics, Vol. V, Cheltenham: Edward Elgar: 63-94. Kim, J. and A.M. Feldman (2006). Victim or Injurer, Small Car or SUV: Tort Liability Rules Under Role-Type Uncertainty. International Review of Law and Economics 26: 455-477. Krauss, M.I. and R.A. Levy (1996). Calculating Tort Damages For Lost Future Earnings: The Puzzles of Tax, Inflation and Risk. Gonzaga Law Review 31: 325-373. Miceli, T.J. (2004). The Economic Approach to Law. Stanford, CA: Stanford University Press. Miceli, T.J. (2006). On Negligence Rules and Self-Selection. Review of Law and Economics 2: Article 1. Polinsky, A.M. and Y.K. Che (1991). Decoupling Liability: Optimal Incentives for Care and Litigation. RAND Journal of Economics 22: 562-570. Polinsky, A.M. and D.L. Rubinfeld (1988). The Welfare Implications of Costly Litigation For The Level of Liability. Journal of Legal Studies 17: 151-164. Rubinfeld, D.L. (1987). The Efficiency of Comparative Negligence. Journal of Legal Studies 16: 375-394. Shavell, S. (1980). Strict Liability Versus Negligence. Journal of Legal Studies 9: 1-25. Shavell, S. (1989). Sharing of Information Prior to Settlement or Litigation. RAND Jour-

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nal of Economics 20: 183-195. Shavell, S. (2004). Foundations of Economic Analysis of Law. Cambridge, MA: Harvard University Press. Spier, K.E. (1994). Settlement Bargaining and the Design of Damage Awards. Journal of Law, Economics, and Organization 10: 84-95. Spier, K.E. (2007). Litigation. In: A. Mitchell Polinsky and Steven Shavell, eds., Handbook of Law and Economics, Vol. I, Amsterdam: North Holland: 259-342.

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Chapter 6

A Note on Judgment Proofness and Risk Aversion∗ Abstract Shavell (1986) established that potentially judgment-proof risk-neutral injurers will take less care than injurers with sufficient funds in the case of strict liability. This note considers strict liability and shows that the reverse may hold if individuals are risk averse, i.e., some potentially judgment-proof injurers expend more on care than some injurers with assets greater than the harm.

Keywords: judgment proofness, care incentives, risk aversion

JEL-Classification: K 13

∗ This chapter is an edited version of: Friehe, T. (2007). A Note on Judgment Proofness and Risk Aversion. European Journal of Law and Economics 24: 109-118.

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1

Introduction

1.1

Motivation and Main Results

Tort law aspires -inter alia- to provide incentives for adequate care in activities that may cause harm. Among the factors that may prevent the optimal workings of tort law in this respect is the possibility that injurers lack sufficient funds to cover the harm caused. Injurers do not internalize the complete social harm, since they can externalize the part that exceeds personal funds, which gives reason to expect insufficient precaution. Indeed, Shavell (1986), using a framework with non-monetary care which reduces the accident probability, gave formal support to this intuition by showing that all injurers with (i) asset levels less than the harm in the case of strict liability and (ii) asset levels less than a critical value (which is less than the harm) in the case of negligence will take less care than the level taken by individuals with sufficient funds. This note uses Shavell’s framework and shows that, in the case of strict liability, the contrary may hold. That is, potentially judgment-proof injurers may take more care than injurers with assets sufficient to compensate the harm caused. We consider risk-averse injurers and contrast the care-taking behavior of injurers with and without binding asset constraint. Our finding can be explained by the asymmetric effect of increasing asset levels on the value of care for the expected utility of injurers with and without sufficient assets. For potentially judgment-proof injurers, care unambiguously increases with assets because higher assets (i) increase the loss in utility due to an accident that can be made less probable by higher care and (ii) decrease the marginal costs of care. However, injurers without binding asset constraint observe a decrease not only in the marginal costs of care but also in the marginal benefit of care as a consequence of an increase in the asset value. The latter effect results because the loss in utility due to an accident decreases owing to diminishing marginal utility. This decrease of the marginal benefit of care contrasts with the case of injurers with insufficient assets due to the fact that, for these individuals, the higher asset level has no effect on the utility in the accident state of the world. Consequently, whereas care unambiguously increases in asset value if the injurer

118

has insufficient assets, care may fall in assets for injurers without binding asset constraint. The literature hitherto has almost entirely focused on risk-neutral individuals. Completing the study of consequences of limited liability in the realm of risk aversion is necessary in order to more fully understand probable effects of the judgment-proofness problem with regard to the workings of the liability system. Bell and O’Connell (1997: 80), for instance, point out that more than 40 percent of the liability insurance market is self-insured, i.e. not actually transacted. This gives reason to believe that individual behavior is often better described by risk aversion than by risk neutrality.

1.2

Relation to the Literature

The literature discussed in the following assumes risk-neutral individuals unless otherwise stated. The discussion concentrates on contributions which leave policy options in view of judgment proofness untouched (on this, see, e.g., Shavell 2005, or Pitchford 1995). In line with Shavell (1986), Beard (1990) presumes that care lowers the accident probability but, in contrast to Shavell, assumes that care expenditures reduce the assets available for compensation, i.e., monetary care. He shows that, in the case of strict liability, potentially judgment-proof injurers may take more care than injurers who are not bankrupted by the compensation of victims. Miceli and Segerson (2003) detail Beard’s results for strict liability and negligence in a simplified framework. If care is monetary and strict liability is the applicable rule, injurers with asset levels less than but close to the harm exert more care than injurers whose asset constraint does not bind in the case of an accident. The fact that care reduces the assets available for compensation de facto reduces the care costs because these only arise as costs in the case where no accident occurs. The possibility that care of potentially judgment-proof individuals exceeds the care level taken by injurers with sufficient assets thus follows if, in comparison to the cost minimization problem of injurers with sufficient assets, the reduction in marginal costs of care overcompensates the reduction in the marginal benefit of care, where the latter reduction is due to limited liability. Our analysis extends to the case of monetary care and, in that case, our model also depicts this result. However, we point to another effect 119

that is the sole driver of our result for the case of non-monetary care. Boyd and Ingberman (1994) augment the analysis by considering the possibility that precaution might impact on the loss magnitude instead of on the accident probability, or that it might lower both. They consider the case of strict liability and find that damages that are noncompensatory can induce efficient incentives in a framework in which the magnitude of losses is affected by precaution. Consequently, in this case, actors with assets less than the harm can be induced to take efficient care. Dari Mattiacci and De Geest (2005, 2006) follow this lead and add a fourth possibility, the separate probability magnitude model in which two different precautions can be taken. In our analysis, we adhere to the convention that care reduces the accident probability. However, we comment on the results of the magnitude model in the conclusion. MacMinn (2002) considers risk-averse individuals and both strict liability and negligence. He shows that injurers who turn out judgment proof in the case of an accident exert more (less) care under negligence than under strict liability if care is non-monetary (monetary).∗ However, he does not touch upon our focus, namely, the comparison of care incentives of individuals with sufficient funds and those of actors whose asset constraint binds in the case of an accident. In the next section, we describe the model and derive our basic results. Section 3 concludes.

2

Model and Analysis

The analysis uses Shavell’s model. Injurers can reduce the probability of an accident by taking care. Victims suffer accident losses and seek remediate action but are passive otherwise. We assume throughout that individuals are risk averse and that strict liability applies.† ∗ In the case of non-monetary care, precaution under strict liability eventually becomes greater than care under negligence since individuals do not exert more than standard care under the latter liability rule. Notably, MacMinn considers the negligence rule that makes injurers liable only for the harm caused by their negligence (see Kahan 1989). † We briefly comment on negligence in our conclusion.

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Let x = injurer care level; x ≥ 0; p(x) = accident probability; 0 < p(x) < 1; p (x) < 0; p (x) > 0; l = magnitude of harm if an accident occurs; l > 0; y = initial assets of injurers; y ≥ 0; u(·) = von Neumann-Morgenstern utility function of wealth of injurers; u (·) > 0; u (·) < 0. In the literature, the assumption that care is non-monetary is widespread. Nonmonetary care can be conceived of as precautionary effort, examples being: slowing for curves or paying attention to cyclists. However, numerous precautionary measures cause monetary costs, e.g., an anti-lock break system. Our main analysis in Section 2.1 assumes non-monetary care, as Shavell (1986). In Section 2.2, we show that the effect, which is specific to the risk aversion framework, is principally unaffected but joined by another effect which is also present in the risk neutrality framework, if we assume monetary care.

2.1

Non-Monetary Care

If care is non-monetary with a monetary equivalent of x, injurers maximize expected utility given by V (x, y) = p(x) u(A) + [1 − p(x)] u(B)

(1)

with A = max{y − l, 0} − x and B = y − x, since injurer’s income given an accident is max{y − l, 0}. Potentially judgment-proof injurers have funds less than the loss, y < l, and their net income given an accident consists only of non-monetary costs x. Expected utility changes with care according to ∂V (x, y) = p (x) [u(A) − u(B)] − p(x) u (A) − [1 − p(x)] u (B). ∂x

(2)

From this follows that expected utility V and the first-order derivative are continuous in x ∀ x ≥ 0. The derivative has a quite intuitive interpretation. Additional care, on the one hand, reduces the probability that the loss in utility due to an accident has to be 121

experienced, which is the marginal benefit of care, and, on the other hand, additional care causes utility to decrease in both contingencies, which is the marginal cost of care. The first-order condition for interior solutions, i.e., in the case of initial income sufficiently greater than zero‡ , ∂V (x, y) =0 ∂x

(3)

is solved by the individually optimal care level, xˆ(y). We assume that the second-order condition

∂ 2 V (x,y) ∂x2

< 0 holds, with

∂ 2 V (x, y) = p (x)[u(A)−u(B)]−2p(x)[u (A)−u (B)]+p(x)u(A)+[1−p(x)]u(B). (4) ∂x2 By application of the implicit function theorem, we find that the change of optimal care with assets is given by

∂ 2 V (x,y)

dˆ x ∂x∂y = − ∂ 2 V (x,y) , dy 2

(5)

∂x

which will be positive if the cross partial by

∂ 2 V (x,y) ∂x∂y

is positive. That cross partial is given

⎧ ⎪    ⎪ if y < l ⎪ ⎪ −p (x) u (y − x) − [1 − p(x)] u (y − x) ∂ 2 V (x, y) ⎨    = {p (x) [u (y − l − x) − u (y − x)] ⎪ ∂x∂y ⎪ ⎪ ⎪ ⎩ −p(x) u (y − l − x) − [1 − p(x)] u (y − x)} if y ≥ l,

(6)

and can be interpreted as follows. For small asset levels, y < l, increases in the asset value increase the utility loss due to an accident that care prevents, u(y − x) − u(−x), and the marginal costs of care are lower at higher net income levels due to diminishing marginal utility of income. Both effects are positive and therefore unambiguously argue for higher precaution in response to increases in asset value. However, once assets are sufficient to at least compensate the harm, y ≥ l, increasing the asset value decreases the utility loss that care prevents as net income in both states is affected and marginal utility is diminishing. This argues for a reduction in care, all else being equal. The decrease in marginal costs, on the contrary, increases incentives for precautionary expenditures. Hence, an increase ‡ Recognize that, since income levels.

∂V (0,0) ∂x

< 0, optimal individual care will be zero for sufficiently small initial

122

in asset value decreases both the marginal benefit and the marginal costs of care if the individual has sufficient assets to compensate the harm caused. These are two opposing effects of which either might be greater than the other. Note that (6) also contains the result for the case of risk neutrality, that is, the individual with sufficient funds does not change care in response to increases in asset value, whereas potentially judgment-proof individuals continuously increase their precaution. This leads to the widely cited result that potentially judgment-proof injurers take less care than agents with sufficient funds. The comparative statics yield the following central result. Proposition 1 Assume strict liability, risk aversion, and non-monetary care. Potentially judgment-proof injurers with assets less than but close to the harm take more care than some of the injurers with assets y ∈ [l, l + Δ] if the following applies to latter individuals: (i) decreasing absolute risk aversion and p(ˆ x) < pc , or (ii) increasing absolute risk aversion and p(ˆ x) > pc . Proof. The claim is proven if (i) potentially judgment-proof injurers with assets equal to y = l − ,  → 0, take care of the same level as individuals with assets equal to y = l, and (ii) if injurers with y ∈ [l, l + Δ] continuously decrease their care in response to rises in asset values under the given conditions. In that case, injurers with assets less than but close to the harm take more care than ’well-to-do’ injurers with assets close to l + Δ. With regard to condition (i), the optimal care increases in y as long as y < l since, in that case,

∂ 2 V (x,y) ∂x∂y

> 0. Note that this result does not need further specifications, e.g.,

with regard to absolute risk aversion, than what we imposed on the functions p(x) and u(·). Thus, the care level xˆ(y = l − ) is equal to xˆ(l) for  → 0. Regarding condition (ii), note that for y ≥ l, the cross partial

∂ 2 V (x,y) ∂x∂y

is not unambigu-

ously signed. Sweeney and Beard (1992) have shown that the sign depends in a complex way on the magnitude of the probability of causing the harm at the optimal care choice 

(N I) over the p(ˆ x), and on the behavior of the absolute risk aversion function r(NI) = − uu (N I)

interval for net income NI, NI ∈ D = [y − x − l, y − x]. We apply their Result A almost verbatim: (1) If r(·) is constant throughout the interval D, then dˆ x/dy = 0; (2) If r(·) is 123

monotonically decreasing throughout D, then there exists a number pc , 0 < pc < 1, such that dˆ x/dy < (>) 0 if p(ˆ x) < (>) pc ; (3) If r(·) is monotonically increasing throughout D, x/dy > () pc . then there exists a number pc , 0 < pc < 1, such that dˆ It is worth noting that the critical probability pc can be approximately equal to zero or to one depending upon the specific absolute risk aversion function.§ This implies that, if pc is close to one, care is reduced in most cases as a consequence of increasing y for the widely used assumption of decreasing absolute risk aversion (e.g., Arrow 1976). Our finding utilizes the fact that, whereas potentially judgment-proof injurers find it unambiguously beneficial to take more care after increases in asset value, injurers without binding asset constraint might find it advantageous to let care decrease with assets. The ambiguity of the comparative statics owes to the fact that care depends on an exogenously given accident probability function and does not in general decrease the riskiness of income prospects.¶ Figure 1 depicts an example in which some injurers, who turn out judgment proof in the accident contingency, are more cautious than other injurers with assets higher than harm. The example is based on the accident probability function p(x) = e−x , as, e.g., in Rubinfeld (1987), and the utility function u(NI) = NI .4 as well as a harm magnitude l = 10. In this example, injurers with assets y ∈ [9, 10), for instance, take more care than injurers who would not be bankrupted in the case of an accident and have assets y ≥ 15.6.

2.2

Monetary Care

If we assume monetary care, the income given that an accident occurs is max{y −l −x, 0}. In this case, the injurer maximizes expected utility given by

Z(x, y) = p(x) u(max{y − l − x, 0}) + [1 − p(x)] u(y − x).

(7)

§ ∼ 1 if r(N I) remains near r(y − x − l) until Sweeney and Beard (1992), for instance, show that pc = the net income almost equals y − x. ¶ In this note, we do not elaborate further but refer to Briys and Schlesinger (1990) and Sweeney and Beard (1992), for instance.  In order to deal with the fact that the utility function is not defined for negative arguments, we add 4 to both contingencies.

124

x ˆ 2.5

2

1.5

1

0.5

5

15

10

20

25

y

=l

Figure 1: Optimal care as a function of the asset value in the case of non-monetary care The first-order derivative is

⎧ ⎪ ⎪ p (x) [u(0) − u(y − x)] − [1 − p(x)] u (y − x) if y − l − x < 0 ⎪ ⎪ ⎨ ∂Z(x, y) = {p (x) [u(y − l − x) − u(y − x)] ⎪ ∂x ⎪ ⎪ ⎪ ⎩ −p(x) u (y − l − x) − [1 − p(x)] u (y − x)} if y − l − x ≥ 0.

(8)

Note that the derivative of Z with respect to care is not continuous in x, which contrasts with V from the section above. This discontinuity is due to the fact that marginal costs of care do not arise in the accident state of the world for potentially judgment-proof injurers. If the individual is bankrupt in the accident state anyway, it is of no relevance for the utility of that state whether she increases precaution even more. Consequently, whereas the marginal benefit of care is continuous, marginal costs of care display a discontinuity, which translates into a discrete fall in optimal care. Recognize that there is a range of asset levels for which it is endogenous which line in (8) applies. This holds as the injurer decides on care, the level of which determines for given l and y whether y − l − x is less than, equal to, or greater than zero. To that extent, this setting of strict liability displays a parallel to the case in which negligence is the liability rule, where discontinuities in absolute and marginal terms are of great 125

importance. The solution to the first-order condition for y − l − x < 0 is individually optimal care x˜1 (y) and x˜2 (y) for the condition resulting for y − l − x ≥ 0.∗∗ At the critical asset value y ) > x˜2 (¯ y ). y¯, it holds that x˜1 (¯ Regarding the question of at which asset value the switch from y − l − x < 0 to y − l − x ≥ 0 occurs, we note that there is no question whether agents with y ≤ l will be bankrupted by compensation requests. Individuals with somewhat greater assets will have binding asset constraints even if they choose the smaller x˜2 (y) instead of x˜1 (y), and therefore pick x˜1 (y) since the derivative for non-binding asset constraints does not apply to them. However, there will be an asset range of individuals who, in the case of an accident, are not bankrupted if they choose x˜2 (y), but are bankrupted if x˜1 (y) is taken. These individuals compare Z(˜ x1 (y), y) with Z(˜ x2 (y), y) to decide on the optimal x2 (y), y) precaution. We define y¯ to be the first asset level for which Z(˜ x1 (y), y) ≤ Z(˜ holds. Individuals with asset values greater than y¯ likewise choose x˜2 (y). Proposition 2 Assume strict liability and monetary care. Potentially judgment-proof injurers with y = y¯ − ,  > 0 take more care than individuals with y = y¯ + γ, γ ≥ 0, if  and γ are sufficiently small. Proof. See the above. This result holds for the case of risk aversion and risk neutrality. In the case of risk neutrality, it is caused by the discontinuity in the marginal costs of care (Miceli and Segerson 2003). We continue and inquire whether individuals with sufficient funds decrease their care further after increases in the asset value, that is, if the effect from Section 2.1 remains after the change in the assumption concerning care. For that, our interest is on the sign of d˜ x2 /dy. In analogy to equation (5), the following cross partial ∗∗ The second-order condition clearly holds for y − l − x < 0, and we assume that it also does in the other case.

126

derivative is critical for this sign. ⎧ ⎪    ⎪ if y − l − x < 0 ⎪ ⎪ −p (x) u (y − x) − [1 − p(x)] u (y − x) ∂ 2 Z(x, y) ⎨    = {p (x) [u (y − l − x) − u (y − x)] ⎪ ∂x∂y ⎪ ⎪ ⎪ ⎩ −p(x) u (y − l − x) − [1 − p(x)] u (y − x)} if y − l − x ≥ 0

(9)

This term has characteristics similar to those of the cross partial derivative for the case of non-monetary care. Increases in y unambiguously call for higher care as long as assets are insufficient with respect to covering harm and care, whereas ’affluent’ injurers have to weigh the two opposing effects, lower marginal benefit and lower marginal costs x2 (y)/dy can be greater than, of care. That is, it holds that d˜ x1 (y)/dy > 0, whereas d˜ equal to, or less than zero, and individuals with sufficient funds decrease monetary care in response to asset value increases if conditions laid out in Proposition 1 apply. This effect can thus further contribute to the difference between care choices of agents with and without binding asset constraint in the case of an accident. For the example given in Section 2.1, we obtain the results depicted in Figure 2. We deduce that injurers with y < 12.616 choose care x˜1 (y), whereas x˜2 (y) is optimal for y = 12.616) = 2.68666 to individuals with y ≥ 12.616. Thus, optimal care falls from x˜1 (¯ x˜2 (¯ y = 12.616) = 2.53773. x ˜1 , x ˜2 x ˜2 2.5

2

x ˜1

1.5

1

0.5

5

15

10

20

25

y

Figure 2: Optimal care as a function of the asset value in the case of monetary care 127

For instance, an individual with y = 12.61 compares the expected utility of the alternative choices. For her, choosing x˜1 (y), which causes y − l − x < 0, yields a higher expected utility than taking x˜2 (y) as precaution, although in that case y − l − x > 0 holds. Furthermore, we can observe that the effect detailed for non-monetary care is present as well since injurers who are not bankrupted by compensation requests decrease their care choice for increases in asset value.

3

Conclusion

Judgment proofness is considered an important impediment to the effectiveness of the liability system in inducing efficient care. By using Shavell’s model, we find that some judgment-proof injurers take more care than some injurers with sufficient assets in certain circumstances. For the case of non-monetary care, this result contrasts pronouncedly with that of the literature heretofore and originates from the risk aversion of individuals in our model. Individuals with sufficient funds may decrease care in response to increases in asset value because this change decreases both the marginal benefit and the marginal costs of care. The lowering of the marginal benefit of care does not result for judgmentproof individuals since the asset value change has no effect on the utility of the accident state. Due to this irrelevance, increases in assets actually increase the marginal benefit of care for individuals who are judgment proof in the accident contingency. For the case of monetary care, this effect combines with a discontinuity in the marginal cost of care. Judgment proofness is very likely to be a valid description of many practical contexts. Following the considerations of Shavell (1986) concerning the consequences of judgment proofness, namely, potentially judgment-proof injurer’s insufficient incentive to take precaution, several policy suggestions, including third-party liability and minimum asset requirements, have been discussed as solutions to the problem (for a discussion see, e.g., Shavell 2005). Our result certainly does not lessen the need to evaluate or to design policy measures to deal with limited liability in an optimal way, yet it certainly dampens the negative conclusions made hitherto with respect to individually optimal choices of

128

injurers with insufficient funds. It is not generally true that potentially judgment-proof injurers exert less caution than other individuals, and this assertion does not hinge upon the nature of care, whether it be monetary or non-monetary, if behavior is best described by risk aversion. Concluding, we comment on two variations to the analysis presented. First, the analysis focused on strict liability. If negligence is the applicable liability rule and the care standard is set at the efficient level, we expect injurers to take due care for sufficiently high assets and to maintain this care level if assets increase, where this sufficient asset level is less than harm irrespective of the care conception.†† The risk aversion we allow for adds to the fact that negligence only implies care costs instead of care costs plus expected harm. Second, care lowers the accident probability in our set-up. If we were to use the model formulation in which care impacts on the magnitude of the loss, the outcome would be characterized by (i) a discrete jump in care from zero to a strictly positive amount at a critical asset level, and (ii) an ambiguous cross partial derivative for further increases in asset value once that jump to strictly positive care values has occurred. If the injurer finds it optimal to choose positive care, the individual’s funds are greater than harm (harm plus care), given optimal care, in the context of non-monetary (monetary) care. Consequently, we cannot transfer our finding to this framework.

4

References

Arrow, K. (1976). Essays in the Theory of Risk-Bearing. 3rd print. Amsterdam: North Holland Publishing. Beard, T.R. (1990). Bankruptcy and Care Choice. RAND Journal of Economics 21: 626-634. Bell, P.A., and J. O’Connell (1997). Accidental Justice: The Dilemmas of Tort Law. New Haven, Connecticut: Yale University Press. Boyd, J., and D.E. Ingberman (1994). Noncompensatory Damages and Potential Insol†† Note that risk aversion complicates the decision on efficient care. See the discussion in Miceli and Segerson (1995). Our reference above is to the standard referred to by Shavell (1986).

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vency. Journal of Legal Studies 23: 895-910. Briys, E. and H. Schlesinger (1990). Risk Aversion and the Propensities for Self-Insurance and Self-Protection. Southern Economic Journal 57: 458-467. Dari Mattiacci, G. and G. De Geest (2005). Judgment Proofness under Four Different Precaution Technologies. Journal of Institutional and Theoretical Economics 161: 38-56. Dari Mattiacci, G. and G. De Geest (2006). When Will Judgment Proof Injurers Take Too Much Precaution? International Review of Law and Economics 26: 336-354. Kahan, M. (1989). Causation and Incentives to Take Care under the Negligence Rule. Journal of Legal Studies 18: 427-447. MacMinn, R. (2002). On the Judgment Proof Problem. Geneva Papers on Risk and Insurance Theory 27: 143-152. Miceli, T.J. and K. Segerson (2003). A Note on Optimal Care by Wealth-Constrained Injurers. International Review of Law and Economics 23: 273-284. Pitchford, R. (1995). How Liable Should a Lender Be? The Case of Judgment-Proof Firms and Environmental Risk. American Economic Review 85: 1171-1186. Rubinfeld, D.L. (1987). The Efficiency of Comparative Negligence. Journal of Legal Studies 16: 375-394. Shavell, S. (1986). The Judgment Proof Problem. International Review of Law and Economics 6: 45-58. Shavell, S. (2005). Minimum Asset Requirements and Compulsory Liability Insurance as Solutions to the Judgment-Proof Problem. RAND Journal of Economics 36: 63-77. Sweeney, G.H. and T.R. Beard (1992). The Comparative Statics of Self-Protection. Journal of Risk and Insurance 59: 301-307.

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Chapter 7

On the Similarity of Bilateral Harm and Unilateral Harm with Role-Type Uncertainty∗ Abstract This paper shows that unilateral-harm contexts with role-type uncertainty can create incentives similar to bilateral-harm contexts. For this purpose, we show that the result of Dharmapala and Hoffmann (2005), namely that standard liability rules do not lead to efficient care choices by injurer and victim if precaution costs are interdependent and harm is unilateral whereas they can in the case of bilateral harm, depends on role-type certainty.

Keywords: bilateral harm, unilateral harm, interdependent costs of care, role-type uncertainty, liability rules

JEL-Classification: K 13, D 62

∗ This chapter is an edited version of: Friehe, T. (2007). On the Similarity of Bilateral Harm and Unilateral Harm with Role-Type Uncertainty. Review of Law and Economics 3: Article 13. This work was presented at the 2005 Annual Meeting of the Scottish Economic Society in Perth and the the 2005 Annual Meeting of the Asian Law and Economics Association in Seoul.

131

1

Introduction

Accidents may be such that only one or both parties to the accident suffer harm, labeled unilateral and bilateral harm, respectively. In the standard framework in which both parties can take precaution, common liability rules lead to efficient behavior in the economic model of tort law with unilateral harm (Shavell 1987) and bilateral harm (Arlen 1990). However, there are circumstances in which incentives are dependent on whether only one or both parties eventually suffer harm. Dharmapala and Hoffman (2005) (DH in the following) establish such a circumstance by showing that efficient care incentives may result in a model characterized by interdependent precaution costs in the bilateral-harm case, whereas the efficient care equilibrium is not attained in the unilateral-harm case. DH show that the equilibrium in efficient care indeed follows under quite reasonable assumptions concerning the magnitude of expected harm of respective parties in the bilateral-harm framework. The outcome concerning the unilateral-harm framework is attributed to a lack of available causes of action. Assume simple negligence as the liability rule and that injurers adhere to the due care level, which is set equal to socially optimal care. The victim bears total expected harm and chooses care, taking into account the effect on her costs and expected harm. Hence, she does not internalize the effect of her care on injurer costs of care. If care by one party lowers precaution costs of the other party, injurer costs of care turn out higher than optimal as a consequence of the incomplete internalization by the victim. The injurer has no cause of action for compensation of the damage in the form of higher precaution costs. Thus, the equilibrium entails an inefficient care choice by at least one party. In contrast, in the bilateral-harm context, standards are directed at both parties and there may be incentives to adhere to the standard for both parties, given due care by the other party. This follows since taking standard care removes the burden of the expected harm of the other party. We show that when only one party suffers harm from the accident but the identity of this party is uncertain, incentives in the unilateral-harm context are similar to incentives in the bilateral-harm context. We do this using the example of DH’s analysis. We generalize the model of DH and, in contrast to their findings, show that standard liability rules can 132

induce efficient care by both parties to the accident in the unilateral-harm framework. Our model allows for uncertainty of individuals concerning their role in an accident.1 There is a wide array of activities where role-type uncertainty is characteristic. If individuals participate in such activities, they may turn out as the harm-inflicting or harm-bearing party in an accident. One prime example is automobile accidents.2 The car-accident setting is also the motivating example of DH concerning interdependent precaution costs which makes our incorporation of role-type uncertainty especially warranted.3 The basic rationale for the fact that role-type uncertainty can make incentives in a unilateral-harm context similar to that of a bilateral-harm context goes as follows. The introduction of role-type uncertainty transforms the ex ante status into one of bilateral harm in expectation terms. Since both parties can turn out to be the victim, both parties bear some expected harm. This can ensure that both parties take efficient care, as in the bilateral-harm framework. Again, assume simple negligence. If party X takes due care and party Y fails to do so, then Y has to compensate individual X in case X turns out as the victim. This threat of expected liability can prove sufficient to make due care an optimal response to due care by the other party. However, we also highlight that, as in the bilateral-harm model, efficient care does not necessarily follow. The similarity of the bilateral-harm case and the unilateral-harm case with role-type uncertainty is not in all cases such that resultant individual incentives are the same. To show this, we provide a comparison of contexts of bilateral harm with independent and interdependent precaution costs and four different unilateral-harm scenarios. In only one of these scenarios is the consideration of unilateral harm with role-type uncertainty equivalent to the consideration of bilateral harm. In the other scenarios, specific conditions need to hold. In Section 2, we detail our model. In the next section, we elaborate on the different 1

This kind of uncertainty is introduced by Kim and Feldman (2006), whose contribution we deal with more extensively below. 2 Automobile accidents are important not only because they are often cited as an example but also because of their empirical significance. For instance, Shavell (2007) states that about half of all tort litigation in the United States concerns automobile accidents. 3 For example, SUVs lower the precaution costs of the driver but raise the precaution costs of others. This causes interdependency in the precaution of drivers and is due to the increased height of the vehicle which improves the sight of the driver but impedes that of others.

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scenarios just referred to, and position the ensuing analysis within that classification. We pinpoint the result of DH for the unilateral-harm setting as a special case of our generalized model in Section 4, and elaborate on why role-type uncertainty can ensure efficient care in this setting in Section 5. The main analysis assumes simple negligence (SN). We provide an illuminating example before we sketch the effect of role-type uncertainty for other standard liability rules. Section 6 concludes.

2

The Model

To exemplify that contexts of unilateral harm combined with role-type uncertainty can create incentives similar to bilateral-harm contexts, we use the framework of DH including notation and assumptions. The population of potential injurers and victims is homogeneous and consists of risk-neutral individuals. We consider two representative individuals X and Y who take care x and y and bear precaution costs C X (x, y) and C Y (x, y), respectively. Parties to the accident have complete information with regard to payoffs and the applying legal rule and standard. The court is well informed and can, for instance, define standards at efficient levels. The care of X and Y reduces expected damages L(x, y) at a decreasing rate.4 Assumption 1: (i) L(x, y) ≥ 0; (ii) Lx < 0; (iii) Ly < 0; (iv) Lxx > 0; (v) Lyy > 0. Each party faces increasing and strictly convex costs of precaution as a consequence of her care choice. X Y > 0; (iv) Cyy > 0. Assumption 2: (i) CxX > 0; (ii) CyY > 0; (iii) Cxx

It is further assumed that care lowers the precaution costs of the other party. Hence, we assume a positive externality which can be illustrated, again, by referring to the case of automobile accidents. The cautious driving of others, such as using the indicator and driving with a safety distance, will normally make it easier for you to hold on to a certain precaution and thereby exert a positive externality. Assumption 3: (i) CyX < 0; (ii) CxY < 0. 4

Subscripts denote partial derivatives.

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Lastly, we follow DH in presuming that expected accident damages L are sufficiently large, relative to the costs of care. Assumption 4: (i) For any y and x < x∗ , L(x, y) > C X (x∗ , y) − C X (x, y), with x∗ being the socially optimal level of care exerted by X; (ii) For any x and y < y ∗ , L(x, y) > C Y (x, y ∗) − C Y (x, y), with y ∗ being the socially optimal level of care exerted by Y. The last assumption ensures that increasing care to the socially optimal level (which shifts expected damages to the residual bearer) is individually rational due to the reduction in expected liability. This maintains the discrete jump at the socially optimal precaution level well known from the literature without interdependent precaution costs (e.g., Shavell 1987). We generalize this model by allowing individuals to be uncertain about their role in an accident ex ante. Individuals X and Y might be injurer or victim in an accident and hold beliefs about the likelihood of being the victim. Interpret α as the subjective probability of X that she will be the victim and take (1 − α) as the subjective probability of X that she will be the injurer, α ∈ [0, 1]. The subjective victim probability of individual Y is denoted β, β ∈ [0, 1].5 We distinguish the following cases: (i) the probabilities are consistent with each other, which implies that the probability with which individual X is the injurer has to be the same as that with which individual Y is the victim, β = 1 − α, and (ii) the probabilities are not restricted to be consistent. The first case is a reasonable starting case. If individuals rationally base their beliefs on past experience, for instance, we would expect that victim probabilities indeed tend to consistency. In the standard case, the number of injurers is equal to the number of victims (e.g., Shavell 1987). This implies an objective victim probability of one half. If subjective victim probabilities were accurate, they would need to be consistent and equal to one half. However, as α and β are subjective, we do not impose an accuracy requirement on them. In the second case, α and β do not have to sum up to one. It is important to make this generalization because 5

For the sake of simplicity, victim probabilities do not depend on the care choice.

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individuals often have difficulties in dealing with probabilistic events and the ranking of their ability in comparison to that of others. For example, Svenson (1981) finds that 90 percent of the automobile drivers in Sweden consider themselves ”above average”. Similar results are reported by Rutter, Quine and Alberry (1998) for motorcyclists in Britain. From that optimism bias, we might deduce that individuals will tend to underestimate the probability of being an injurer and exaggerate the victim probability. In such a case, victim probabilities would no longer sum to one. As usual, welfare is assumed to be maximal if expected social costs are minimized. Hence, the social goal is the minimization of the sum of precaution costs and expected harm. min SC(x, y) = C X (x, y) + C Y (x, y) + L(x, y) x, y

(1)

The socially optimal care levels x∗ and y ∗ simultaneously solve6 CxX (x, y) + CxY (x, y) + Lx (x, y) = 0

(2)

CyX (x, y) + CyY (x, y) + Ly (x, y) = 0.

(3)

Turning from the social to the individual optimization, parties take care to minimize individual costs consisting of care costs and expected liability. As we consider simple negligence, the court uses x∗ (y ∗ ) as the care standard if actor X (Y) is the injurer. We now derive the individual costs of X: (i) If x < x∗ and y < y ∗, she bears expected damages only as injurer. Her costs for the given restriction on care then are C X (x, y) + (1 − α)L(x, y). (ii) If x ≥ x∗ and y < y ∗ , individual X will be compensated as a victim. In the role of the injurer, individual X adheres to the standard and therefore does not have to compensate individual Y. Consequently, individual X’s costs in this circumstance consist only of precaution costs, C X (x, y). (iii) If x < x∗ and y ≥ y ∗ holds, X receives no compensation as victim because individual 6 The second-order conditions are ensured by the assumptions of DH, namely that the mixed cross partials are sufficiently small.

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Y follows the care standard. If X turns out as the injurer, she has to compensate Y because X’s care falls short of due care. Hence, costs are C X (x, y) + L(x, y). (iv) If x ≥ x∗ and y ≥ y ∗ holds, individual X receives no compensation as victim because Y complies. If X is the injurer, X also pays no compensation to Y. Thus, X bears expected damages in the role of the victim, leading to expected costs of C X (x, y) + αL(x, y). These considerations yield care-dependent individual costs of individual X, labeled XC,

XC(x, y) =

⎧ ⎪ ⎪ C X (x, y) + (1 − α)L(x, y) if x < x∗ and y < y ∗ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ C X (x, y) if x ≥ x∗ and y < y ∗ ⎪ ⎪ C X (x, y) + L(x, y) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ C X (x, y) + αL(x, y)

if x < x∗ and y ≥ y ∗

(4)

if x ≥ x∗ and y ≥ y ∗.

In analogy, we obtain individual costs of actor Y, labeled Y C, ⎧ ⎪ ⎪ C Y (x, y) + (1 − β)L(x, y) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ C Y (x, y) Y C(x, y) = ⎪ ⎪ C Y (x, y) + L(x, y) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ C Y (x, y) + βL(x, y)

3

if y < y ∗ and x < x∗ if y ≥ y ∗and x < x∗ if y < y ∗ and x ≥ x∗

(5)

if y ≥ y ∗and x ≥ x∗ .

Bilateral Harm and Unilateral Harm with RoleType Uncertainty Compared

In reality, it often holds that both parties to the accident suffer harm. The social costs to be minimized in such circumstances of bilateral harm (BH) can be stated as

SC BH (x, y) = C X (x, y) + C Y (x, y) + LX (x, y) + LY (x, y),

(6)

where LX (x, y) [LY (x, y)] is the expected harm of individual X [Y] and (xBH , y BH ) are socially optimal care levels. Bilateral-harm contexts may be characterized by independent or interdependent precaution costs. For the case of independent precaution costs, Arlen (1990) shows that

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standard liability rules induce efficient care. Let us briefly reproduce the reasoning in order to be able to contrast it to the case of interdependent precaution costs. In the case of simple negligence, individual costs of, for instance, actor X can be stated as

XC BH (x, y) =

⎧ ⎪ ⎪ C X (x, y) + LY (x, y) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ C X (x, y)

if x < xBH and y < y BH if x ≥ xBH and y < y BH

⎪ ⎪ C X (x, y) + LX (x, y) + LY (x, y) if x < xBH and y ≥ y BH ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ C X (x, y) + LX (x, y) if x ≥ xBH and y ≥ y BH .

(7)

First of all, note that independency of precaution costs implies CyX , CxY = 0 ∀ (x, y). Actor X’s individual costs are C X (xBH , y BH ) + LX (xBH , y BH ) should she and Y take due care, which is set equal to the optimum level, or the minimum of C X (x, y BH ) + LX (x, y BH ) + LY (x, y BH ) using x < x∗ if only Y takes due care. Since the latter term is minimized by due care, there is an equilibrium in standard care. Furthermore, because C X (xBH , y BH ) + C Y (xBH , y BH ) < C X (x, y) + C Y (x, y) + LX (x, y) + LY (x, y) for any (x, y) not equal to (xBH , y BH ) by definition of (xBH , y BH ), there is no equilibrium in which both parties are negligent. For the case of interdependent precaution costs, it is less clear that only an efficient care equilibrium should result under standard liability rules. DH can establish this result given a number of assumptions.7 To see this need for assumptions, note that, for instance, given negligence as liability rule and due care by Y, X can either bear C X (xBH , y BH ) + LX (xBH , y BH ) should she also take due care or the minimum of C X (x, y BH ) + LX (x, y BH ) + LY (x, y BH ) if only Y takes due care. However, the last term is minimized by x < xBH due to the cost interdependency. To make due care the optimal response to due care, DH assume that the difference in expected costs, and especially the magnitude of LY (x, y BH ), is sufficiently large to make taking xBH better than any other x < xBH for actor X. This ensures the equilibrium in efficient care and a similar assumption excludes an equilibrium in substandard care. Now, let us turn to the unilateral-harm context with role-type uncertainty. Note that if both individuals are uncertain as to their role in an accident, both parties bear positive 7

The requirements are laid out in Assumptions 5-9 in DH.

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expected harm. To that extent, the ex ante description is similar to the bilateral-harm case despite the fact that only one party suffers harm ex post. There are four different scenarios for the circumstances in which an activity may take place. These are: (i) [(ii)] role-type probabilities are consistent [inconsistent] and precaution costs are independent, and (iii) [(iv)] role-type probabilities are consistent [inconsistent] and precaution costs are interdependent. Social costs in all of these instances can be stated as in (6), as long as LX and LY are appropriately defined. For this purpose, take η as the objective victim probability of individual X. In that case, LX (x, y) = ηL(x, y) and LY (x, y) = (1 − η)L(x, y), so that social costs can be stated as C X (x, y) + C Y (x, y) + ηL(x, y) + (1 − η)L(x, y) or as in (6). The former specification obviously collapses to our definition in (1). This establishes a structural analogue to the bilateral-harm case. However, it is of importance to notice that individual incentives can in fact be quite different from those in the bilateral-harm case. This difference comes from the fact that victim probabilities are subjective and do not necessarily concur with objective ones, and that the former are important for the individual optimization whereas the latter are relevant for social costs. If role-type probabilities are consistent and precaution costs independent, only the efficient care equilibrium exists under standard liability rules.

Simply by redefining

(1 − α)L(x, y) [αL(x, y)] as L (x, y) [L (x, y)], where L (x, y) + L (x, y) = L(x, y), that Y

X

X

Y

is, using the consistent subjective victim probabilities, one can cast the argumentation of the unilateral-harm case with role-type uncertainty as one in a framework of bilateral harm. If role-type probabilities are inconsistent and precaution costs independent, there may be a second equilibrium in suboptimal care apart from the efficient care equilibrium under standard liability rules. Both results are derived by Kim and Feldman (2006), who indeed need inconsistent probabilities for their propositions.8 Consequently, if role-type probabilities depart in some fashion from being consistent, then resultant incentives are different from those in the bilateral-harm framework, testified by the fact that only the efficient outcome results in the latter framework given independent precaution costs. For 8 The first result concerning the existence and uniqueness of the efficient care equilibrium is not stated explicitly but follows immediately.

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instance, if individuals attach a too high probability to ending up as the victim in an accident, i.e., if α + β > 1, the second equilibrium in suboptimal care exists under simple negligence. Individuals act to minimize individual costs, which are dependent on the subjective victim probability. The equilibrium in suboptimal care can therefore only exist in the context of unilateral harm and role-type uncertainty if XC(x, y)+Y C(x, y) < SC(x, y) holds. In contrast, in the bilateral-harm case, the sum of individual costs always adds up to social costs, which rules out an equilibrium in suboptimal precaution. Consequently, if individuals combined were to consider exactly total expected harm, only the efficient outcome results in the model by Kim and Feldman (2006). Our ensuing analysis considers interdependent precaution costs with consistent and inconsistent role-type probabilities, that is, scenarios (iii) and (iv). In the bilateral-harm case, the efficient care equilibrium results if certain assumptions, alluded to above, are made. We will show that there are circumstances in which the efficient care equilibrium results in the unilateral-harm framework with role-type uncertainty if certain assumptions for the role-type probabilities other than concerning consistency are met. Consequently, these conditions on victim probabilities compare to the assumptions of DH on the magnitude of expected harm of respective parties. As in the case of independent precaution costs, the scenario of consistent victim probabilities is more comparable to the bilateralharm framework than the scenario of inconsistent probabilities.

4

The Special Case of Dharmapala and Hoffman

Our model allows for the whole range of probabilities, α, β ∈ [0, 1]. DH assume that the identities of the injurer and the victim are determined. This means that, for instance, α = 0 and β = 1, so that person X will be injurer and individual Y the victim with certainty if an accident occurs. With reference to (4) and (5), we retrace the result that there is no equilibrium in efficient care by both parties. If injurer X chooses substandard care, victim Y will choose a care level of zero since under that circumstance, there are only costs and no benefits to care expenditures. If the injurer chooses x = x∗ , the victim

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minimizes her costs by choosing y˜ = arg min{C Y (x∗ , y) + L(x∗ , y)} < y ∗ . The injurer will never choose supraoptimal care, since she thereby only increases her costs. The fact that the victim, as the residual risk bearer, does not take socially optimal care owes to the decreasing effect of victim care on the injurer costs of care that is not internalized by the victim. Hence, the equilibrium is (x∗ , y˜), since the injurer takes due care as C X (x, y) + L(x, y) > C X (x∗ , y) holds for all x < x∗ and y by Assumption 4 (i).

5

Role-Type Uncertainty Can Enable Efficiency

We now turn to our setting in which the role probabilities of actors are not determined and illustrate that role-type uncertainty can induce efficiency in the presence of interdependent costs of care. The role-type uncertainty creates expected harm for both parties, which can, in analogy to the bilateral-harm case, motivate efficient care.9 The claim is that for certain combinations of respective victim probabilities, only an efficient-care equilibrium exists. To show this, we need to establish conditions for which the equilibrium in efficient care exists and for which it is unique. Hence, we proceed as follows: (i) We define conditions for the victim probabilities that make standard care optimal, given standard care by the other actor. This establishes the existence of the equilibrium in efficient care (Section 5.1). (ii) In Section 5.2, we turn to a possible equilibrium in substandard care.10 Regarding (i), we need to establish that both individuals exert standard care given standard care by the other individual, whereas, with regard to (ii), we only need to identify victim probabilities that make at least one individual respond with standard care, given substandard care by the other individual, to ensure the uniqueness of the equilibrium in efficient care. This second matter will be elaborated on in Subsection 5.2.1 for the case of consistent probabilities and in Subsection 5.2.2 when there is no such restriction on the probabilities. 9 Note that even if, for instance, individual X is certain to be the injurer, whereas individual Y is uncertain and ascribes a probability 0 < β < 1 to being the victim, individual Y perceives an expected harm of (1 − β)L(x, y) for individual X. The promise of avoiding this expected harm by taking standard care may suffice to motivate efficient care by Y. 10 In the appendix, we touch upon why, under our assumptions, there will be no supraoptimal care.

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5.1

Equilibrium in Efficient Care

In this section, we detail conditions under which both parties to the accident choose efficient care for the case of consistent role probabilities, and remark on the similarity to the more general case without the consistency requirement at the end. Assume that X chooses x = x∗ and consider the costs of Y in (5). Since we consider consistent role probabilities, we can express the victim probability of actor Y as the injurer probability of individual X, (1 − α). Individual Y will choose efficient care if11 C Y (x∗ , y˜) + L(x∗ , y˜) ≥ C Y (x∗ , y ∗) + (1 − α)L(x∗ , y ∗). ∗ There is a critical victim probability (1 − αCP Y ) which makes individual Y indifferent

between choosing y ∗ and y˜.12 This is given by ∗ 0 < (1 − αCP Y) =

C Y (x∗ , y˜) − C Y (x∗ , y ∗) + L(x∗ , y˜) < 1. L(x∗ , y ∗)

(8)

∗ Recognize that 0 < 1 − αCP Y < 1 since the numerator is strictly positive by Assumption

4 (ii) and C Y (x∗ , y˜) + L(x∗ , y˜) < C Y (x∗ , y ∗ ) + L(x∗ , y ∗) is true because otherwise y˜ would not be cost-minimizing. As long as individual Y holds a victim probability less than the ∗ critical level, (1−αCP Y ) < (1−αCP Y ), she takes due care, given individual X complies with

standard care. Since individual Y trades off the role-independent increase in precaution costs, C Y (x∗ , y ∗ ) − C Y (x∗ , y˜), against expected liability savings in case individual Y is the injurer, L(x∗ , y˜) − (1 − α)L(x∗ , y ∗), the investment in care is worthwhile for low victim probabilities (1 − α). If individual Y takes efficient care given that X takes x∗ , she does so to reduce the expected harm borne. In the bilateral-harm framework, there exists a parallel in the expected harm of the other party. In that context, given that individual X adheres to the standard, individual Y bears expected harm of both Y and X in the case of substandard care and only the own expected harm in the case of standard care. Consequently, the 11

Recall from above that y˜ = arg min{C Y (x∗ , y) + L(x∗ , y)} < y ∗ . We use the index CP to denote that the values of critical respective probabilities apply to consistent probabilities. 12

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avoidance of the expected harm of X can motivate to comply with the high due care standard if it is sufficiently large.13 Similarly, the avoidance of αL motivates Y to exert due care if α is sufficiently large.14 Thus, our requirements for the subjective victim probabilities find equivalents in DH’s assumptions on the size of expected harm of the other party. We proceed likewise for individual X, with y = y ∗ given, and find ∗ 0 < αCP X =

C X (˜ x, y ∗ ) − C X (x∗ , y ∗) + L(˜ x, y ∗ ) < 1. L(x∗ , y ∗)

(9)

∗ ∗ Victim probability αCP ˜.15 X makes individual X indifferent between choosing x and x

Hence, individual X will, just as actor Y, take due care if the victim probability is suffi∗ ∗ ciently small. Note that the sum of critical victim probabilities, αCP X and (1 − αCP Y ),

is greater than one.16 If the victim probability of actor X, for instance, equals exactly the critical level, it is ensured that the victim probability of Y will fall below the value defined in (8). Now, we turn to the case in which role probabilities are not restricted to being consistent. The reasoning parallels that from above. Hence, we mainly relabel measures for later reference. Individual Y chooses efficient care, given that X chooses standard care, if the subjective victim probability is less than the critical victim probability β ∗ , defined ∗ ∗ ∗ as (1 − αCP Y ) in (8). For individual X, we find α , defined like αCP X in (9). For all

α < α∗ , individual X adheres to the standard because the expected individual costs are lower, given individual Y exerts due care. Proposition 1 Assume Assumptions 1-4 hold. An equilibrium in efficient care exists in a model with interdependent costs of care and simple negligence as the liability rule, if α < α∗ and β < β ∗ holds. 13

See Assumptions 5 and 6 in DH for requirements on the size of the expected harm. This simplifies the facts as the total change in expected harm borne by individual Y amounts to L(x∗ , y˜) − (1 − α)L(x∗ , y ∗ ). 15 x˜ is the individually optimal care given due care by the other actor and full responsibility for expected damages, x ˜ = arg min{C X (x, y ∗ ) + L(x, y ∗ )} < x∗ . X ∗ X Y ∗ Y (x∗ ,y ∗ )+L(˜ x,y ∗ ) (x∗ ,y ∗ )+L(x∗ ,˜ y) 16 This can be seen by rearranging the claim C (˜x,y )−C + C (x ,˜y)−C >1 L(x∗ ,y ∗ ) L(x∗ ,y ∗ ) x, y ∗ ) + L(˜ x, y ∗ ) + C Y (x∗ , y˜) + L(x∗ , y˜) > C X (x∗ , y ∗ ) + C Y (x∗ , y ∗ ) + L(x∗ , y ∗ ). to C X (˜ 14

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Proof. Follows from the above. In sum, there is an equilibrium in efficient care if both individuals have a sufficiently low victim probability. This holds because the standard care level under negligence is directed at the injurer. Injurer care will be compared with standard care and the injurer will be sentenced to compensate the victim if she is found negligent. Hence, given a sufficiently high probability of being the injurer, it minimizes expected costs to take standard care and thereby avoid liability in the subset of events in which one is the injurer.

5.2 5.2.1

No Care Equilibrium at Inefficient Care Levels Consistent Victim Probabilities

For there to be a substandard-care equilibrium, both parties must have weakly higher individual costs if they unilaterally deviate to standard care. Consequently, it must simultaneously hold that x, y¯) + (1 − α)L(¯ x, y¯) ≤ C X (x∗ , y¯) C X (¯

(10)

x, y¯) + αL(¯ x, y¯) ≤ C Y (¯ x, y ∗), C Y (¯

(11)

and

where (¯ x, y¯) denotes care in an equilibrium with substandard care levels in the case of consistent victim probabilities. From this we can deduce that such a substandard-care equilibrium can only result if x, y¯) + C Y (¯ x, y¯) + L(¯ x, y¯) ≤ C X (x∗ , y¯) + C Y (¯ x, y ∗) C X (¯

(12)

holds. In the standard case of independent precaution costs, this argumentation would already suffice to exclude the possibility of an inefficient equilibrium (see Section 3). However, it does not suffice in this frame, as we elaborate in the following. The central x < x∗ ) in relation to its difference is that C X (C Y ) is elevated by the choice y¯ < y ∗ (¯ level at the first-best outcome, whereas it is usually independent of the other’s action. It 144

holds by definition of (x∗ , y ∗) that x, y¯) + C Y (¯ x, y¯) + L(¯ x, y¯) > C X (¯ C X (x∗ , y ∗ ) + C Y (x∗ , y ∗) + L(x∗ , y ∗) > C X (x∗ , y ∗ ) + C Y (x∗ , y ∗).

(13)

x, y ∗ ) > Yet, because of the cost interdependency, we have C X (x∗ , y¯) > C X (x∗ , y ∗) and C Y (¯ C Y (x∗ , y ∗ ) in inequality (12) due to x¯ < x∗ and y¯ < y ∗ , respectively. This fact prevents us from deriving a contradiction to the inequality in (12) directly, as is done in the standard x, y ∗ ) case (exemplified by Kim and Feldman 2006), since the ranking of C X (x∗ , y¯) + C Y (¯ and C X (x∗ , y ∗) + C Y (x∗ , y ∗) + L(x∗ , y ∗) is not unambiguous. However, given that DH emphasize in their Assumption 4 concerning the unilateral-harm context that especially large expected damages are considered, it will often be the case that the sum of the differences in precaution costs provoked by the deviation from the optimum is not as large as expected damages.17 We fix this idea in Assumption 5.18 Assumption 5: For any y < y ∗ and x < x∗ , L(x, y) ≥ C X (x∗ , y)−C X (x, y)+C Y (x, y ∗ )− C Y (x, y). If we state this assumption as L(¯ x, y¯) ≥ C X (x∗ , y¯) − C X (¯ x, y¯) + C Y (¯ x, y ∗) − C Y (¯ x, y¯),

(14)

in reference to (12), it becomes obvious that it allows the exclusion of a substandard-care equilibrium. Proposition 2 Assume Assumptions 1-5 hold. The only equilibrium in a model with interdependent costs of care, consistent role probabilities, and simple negligence as the ∗ ∗ liability rule is in efficient care if αCP X < αCP X and 1 − αCP Y < 1 − αCP Y hold. 17

This will depend on the intensity of the interdependence in precaution costs, among other factors. This assumption actually finds its analog in Assumption 6 of DH, an assumption they utilize only in the bilateral-harm setting. It requires LY (x, y) > C X (x∗ , y) − C X (x, y) ∀ x < x∗ , y < y ∗ and X L (x, y) > C Y (x, y ∗ ) − C Y (x, y) ∀ x < x∗ , y < y ∗ . Our assumption utilizes the sum of these expressions and substitutes LY (x, y) [LX (x, y)] for (1 − α)L(x, y) [αL(x, y)]. 18

145

Proof. Follows from the above. If Assumption 5 holds, there is no substandard-care equilibrium if probabilities of individuals X and Y are consistent, because it is beneficial for at least one actor to deviate by exerting due care. That is why in this case, no further conditions than the ones from Section 5.1 need to be added to ensure the existence and uniqueness of the equilibrium in efficient care. However, recognize that Assumption 5 is by no means necessary to attain our more general results. The case of consistent probabilities is a special instance of the case without restrictions on role probabilities. To that extent, the analysis of the next section, in which we allow for more general role probabilities, is applicable to the case of consistent probabilities as well. The case of consistent role probabilities is obviously quite similar to the bilateralharm framework. This is apparent in the comparability of assumptions needed to support the efficient care equilibrium as the only outcome. The ensuing case of inconsistent role probabilities departs somewhat more. It is, for instance, possible that (α + β)L(x, y) < L(x, y), since role probabilities are unrestricted. This clearly introduces the need for more conditions on role probabilities than what Proposition 2 requires. 5.2.2

Victim Probabilities Do Not Have to Be Compatible

In the following, we provide the intuition for the conditions on role probabilities that exclude an equilibrium in substandard care. The formal analysis is somewhat more laborious than what we have delivered so far. The complete discussion of this case as well as the precise statement of results in Proposition 3 are therefore included in Appendix A, whereas the exposition in this subsection is rather brief. Suppose an equilibrium in substandard care exists and consider individual X more closely.19 Given that individual Y takes less than due care, individual X has costs given

19 By symmetry, the same argumentation applies to individual Y. However, recall that to support our claim, we only need to find conditions that make it preferable for at least one individual to exert due care as a response to substandard care.

146

by

⎧ ⎪ ⎨ C X (x, y) + (1 − α)L(x, y) if x < x∗ XC(x, y) = ⎪ X ⎩ C (x, y) if x ≥ x∗ .

(15)

If it is optimal for individual X to take less than due care, this implies that a unilateral deviation to standard care causes larger individual costs. If we denote care levels in a substandard-care equilibrium for the case of unrestricted probabilities by (ˆ x, yˆ), this implies that D defined as x(α, β), yˆ(α, β)) + (1 − α)L (ˆ x(α, β), yˆ(α, β)) − C X (x∗ , yˆ(α, β)) D(α, β) = C X (ˆ

(16)

is negative, where D represents the cost consequence of not deviating unilaterally to standard care from the substandard-care equilibrium. Now, we inquire whether there are role probabilities for which D is positive, indicating that there is no longer a substandardcare equilibrium since actor X rather exerts standard care. Critical role probabilities turn D equal to zero as actor X is then indifferent between exerting substandard and due care. The assumption that care by X and Y are substitutes and a simplification made, which is detailed in the appendix, ensure that D is a continuous and monotonous function. It can be shown that xˆα , yˆβ < 0 and xˆβ , yˆα > 0 holds. Individual X compares the cost level of the substandard care case with the unilateral deviation to standard care which is expressed by function D. Function D is monotonous and continuous, and (i) D(0, β) > 0 as well as (ii) D(1, β) < 0 holds. The sign under (i) holds by Assumption 4 (i) and expresses that it is always advantageous to take standard care if the actor holds the belief to certainly be the injurer. This is due to the fact that the magnitude of expected liability which can be avoided by exerting due care is sufficient. The sign under (ii) holds, as zero care has no expected consequence if the actor expects to be the victim, since the negligent other party will always fully compensate. Because ¯ = 0. of (i) and (ii), there must be one value α∗∗ for every β¯ which yields D(α∗∗ , β) ¯ is the set of critical role probabilities that we look for in this subsecThis pair (α∗∗ , β) tion to be in a position to impose conditions ensuring the uniqueness of the equilibrium in

147

D D(α, β2 ) D(α, β1 )

α∗∗ (β1 )

α∗∗ (β2 )

α

Figure 1: Stylized function D for different β, β2 > β1 efficient care. For α < α∗∗ , individual X rather expends on standard care as response to substandard care by V as long as β ≥ β¯ since Dα < 0 and Dβ > 0 holds. An increase in the victim probability of actor X makes exerting due care less attractive, Dα < 0, because actor X (Y) lowers (increases) her care in response. In other terms, individual X finds any substandard care equilibrium the more appealing, the more likely it is that X is the victim who will not be punished for her substandard care by compensation requirements. In contrast, an increase in the victim probability of actor Y increases the payoff of the unilateral deviation to standard care from X’s perspective, Dβ > 0. Thus, the critical level α∗∗ is an increasing function of the value β¯ as the substandard equilibrium becomes increasingly less attractive for X if individual Y considers it increasingly likely to be the victim. Figure 1 illustrates the fact that D falls in α and that larger values of β shift D to the right. These intervals for α and β, respectively, which enable efficiency, always exist. For example, the borderline level for actor X’s victim probability which ensures that he takes ¯ is strictly positive. due care in response to standard or substandard care, min{α∗ , α∗∗ (β)}, Therefore, considering interdependent precaution costs in the unilateral-harm context as a realistic feature no longer carries the drastic connotation with respect to standard liability rules implied by DH.

148

5.3

Discussion

Let us compare this finding to the benchmark of DH detailed in Section 4. There, it holds that α = 0 and β = 1. Starting from this benchmark, we argue that if β is sufficiently lower than one, the only outcome possible is an equilibrium in efficient care. The rationale goes as follows. First, the individual X, certain to be the injurer, exerts standard care even in the benchmark. In this special case, the problem lies with individual Y who, certain to be the victim, minimizes the sum of personal precaution costs and expected damages. Hence, because individual Y does not internalize the effect of her precaution on the precaution costs of X, she takes too little care. Now, we reduce the probability that individual Y attaches to the victim role. This has the effect that individual Y also considers the eventuality that she is the injurer who is free from liability as long as care complies with the standard. This can incentivize the exertion of standard care if the victim probability is sufficiently below one, i.e., the injurer probability is sufficiently above zero. A parallel to the bilateral-harm framework can be drawn. In our framework, the exertion of standard care has its virtue compared to substandard care due to the fact that liability is avoided whenever individual Y turns out as the injurer. This avertable liability is the expected harm of actor X from Y’s perspective, (1 − β)L(x, y). In the bilateral-harm framework, this function is fulfilled by the expected harm of the other party, LX (x, y). The intuition for our general result is, accordingly, pretty straightforward. If an individual expects to be the injurer, i.e., the party at which the behavioral standard is directed, with a probability of a specific magnitude, she rather bears the additional expenditure on care. The alternative is to possibly bear the expected liability because of substandard care. In other terms, it is a trade-off of certain additional precaution costs against a saving in expected liability. Given role-type certainty, there is not an argument for both individuals to take efficient care. Injurers are free from liability as they comply with the standard and the only consideration of the victim in that context is the minimization of the sum of private precaution costs and expected damages. In our framework with simple negligence as the liability rule, small victim probabilities achieve both necessary effects for the existence and uniqueness of the equilibrium in efficient care, namely (i) 149

that standard care is the best response to standard care and (ii) that standard care is the best response to substandard care. Small victim probabilities create a threat in the realm of substandard-care contexts, in the form of expected liability of a sufficient magnitude to exclude an equilibrium in substandard care. The simultaneous requirement as to the victim probability of the other individual owes to two reasons. First, it is a consequence of the fact that standard care ought to be the best response to standard care for both actors. Second, taking standard care as a response to substandard care is more desirable if the other party considers it likely to be the victim, because this evokes lower care by this party.

5.4

An Example

A better understanding of the requirements concerning the subjective victim probabilities and the power of our results concerning contexts of interdependent precaution costs and unilateral harm can be generated by a simple example. For that purpose, we take C X (x, y) = 2x−y and C Y (x, y) = 2y−x as individual precaution costs, P (x, y) =

1 1+x+xy+y

as the accident probability, as in Kim and Feldman (2006), and H = 100 as the level of harm, where P (x, y)H = L(x, y). Recognize the cost interdependency in the respective precaution cost functions and the fact that, inter alia, Px H < 0, Pxx H > 0 and X + (1 − α)Pxy H > 0 for α ∈ [0, 1) hold, as required by our assumptions.20 Social costs Cxy

as a sum of individual precaution costs and expected damages are thus given by

SC(x, y) = C X (x, y) + C Y (x, y) + L(x, y) = x + y +

100 . 1 + x + xy + y

(17)

The socially optimal care levels for this minimization problem are (x∗ , y ∗) = (3.642, 3.642). Proceeding as above, we (i) detail the requirements for the existence of an efficientcare equilibrium and (ii) illuminate the conditions with respect to any substandard-care 20 X Y For the sake of simplicity, we choose cost functions with Cxx , Cyy = 0 which is excluded by the assumptions of DH (see Assumption 2 (iii) and (iv) above). However, this is no restriction because Lxx (Lyy ) > 0 ensures second-order conditions except in the case of α(β) = 0, in which case the discontinuity makes due care obviously cost-minimizing, or in case α(β) = 1 and y < y ∗ (x < x∗ ) hold, in which case zero care is obviously cost-minimizing. Thanks to Jeonghyun Kim for pointing this out.

150

equilibrium. Regarding (i), we search for critical values α∗ and β ∗ that make standard care by both individuals a care equilibrium. For this, we take standard care by actor Y as given and inquire what victim probability α makes actor X take standard care instead of individually optimal care, where the latter is smaller due to the cost externality. Genx, y ∗ ) + L(˜ x, y ∗ ). With erally, α∗ solves the equation C X (x∗ , y ∗ ) + α∗ L(x∗ , y ∗) = C X (˜ x˜ = 2.28 = arg min{2x − y ∗ +

100 }, 1+x+xy ∗ +y ∗

we find α∗ = .8847. The symmetry of the

problem yields the same for β ∗ , i.e., the critical victim probability for actor Y responding to x = x∗ with y = y ∗. In Figure 2, we highlight how role-type uncertainty reintroduces a discontinuity that can make due care advantageous despite the positive externality on precaution costs. For y < y ∗ = 3.642, individual Y bears full expected damages, as x = x∗ is given. Once individual Y also complies with the standard, she bears damages only in the event she turns out as the victim. VC

y∗

y˜

y

Figure 2: Discontinuity in individual costs for β = .5 and x = x∗ With regard to (ii), behavior is governed by the respective first-order conditions (1 − α)100(1 + y) =0 1 + x + xy + y (1 − β)100(1 + x) = 0. V Cy (x, y) = 2 − 1 + x + xy + y ICx (x, y) = 2 −

(18) (19)

That both parties appoint equal probability to both states, i.e., (α, β) = (.5, .5), is a 151

reasonable starting case. Reaction curves intersect at (ˆ x, yˆ) = (1.924, 1.924). Would this actually result as substandard-care equilibrium? To answer this question, we insert the relevant values into D(α, β), to find that, given these victim probabilities, individual X rather exerts standard care.

D(.5, .5) =C X (1.924, 1.924) + (1 − .5)L(1.924, 1.924) − C X (3.642, 1.924) =2.413 > 0

(20)

In fact, for β¯ = .5, the corresponding critical victim probability for actor X is α∗∗ = .64. This α∗∗ increases in β¯ as found in the text. For example, whereas given β¯ = .7, the corresponding α∗∗ is .725, the comparison yields α∗∗ = .55 for β¯ = .2. In sum, when presenting the above in the terms of Proposition 3, we obtain: An equilibrium in efficient care exists in this model with interdependent costs of care and simple negligence as the liability rule, if α < .8847 and β < .8847 holds. The efficient care equilibrium is unique if, for instance, α < min{.55; .8847} and .2 ≤ β < .8847 or .2 ≤ α < .8847 and β < min{.55; .8847}. Note that we could have used other pairs of critical role probabilities to exclude the possibility of an equilibrium in substandard care, for instance, one of those mentioned in the next to last paragraph. This example demonstrates that the conditions regarding victim probabilities are not necessarily extreme.

5.5

Role-Type Uncertainty and other Liability Rules

Liability rules affect the apportionment of the damage burden dependent on care. For simple negligence as the liability rule, we have established that role-type uncertainty can ameliorate the impact of interdependent costs of care. In the following, we sketch how our result carries over to strict liability with a defense of contributory negligence, negligence with a defense of contributory negligence, and comparative negligence.21 Strict liability with a defense of contributory negligence (SLCN) is the mirror image 21

The respective propositions and formal proofs of this section are relegated to Appendix B.

152

of simple negligence (SN), as this rule entails one standard directed at the victim. DH found with their assumptions concerning (α, β) that (˜ x, y ∗) is the equilibrium, with x˜ < x∗ . This results because, whereas the victim complies with the standard, the injurer optimizes continuously, taking into account only her precaution costs and expected damages. The cost functions in our general framework are similar to the case of SN except for the fact that negligent victims pay expected damages irrespective of the behavior of the injurer, whereas under SN, the negligent injurer pays expected damages irrespective of the behavior of the victim. The fact that SLCN is like SN reversed leads to the observation that the terms α [β] and (1 − α) [(1 − β)] appear under reversed conditions for the care choices in the respective individual costs, (4) and (5). This reasoning leads to the conclusion that role-type uncertainty can further efficiency under SLCN as well. Yet victim probabilities need to be above critical values instead of below. Under negligence with a defense of contributory negligence (NCN), the victim bears expected damages in case both parties are negligent, in case only she is negligent and in case both parties are non-negligent. Individual costs reveal that, on the one hand, for role-type uncertainty to enable efficient care choices, a small victim probability is needed in order to make standard care the best response to standard care. On the other hand, too small victim probabilities make potential substandard equilibria attractive. Thus, the victim probabilities that allow for pure strategy equilibria in efficient care need to fall between two critical values. The case of NCN differs from SN and SLCN in that the same party is responsible if both parties are either non-negligent or negligent. For SN and SLCN, one role bears expected damages if both parties are negligent and the other party is responsible if both parties take standard care. For NCN, this creates a tension between a required minimum and a maximum level, where it is not assured that this ranking of levels ensues. Finally, we consider comparative negligence (CN). Under this liability rule, individual expected liability is independent from the victim probability if x < x∗ and y < y ∗, because it varies with the own negligence in comparison with the negligence of the other individual. This makes it impossible to exclude any substandard equilibrium by requirements on

153

the respective victim probabilities. However, if Assumption 5 holds, there is no possible substandard-care equilibrium. This is the same result as in the case of consistent probabilities and relies on the same rationale. Respective fractions borne by agents if both are negligent always sum to one, irrespective of the subjective victim probabilities.22 Conditions similar to those of Section 5.1 assure the existence of the equilibrium in efficient care.

6

Conclusion

Accident situations can be distinguished into those in which only one party to the accident suffers harm and those in which both do. We show that, although the situation is one of unilateral harm as only one side is harmed ex post, it often holds that ex ante incentives are similar to those of the bilateral-harm case. This holds because role-type uncertainty creates positive expected harm for both parties. Our analysis took the analysis of Dharmapala and Hoffmann (2005) as a case in point. They show that no standard liability rule ensures the equilibrium in efficient care in unilateral-harm models if precaution costs are interdependent, whereas this problem does not necessarily emerge if the model is one of bilateral harm. We allow for the possibility that individuals are uncertain as to their role in an accident and show for simple negligence that for a range of role-type probabilities, efficient care by both parties results as the unique equilibrium in the unilateral-harm context. Moreover, in an example, we demonstrate that this range is broad enough to be of practical significance. To show the generality of our findings, we also consider role-type uncertainty with other liability rules, namely strict liability with a defense of contributory negligence, negligence with a defense of contributory negligence, and comparative negligence. For the former we find that role-type efficiency can similarly enable the attainment of the efficient outcome, although the conditions for the victim probabilities are reversed as the liability 22 This is also why Kim and Feldman (2006) find a relative advantage of comparative negligence in comparison to the other liability rules. In their model and here, this advantage originates from the role-independent payoffs if both parties are negligent.

154

rule reverses the apportionment in comparison to simple negligence. For negligence with a defense of contributory negligence, we find that there may be circumstances in which roletype uncertainty cannot enable the efficient care equilibrium because the requirements set up by critical probabilities might not be fulfilled. For the liability rule of comparative negligence, payoffs for substandard care levels do not depend on victim probabilities. This eases ensuring the uniqueness of the equilibrium in efficient care under relatively general conditions. Conditions on role probabilities are far less restrictive under relatively general conditions if role probabilities of individuals are consistent. It is an empirical question whether consistent beliefs are realistic. It indeed has been established empirically that individuals tend to be too optimistic concerning their abilities in comparison to that of others. Hence, a possible transfer to our context is that, as a consequence of this optimism bias, the subjective victim probabilities will not be consistent. In fact, this can imply that the sum of subjective victim probabilities is greater than one, as one does not expect to be the cause of an accident. Given that, strict liability with a defense of contributory negligence is appealing as it requires high victim probabilities to induce the efficient outcome. In concluding, the following is worthy of note: This analysis shows that the bilateralharm case may actually be more important than previously believed. Some situations described by unilateral harm ex post might be better characterized as ones of bilateral harm due to role-type uncertainty. Another important aspect of this study is that uncertainty can further efficiency, although it is usually conceived as an obstacle to efficiency (see, e.g., Dari Mattiacci forthcoming). However, the uncertainty in this framework is not due to blurred legal standards or uncertain compensation levels but results as a consequence of the inherent nature of the given activity.

155

Appendices Appendix A In this addendum, we restate the arguments of Subsection 5.2.2 together with the associated formal apparatus. Suppose an equilibrium in substandard care exists. Given that individual Y takes less than due care, individual X has costs given by (15). We search for critical levels of the subjective victim probabilities that make X indifferent between the exertion of substandard care and standard care, given the substandard care of individual Y. For this purpose, D(α, β) = 0 ought to hold in (16) at the critical role probabilities. The care values in a substandard equilibrium are, for simplicity, derived as the intersection of respective reaction curves resulting from the cost functions relevant for care levels x < x∗ and y < y ∗. This simplification may generate stricter conditions on role probabilities than are necessary for the intended exclusion, but ensures that D is a continuous function and thereby eases the analysis.23 The first-order condition for care by individual X CxX (x, y) + (1 − α)Lx (x, y) = 0

(21)

yields x = x(y, α). We apply the implicit function theorem to see that X (x, y) + (1 − α)Lxy (x, y) Cxy dx =− X dy Cxx (x, y) + (1 − α)Lxx (x, y)

(22)

23 The set of combinations of (ˆ x, yˆ) that we use to evaluate function D is greater than the set of care combinations that actually are equilibria in substandard care. This results because we do not reflect that individual Y might for some considered combination of role probabilities rather deviate discretely to y ∗ instead of the continuous deviation considered. We further do not impose a restriction which ensures ˆ(α, β) < x∗ always hold. This affects our results only in so far as conditions that yˆ(α, β) < y ∗ and x defined may be stricter than necessary because the function D assumes more substandard-care equilibria possible than there are. In other terms, it can occur that we employ yˆ(α, β) in function D and continue to search for critical victim probabilities although individual Y has already deviated unilaterally from the substandard care at less restrictive critical victim probabilities. We indeed continue to search and do not stop short as higher care by individual Y makes the substandard equilibrium relatively more attractive for X, as can be seen by differentiating D with respect to y. Given 0 ≤ α < α∗ and 0 ≤ β < β ∗ hold, we know that once one individual takes standard care, the other responds with standard care. Thus, there is no problem of circularity treated extensively in Endres and Querner (1995).

156

and dx Lx (x, y) = X < 0. dα Cxx (x, y) + (1 − α)Lxx (x, y)

(23)

Proceeding for Y as for X, we find y = y(x, β) for which it holds that Y Cxy (x, y) + (1 − β)Lxy (x, y) dy =− Y dx Cyy (x, y) + (1 − β)Lyy (x, y)

(24)

dy Ly (x, y) = Y < 0. dβ Cyy (x, y) + (1 − β)Lyy (x, y)

(25)

and

We need to know about the effect of changes in role probabilities on individual precautionary behavior. Since a change in β affects the choice of X via a change in y and vice versa, we need to describe the reaction of individuals to changes in the precaution of the other actor,

dx dy

and

dy . dx

For that, we need information regarding the mixed cross partials,

X Y Cxy , Cxy and Lxy . DH’s only statement in this respect is that these are sufficiently small

to ensure that the social optimum is indeed a minimum of the social cost function. That is why we add an assumption concerning the relation of x and y. Assumption 6: The care of X and Y are substitutes, i.e., for individual X it holds that X (x, y) + (1 − α)Lxy (x, y) > 0 for α ∈ [0, 1) is true and analogously so for Y.24 Cxy

The argumentation does not crucially depend on this common assumption that care of the individuals are substitutes. Principally, we assure monotonicity of D by our assumption and keep our analysis simple.25 As regards contents, Assumption 6 states that the diminution in care incentives due to the effect of care by the other party on the marginal benefit of the own care is not overturned by the effect on precaution costs.26 Assumption 6 also allows to conclude that there is no equilibrium with more than due care.27 24

Note that for α[β] = 1, individual X [Y] chooses no care in response to any y[x] < y ∗ [x∗ ]. The primary consequence of the reversed sign, i.e., care by both actors being complements, is that not all terms in the derivatives of the function D, equations (31) and (32) below, have the same sign. Yet, for instance, the sign of Dα should in principle be unaffected as D(0, β) > 0 by Assumption 4 (i), and D(1, β) < 0 holds as zero care has no consequence for negligent victims given negligent injurers. And the sign of Dβ will also be unaffected as long as x > 0. 26 It is usually assumed that more care by one party diminishes the effect of care by the other party, i.e., Lxy > 0. The above assumption states that the undetermined effect on precaution costs, Cxy , cannot dominate the first effect. 27 The proof is similar to that by Kim and Feldman (2006) to which we refer. 25

157

Both reaction curves fall in the (x, y)-plane and shift toward the origin for larger values of α and β, respectively. For (α, β) = (1, 1), the substandard-care equilibrium of (x, y) = (0, 0) results. We presume that the smallness of mixed partials assumed by DH suffices for

X Y X Y + (1 − α)Lxx ][Cyy + (1 − β)Lyy ] > [Cxy + (1 − α)Lxy ][Cxy + (1 − β)Lxy ] [Cxx

(26)

to hold. In that case, the slope of the reaction curve of individual X has a larger absolute value than that of Y, as (26) is a rearrangement of the respective slopes of the reaction curves. Thereby, this inequality ascertains that if there is a substandard-care equilibrium, it is the unique substandard-care equilibrium. Since we seek critical values for the subjective victim probabilities, we inquire how the equilibrium care levels, xˆ = xˆ(α, β) and yˆ = yˆ(α, β), change in response to varying victim probabilities. Application of Cramer’s rule to the system of first-order conditions with respect to care of both individuals yields ∂ xˆ ∂α ∂ xˆ ∂β ∂ yˆ ∂β ∂ yˆ ∂α

Y Lx (Cyy + (1 − β)Lyy ) 0 T X Ly (Cxx + (1 − α)Lxx ) = 0 T

=

(27) (28) (29) (30)

X Y X Y with T = [Cxx + (1 − α)Lxx ][Cyy + (1 − β)Lyy ] − [Cxy + (1 − α)Lxy ][Cxy + (1 − β)Lxy ] > 0

by (26). The function D varies continuously because costs and expected damages vary continuously in care, and care varies continuously in the victim probabilities. Due to the fact that D is continuous and D(0, β) > 0 as well as D(1, β) < 0 hold, we can use the intermediate value theorem. There must be at least one value α∗∗ for every β¯ which yields ¯ = 0. D(α∗∗ , β) ¯ we derive D with respect To see that there is exactly one critical level α∗∗ for every β, 158

to α

Dα (α, β) =

 ∂ xˆ  X C (ˆ x, yˆ) + (1 − α)Lx (ˆ x, yˆ) ∂α x  ∂ yˆ  X C (ˆ x, yˆ) − CyX (x∗ , yˆ) + (1 − α)Ly (ˆ x, yˆ) − L(ˆ x, yˆ) < 0. + ∂α y

(31)

The first term is either equal to zero for interior optima or negative by (27). The second term is negative by the smallness of mixed cross partials and (30). Basically the same applies to the derivative with respect to β

Dβ (α, β) =

 ∂ xˆ  X C (ˆ x, yˆ) + (1 − α)Lx (ˆ x, yˆ) ∂β x  ∂ yˆ  X C (ˆ x, yˆ) − CyX (x∗ , yˆ) + (1 − α)Ly (ˆ x, yˆ) > 0. + ∂β y

(32)

We find that for α < α∗∗ , actor X takes due care as a response to substandard care ¯ where α∗∗ = α∗∗ (β) ¯ and α∗∗ > 0. by Y as long as β ≥ β, β¯ ∂y [CyX (ˆ x, yˆ) − CyX (x∗ , yˆ) + (1 − α)Ly (ˆ x, yˆ)] dα∗∗ ∂β >0 = (−1) ∂y ¯ X X ∗ dβ [Cy (ˆ x, yˆ) − Cy (x , yˆ) + (1 − α)Ly (ˆ x, yˆ)] − L(ˆ x, yˆ) ∂α

(33)

Given such an α∗∗ for which D = 0 holds, we can rearrange the expression for D in (16) to implicitly define ¯ = 0 < α∗∗ (β)

C X (ˆ x, yˆ) − C X (x∗ , yˆ) + L(ˆ x, yˆ) βSLCN holds. The efficient care equilibrium is unique if α > α > αSLCN ∗ ∗∗ ∗ ∗ , αSLCN (β¯SLCN )} and β¯SLCN ≥ β > βSLCN or α ¯ SLCN ≥ α > αSLCN and max{αSLCN ∗ ∗∗ β > max{βSLCN , βSLCN (α ¯ SLCN )}.

Proof. For individual X [Y], we refer to (4) [(5)] in which the first entry contains α [β] instead of (1 − α) [(1 − β)] and the fourth entry contains the weight (1 − α) [(1 − β)] instead of α [β]. We first consider the efficient equilibrium. Given y = y ∗ , individual X ∗ takes standard care if α > αSLCN with

∗ =1− 0 < αSLCN

x, y ∗) + L(˜ x, y ∗) − C I (x∗ , y ∗ ) C I (˜ < 1. L(x∗ , y ∗)

(35)

∗ ∗ , so that individual Y takes standard care for β > βSLCN and Similarly, we find a βSLCN

x = x∗ . 160

Next, we consider an equilibrium in substandard care with equilibrium values x = x(α, β) and y = y(α, β). These values vary with α and β according to ∂x ∂α ∂x ∂β ∂y ∂β ∂y ∂α

Y Lx (Cyy + βLyy ) >0 T X Ly (Cxy + αLxy ) = 0 T Y + βLxy ) Lx (Cxy = 0 with reference to the

sufficiently small mixed partials assumed by DH. To ascertain the signs, we again impose the assumption 6, modified for this and the next liability rule. Assumption 6’: The care of X and Y are substitutes, i.e., for individual X it holds that X (x, y) + αLxy (x, y) > 0 for α ∈ (0, 1] is true and analogously so for Y.29 Cxy

The equilibrium values change in directions opposite to the case of simple negligence, as the weight attached to the expected damage L is α [β] instead of (1 − α) [(1 − β)]. The signs of partial derivatives of function D change accordingly, so that

Dα (α, β) > 0

(40)

Dβ (α, β) < 0.

(41)

As D(0, β) < 0 is true because negligent injurers do not have to compensate negligent victims and the choice of x = 0 is therefore without consequence and D(1, β) > 0 is ∗∗ for every β¯SLCN which makes true by Assumption 4 (i), there is a critical value αSLCN

individual X indifferent. ∗∗ (β¯SLCN ) = 0 < αSLCN

C X (x∗ , y) − C X (x, y) αSLCN (β¯SLCN ) and β ≤ β¯SLCN .30 29 30

Note that for α(β) = 0, individual X (Y) chooses due care in response to any y(x) < y ∗ (x∗ ). ∗ Mirroring the above for individual Y would yield critical values βSLCN to make standard care a best

161

Negligence with a defense of contributory negligence Proposition 5 Assume Assumptions 1-4 and 6’ hold. An equilibrium in efficient care exists in a model with interdependent costs of care and NCN as the liability rule, if α < α∗ ∗∗ and β < β ∗ holds. The efficient care equilibrium is unique if αSLCN (β¯SLCN ) < α < α∗ ∗∗ and β < min{β¯SLCN , β ∗ } or βSLCN (α ¯ SLCN ) < β < β ∗ and α < min{α ¯ SLCN , α∗ }.

Proof. NCN directs standards at the injurer as well as at the victim. The respective cost functions are similar to (4) and (5) except for the fact that entries in the first and fourth row contain α (β) as weight attached to expected damages. An α smaller than α∗ ∗ defined like αCP X in (9) makes individual X respond to due care by Y with standard care.

How does X respond to y < y ∗ ? The argumentation is the same as for the case of SLCN, as the relevant costs are identical. Putting both requirements on victim probabilities for NCN together shows that we need α to fall in the interval ∗∗ (β¯SLCN ) = 0 < αSLCN

C X (x∗ , y) − C X (x, y) C X (˜ x, y ∗) − C X (x∗ , y ∗) + L(˜ x, y ∗ ) 0. Thus, higher e stand for lower avoidance effectiveness, which shows in the absolute level of π(a, e) as well as at the margin πa (a, e). It is easily imagined that there is some scope for avoidance in many accident contexts, but not so in others. This effectiveness can therefore be imagined as factors that may be harnessed, e.g., a given uncertainty over causation or foreseeability. To simplify our analysis, we make the following assumptions Assumption 1: lima→0 πa (a, e) = −∞

Assumption 2: limx→0 px (x) = −∞

Assumption 3: lime→el π(a, e) = 0 ∀a 4

Subscripts denote derivatives. The assumption that avoidance is undertaken only in the accident contingency is most reasonable and without qualitative importance. Assuming that avoidance is decided on ex ante yields comparable results. Nussim and Tabbach (2007) assume that avoidance is taken in all cases. 5

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Assumption 4: lime→eh π(a, e) = 1 ∀a We assume that welfare maximization can be approximated by wealth maximization. Given constant activity, the social objective is the minimization of total social costs SC, being the sum of precaution costs, expected harm, and avoidance costs.

SC = x + p(x)(h + a)

(1)

In the social optimum, avoidance is nil since it is pure waste and optimal care x∗ accords with 1 + px (x∗ )h = 0.

(2)

The optimal care levels changes with harm according to x∗h =

3 3.1

−px (x∗ ) > 0. pxx (x∗ )h

(3)

The Analysis Strict Liability versus Negligence

Under strict liability, injurers are always required to compensate victim harm. Consequently, injurers minimize injurer costs C, which can be stated as

C = x + p(x)[a + π(a, e)h]

(4)

The following first-order conditions describe the private optimum

1 + px (ˆ x)[ˆ a + π(ˆ a, e)h] = 0

(5)

1 + πa (ˆ a, e)h = 0

(6)

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The comparative statics for the individually optimal injurer choice yield xˆh , a ˆh , xˆe > 0 and a ˆe < 0. These signs give support to the proposition of Sanchirico (2006), that it is sensible to change procedural rules and the like to reduce the effectiveness of avoidance. Care increases with a decrease in the effectiveness of avoidance, an increase in e, which is always desirable as long as x < x∗ , and avoidance activities, which are waste itself and deteriorate enforcement, decline with a decrease in effectiveness. The injurer cost function for a given harm level and avoidance effectiveness is thus

C A (e, h) = xˆ + p(ˆ x)[ˆ a + π(ˆ a, e)h]

(7)

Due to Assumption 1, we know that at least some effort is taken concerning the aversion of detection, aˆ > 0. This leads to our first observation. Proposition 1 Given Assumption 1, it holds that a ˆ > 0 and xˆ < x∗ under strict liability. Proof. Assumption 1 ensures aˆ > 0. Then, [ˆ a + π(ˆ a, e)h] < h follows. Since xˆD > 0 with D=a ˆ + π(ˆ a, e)h, xˆ < x∗ obtains. Negligence, as the other principal liability rule, may release the injurer from liability but requires the taking of some standard of care, usually set equal to socially optimal care. Consequently, injurer costs in this case are given as ⎧ ⎪ ⎨ x + p(x)[a + π(a, e)h] if x < x∗ C= ⎪ ⎩ x + p(x)a if x ≥ x∗

(8)

Obviously, if the injurer takes due care, there is no longer a rationale to engage in avoidance activities so that a = 0 is individually optimal given x ≥ x∗ . Likewise, it is never optimal to choose more than due care. However, in contrast to the standard framework, there are always instances in which injurers choose to be negligent. Proposition 2 Given Assumption 3 and 4, and due care set equal to first-best care, injurers do not abide by the care standard under negligence if avoidance is highly effective, e, h) holds. i.e., if e < e¯, where e¯ is the effectiveness for which x∗ = C A (¯ 170

Proof. The cost of adhering to due care are C O = x∗ , which do not change with the effectiveness of avoidance. If care falls below due care, total injurer costs are given by a, e)h. Assumption 3 and 4 (7). These increase with a decrease in effectiveness by p(ˆ x)πe (ˆ ensure that there is a level of effectiveness e¯. Thus, C A (e, h) < x∗ ∀e < e¯. The above details that the distinction between strict liability and negligence vanishes as far as the behavioral impact is concerned if the standard of care is perceived as excessive by injurers, given the opportunity and effectiveness of avoidance. A parallel can be found in the standard framework if the due care standard is set excessively high. However, for medium and low effectiveness of avoidance, we obtain a clear ranking of liability rules according to their effects on social costs. Corollary 1 Given Assumption 3 and 4, and due care set equal to first-best care, negligence implies lower social costs than strict liability for e ≥ e¯. Proof. By definition of e¯, the injurer chooses x∗ for e ≥ e¯ under negligence. In contrast, injurers choose xˆ < x∗ and a ˆ > 0 ∀e ∈ (el , eh ) with obvious implications for social costs.

3.2

Uncertain Due Care Standard

The standard framework may be attacked due to the assumptions on information available to individuals and/ or courts. Negligence ensures efficient care if due care equals efficient care in the standard model. For this to work, the injurer must have perfect information on the standard of care and the court must be in the position to accurately assess care taken. Weakening these presumptions questions the efficient outcome under negligence in the standard model. We present the consequences of weakening these presumptions for our model and find that uncertainty may improve upon the outcome achieved under perfect information.6 For this section, assume that injurers are uncertain as to which level of care is proclaimed as standard of care. Due care xs = x∗ +  is a random variable due to  being 6 Uncertainty on the due care standard and imperfect information of the court on the care taken are very similar in structure. Thus, our analysis would apply analogously if we were to assume that courts observe care only with error but injurers are certain on due care.

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random on support [−Δ, Δ] with f as density function of . We assume that the expected value of  is equal to zero and that Δ is large enough so that taking care x∗ + Δ is always dominated by some lower care level (so that uncertainty keeps its bite).7 The injurer will be judged negligent if care taken falls below due care. Consequently, being judged negligent has a probability of F (x∗ − x), where F is the distribution of , and injurer costs can be stated as C U = x + p(x)[a + F (x∗ − x)π(a, e)h]

(9)

Note that we assume that avoidance is chosen before the uncertainty on due care is resolved. If that does not hold, injurers invest in avoidance only in the contingencies in which care taken turned out to fall below due care. The following first-order conditions describe the private optimum (˜ x, a ˜) and give a clear indication of the changes to the standard case. x)[˜ a + π(˜ a, e)F (x∗ − x˜)h] − p(˜ x)π(˜ a, e)f (x∗ − x˜)h = 0 1 + px (˜

(10)

a, e)F (x∗ − x˜)h = 0 1 + πa (˜

(11)

Concerning the incentives for care, there are two aspects that enter the analysis due to uncertainty on the due care level. First of all, there is a discount effect since harm is not borne by the injurer if the care taken is judged to be sufficient, i.e., because only F (x∗ − x)h is relevant instead of h itself. Counteracting the first effect on marginal care incentives, there is a liability prevention effect because an additional unit of care makes it less likely that a negligence finding will occur. Without further assumptions, it is not clear whether the resulting individually optimal care will be less than or greater than xˆ. Still, the literature takes the conclusion that this uncertainty increases care, i.e., that the second effect is stronger than the first, as fairly general (Shavell 2004: 227). It however is unambiguous that the incentives for avoidance are diminished since F (x∗ − x˜) < 1 holds. This follows since there is no liability prevention effect with respect to avoidance, whereas the discount due to F < 1 works in full force. 7

This in turn will usually imply that xˆ ∈ [x∗ − Δ, x∗ + Δ].

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Proposition 3 Suppose that e < e¯ and expected due care equal to first-best care. Then, social costs are lower in the case that injurers are uncertain regarding the due care level than in the case that they are certain if a + π(˜ a, e)F (0)h] + p(x∗ )π(˜ a, e)f (0)h ≤ −px (x∗ )h = 1 −px (x∗ )[˜

(12)

x)[˜ a + π(˜ a, e)F (x∗ − xˆ)h] + p(ˆ x)π(˜ a, e)f (x∗ − xˆ)h ≥ −px (ˆ x)[ˆ a + π(ˆ a, e)h] = 1 (13) −px (ˆ

Proof. Given that e < e¯, injurers under negligence with certainty on due care do not comply with due care but choose (ˆ x, a ˆ), with xˆ < x∗ and a ˆ > 0. As established, avoidance will fall due to uncertainty, i.e., a˜ < a ˆ which lowers social costs from a stand-alone perspective. However, care incentives also change. Craswell and Calfee (1986) show that injurers may take more or less than due care depending on assumptions on the error term and its distribution in the model without avoidance. Similar considerations determine whether x˜ is less or greater than xˆ. Consequently, if care incentives are not reduced and are not boosted too much by the effect that liability can be prevented by a little more care, social costs will be lower in the setting with uncertainty. A sufficient formal condition for this improvement can be stated as a condition on the marginal benefit from additional care. It has to hold that (12) and (13) hold simultaneously since in that case xˆ ≤ x˜ ≤ x∗ . To see why this holds, note the following. The inequality in (12) ensures that care x˜ does not surpass socially optimal care x∗ because the marginal benefit of care at x˜ = x∗ in the uncertainty setting is weakly less than one. Similarly, the inequality in (13) ensures that care x˜ does not fall below xˆ because the marginal benefit of care at x˜ = xˆ in the uncertainty setting is weakly greater than one. Taken together, a˜ < a ˆ and x˜ ∈ [ˆ x, x∗ ] implies that social costs are lower in the setting in which individuals are uncertain on due care (this follows from the definition of expected social costs in (1)). There is an earlier contribution to the literature, which pointed to the possibility of uncertainty having beneficial effects. Bartsch (1997) considers the case in which care is

173

two-dimensional but only one dimension is observable. In that setting, uncertainty on due care may be socially advantageous since it induces incentives for both accident prevention measures as the harm is borne with positive probability irrespective of actual care taken. If there is certainty with respect to due care, injurers might be content with satisfying the care standard with respect to the observable care measure and not at all invest in the second care type. In our setting, the potential advantage of uncertainty from a social stance arises from the fact that the expected liability is discounted by the probability that the care taken will be judged insufficient relative to due care, which reduces incentives for avoidance.

3.3

Second-Best Considerations on Negligence

In circumstances of high avoidance effectiveness, injurers deviate from the socially desired outcome by exerting suboptimal care and positive avoidance effort. In acknowledgment of this fact, policy makers may think about changing the setting in some way to obtain a second-best outcome. Interestingly, in the light of the results from the previous section, it may be optimal for the policy maker to introduce uncertainty on the care standard. We focus in the following on the possibility of adapting the care standard in some fashion. Our considerations are in the spirit of Gomez and Ganuza (forthcoming), who consider second-best care standards in the context of judgment proofness. x) as long as x∗ ≤ (> From Proposition 2, we know that the injurer chooses x∗ (ˆ )C A (e, h). Assume that the effectiveness of avoidance is some e < e¯ so that the injurer does not take due care. The resources spend on avoidance have no social value. Consequently, C A (e, h) contains positive components without any value to the social problem. It is then possible to transform the outcome into something more desirable from a societal perspective. Let x∗∗ (e, h) be care such that x∗∗ (e, h) = C A (e, h). Proposition 4 The optimal second-best negligence rule chooses x¯(e, h) = min{x∗ , x∗∗ (e, h)} as due care standard. Proof. Social costs under the standard negligence rule and the second-best rule are the 174

same for e ≥ e¯. However, for e < e¯, applying solely x∗ implies social costs of xˆ +p(ˆ x)[ˆ a +h], whereas the use of x¯ implies x∗∗ + p(x∗∗ )h. It holds that xˆ < x∗∗ by definition of x∗∗ for given e. An increase in care is definitely desirable since xˆ, x∗∗ < x∗ . In addition, social costs fall due to the fact that no resources are spent on avoidance. The modified negligence rule asserts that injurers have alternative means to attain their end of minimized individual costs. In cases of medium and low effectiveness, the first-best due care standard is an offer which is sufficiently attractive to individuals. However, for high effectiveness, combining care with avoidance is more attractive than taking first-best care. The second-best rule acknowledges this fact and elicits as much care from injurers as possible, while achieving the socially desired level of avoidance. After discussing adjustments to due care, we turn our attention to another secondbest consideration which concerns the interpretation of causation. Whereas it is usually assumed that injurers need to compensate victims fully if they breach their duty of care, another view argues that only the share of harm caused by the deviation from due care ought to be compensated. Both interpretations induce the same outcome in the standard model of tort law without imperfections. However, Kahan (1989) covers the reducedcompensation interpretation and shows its advantage over the usual conception if due care is chosen excessively, for instance.8 The reduced-compensation interpretation gives the following injurer cost function ⎧ ⎪ ⎨ x + p(x)[a + π(a, e){h − C= ⎪ ⎩ x + p(x)a

p(x∗ ) h}] p(x)

if x < x∗ if x ≥ x∗

(14)

Note that subject to x < x∗ , the first-order conditions are solved by (xK , aK ), where 1 + px (xK )[aK + π(aK , e)h] = 0

(15)

1 + πa (aK , e){h − p(x∗ )/p(xK )h} = 0.

(16)

8 We label one of the interpretations reduced-compensation only to have a shorthand and not to imply any judgment on when the victim is made full.

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Given these alternative interpretations, we first inquire whether efficient care can be induced more often under either of the two. Proposition 5 Assume due care is set equal to first-best care. Injurers abide by the standard under the reduced-compensation interpretation for fewer levels of effectiveness than under the full-compensation interpretation of negligence. Proof. Again, the cost level associated with adhering to due care is C O = x∗ , which does not change with the effectiveness of avoidance and is the same across both interpretations. x)[ˆ a + π(ˆ a, e¯)h] = C A (¯ e, h). Now, the injurer As defined earlier, it holds that x∗ = xˆ + p(ˆ minimizes her costs given the different objective function under reduced-compensation by choosing (aK , xK ) instead of (ˆ a, xˆ). However, even if the injurer were to choose the latter activity vector, it holds that xˆ + p(ˆ x)[ˆ a + π(ˆ a, e¯)h] > xˆ + p(ˆ x)[ˆ a + π(ˆ a, e¯)h] − a, e¯)h. Consequently, for the strategy of choosing due care to become profitable p(x∗ )π(ˆ under the reduced-compensation interpretation, effectiveness has to fall further. Next, we want to establish the relative magnitude of avoidance and care given that the standard of care is not chosen under both interpretations. ˆ and xK > xˆ. Proposition 6 Suppose that e < e¯. Then, it holds that aK < a Proof. Comparison of (6) and (16) shows that the reduced-compensation interpretation provokes less avoidance. Given this, evaluating (5) and (15) yields the result on care. The last two results conflict with respect to the evaluation of the interpretations of causation from a social cost perspective. Whereas the reduced-compensation interpretation more often invokes injurers trying to avoid enforcement, the activity choices are more favorable than those under the full-compensation interpretation as soon as the latter also induces avoidance. Before we conclude this subsection, we can combine the two different second-best considerations. Given that it is possible to adjust the negligence rule as detailed earlier, is it desirable for the negligence rule to be of the full-compensation or reduced-compensation interpretation initially? 176

Corollary 2 The second-best negligence rule attains weakly higher care if negligence follows the full-compensation interpretation. Proof. We consider the respective performance for all e ∈ (el , eh ). First, injurers adhere to x∗ for more levels of e under full-compensation negligence. Next, for levels of effectiveness e < e¯, care is always at least weakly higher under full-compensation since the second-best standard given in Proposition 4 amounts to the level of individual costs, and, as is argued in the proof to Proposition 5, the level of total injurer costs under the reduced-compensation interpretation is always at least weakly below the level of the full-compensation interpretation. Consequently, the ambiguous ranking of the different interpretations elaborated on above is no longer present as soon as we allow for a second-best adjustment to due care. Given that is possible, social costs are lower under the full-compensation interpretation.

3.4

Avoidance and Punitive Damages

Punitive damages are damages that exceed the harm to the victim. These are of importance especially in the United States, where punitive damages are awarded in roughly six percent of all cases in which plaintiffs prevail (Polinsky and Shavell 2000). Since incentives for care are optimal if damages equal harm in the standard framework, some additional factor needs to be accounted for to give punitive damages an economic rationale. An important justification resides in the possibility of injurers escaping from suit. For instance, Polinsky and Shavell (1998) argue that this possibility may be due to the fact that (i) it may be difficult for the victim to determine that the harm was the result of some party’s act, (ii) it might be difficult for the victim to prove who caused the harm, and (iii) the victim might not sue because of expected litigation costs. Note that the extent to which causal factors (i) and (ii) affect the probability of escaping suit can, at least to some extent, be influenced by the injurer. We may thus reasonably argue that injurer avoidance activities can aggravate these factors. Polinsky and Shavell (1998) continue by arguing that there is an optimal damage multiplier, being the reciprocal of the probability of liability. This is the usual recom177

mendation of the literature; increase the damage payment to such an extent that care incentives are such that first-best care is induced.9 We will evaluate this approach in the light of injurer’s option to choose avoidance activities.10 Assume strict liability. The objective function of the injurer with damages d, where d might be greater than h, reads C P D = x + p(x)[a + π(a, e)d]

(17)

and gives the following first-order conditions

1 + px (xP D )[aP D + π(aP D , e)d] = 0

(18)

1 + πa (aP D , e)d = 0

(19)

Lemma 1 The first-best care level results if damages equal d∗ (h, e) =

h−aP D π(aP D ,e)

> h.

Proof. Optimality condition (18) yields x∗ if aP D + π(aP D , e)d = h from which d∗ (h, e) follows. It holds that d∗ (h, e) > h since this inequality can be rearranged to h > aP D + π(aP D , e)h, which holds due to Assumption 1. We confirm the result of the literature in so far as the level of damages needs to be elevated in order to obtain first-best care. A first difference in relation to the literature already obtains from the observation that d∗ < h/π, i.e. the damage level inducing firstbest care is less than harm multiplied by the reciprocal of the probability of the injurer being made to compensate the victim, π. This holds because the injurer expects to expend on avoidance given the accident contingency, which allows d∗ to be less than h/π. In the presence of avoidance activities, social costs are the sum of precaution costs, costs of avoidance, and expected harm. This is a factor that may argue against the optimality of the use of d∗ (h, e) as damage measure, since injurers would expend p(x∗ )aP D (d∗ , e) 9 Craswell (1999) presents an evaluation and considers alternative opportunities to restore care incentives. 10 The traditional wisdom is succinctly stated by Rubin (2005: 227): ”Firms will sometimes make efforts to hide their wrongful behavior. If they succeed, then there is insufficient deterrence. Therefore, multiplied damages can be useful in preventing such efforts at concealment. The optimal damage multiplier should be the inverse of the probability of detection.”

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on avoidance if d∗ (h, e) were chosen. One may therefore expect that the optimal damage level may be lower than d∗ (h, e) to reflect the latter fact, on the one hand, but still surpass h to instill ’sufficient’ care, on the other hand. We find that optimality considerations may restrict the damage measure even further. Proposition 7 The optimal level of damages is generally different from d∗ (h, e) and may even fall short of harm. Proof. The policy maker solves the following constrained minimization problem to find the optimal level of damages. min x + p(x)[a + h]

(20)

x = argminx {x + p(x)[a + π(a, e)d]}

(21)

a = argmina {a + π(a, e)d}

(22)

x, a, d

subject to

Setting up the Langrangean with λi , i = 1, 2, as multipliers, the set of conditions describing the optimum is

1 + px [a + h] + λ1 pxx [a + πd] = 0

(23)

p + λ2 πaa d = 0

(24)

λ1 px π + λ2 πa = 0

(25)

From this, we can derive an expression which implicitly defines the optimal level of damages11 dopt =

pπa pxx [a + πdopt ] πaa px π[−1 − px [a + h]]

(26)

There is no reason to assume that the level of damages defined by (26) equals d∗ . Note for instance, that, whereas dopt is a function - inter alia - of the productivity of injurer 11 Note that the cost minimizing choices by the injurer are functions of dopt , which prevents solving (26) explicitly.

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care with respect to the accident probability, d∗ is totally unaffected by changes in the care effectiveness. To prove the possibility that dopt may fall short of the magnitude of harm and show the dependence of dopt on the effectiveness, we rely on an example. Assume p(x) = whereas π(a, e) =

10−(a/e).9 10

1 , 1+x

. Note that this function π(a, e) complies with πa < 0 and

πe , πaa , πae > 0. Harm amounts to 200. Given these specifications, we obtain optimal damages as low as 24.7513 for an effectiveness e = 2. Consequently, damages are far below harm if avoidance is very effective. A decrease in the effectiveness decreases the attractiveness of avoidance as a method to minimize the expected payment. Consequently, the level of damages can increase. For instance, assuming that e = 4, one obtains dopt = 46.9. As a result of the decrease in avoidance effectiveness, care incentives improve. We find that punitive damages might indeed turn out as counterproductive concerning the minimization of total social costs. In analogy to the finding of Malik (1990), an increase in the payment due upon detection increases the efforts to avoid detection. The optimal level of damages needs to balance the desire to increase care incentives and lower avoidance activities. The traditional wisdom of the damage multiplier as the reciprocal of the liability probability has been questioned already, as alluded to in the introduction. Hylton and Miceli (2005) enlarge the setting in which punitive damages are evaluated by allowing for incentives to bring suits in a costly civil litigation system. Increasing the damage level implies that more victims find it individually advantageous to file suit.12 This increase in the number of victims filing suit increases care incentives of the injurer but, at the same time, is costly due to positive litigation costs. As a consequence, the optimal multiplier in their setting equates the marginal social benefit from deterrence with the marginal costs of litigation. In our analysis, we likewise obtain incentives for higher care upon increasing damages but, at the same time, induce an increase in avoidance. Consequently, optimal damages in our setting trade-off these marginal benefits and marginal costs. Interpreting avoidance as legal expenses, we can say that an increase in the damage level achieves 12 Victims have an exogenous cost of litigation and a loss level drawn from a distribution. If the level of damages is increased, victims with slightly lower loss levels will also find it worthwhile to sue.

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that injurers invest more in care to prevent the accident contingency but will also invest more in avoidance to fight off the judgment given the accident contingency. In Hylton and Miceli (2005), an increase in the damage level makes injurers invest more in accident prevention but also implies that fixed litigation costs will be incurred more often. Boyd and Ingberman (1999) analyze punitive damages in a setting in which the assets of the potentially liable injurer are a decision variable. It often holds that compensation cannot be full due to the limited injurer assets available for making the victim whole. Consequently, two ways of reducing expected liabilities present themselves, being increasing precaution or reducing the asset level which will be asked for in order to compensate victims. It is established that punitive damages can have the counterintentional effect of reducing precaution due to the fact that injurers reduce their exposure. In our study, there is no consideration of potential insolvency.

4

Conclusion

Avoidance activities are an important avenue for the minimization of individual costs. This has been recognized within the literature on criminal law. However, it is of import in the accident setting as well. There are various means that injurers can utilize to decrease the probability that a legally binding request for compensation comes into effect. Admitting the possibility of avoidance introduces complications into the standard model of accidents. We established that the distinction between strict liability and negligence in effect vanishes for highly effective avoidance opportunities. In view of this fact, it may be best to use the negligence rule, however, only after adjusting due care to a secondbest level. Enriching the model by avoidance importantly casts doubt on the optimality of punitive damages. Since avoidance activities are socially costly and increase with the payment requested from the injurer, avoidance presents a nonnegligible counterargument to the traditional multiplier principle. We also established that uncertainty on due care can be welfare-improving in a setting which allows for avoidance. The rationale is that behavioral adaptations due to given uncertainty will tend to be in line with the social

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interest. Undoubtedly, the preceding analysis leaves topics relating to avoidance untouched. Work that lies ahead of us comprises, for instance, the sanctionability of avoidance. It can be expected that results along these lines will critically depend on assumptions on the verifiability and the recursiveness of avoidance. It is usually assumed that avoidance is unverifiable which precludes conditioning a sanction on avoidance (Malik 1990, Innes 2001). If it is assumed that avoidance is verifiable at reasonable cost, the possible recursiveness of avoidance needs to be considered when determining the optimal policy. Sanchirico (2006) argues that sanctioning first-order avoidance will increase second-order avoidance efforts, and so forth. This may question the desirability of sanctioning avoidance.

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Langlais, E. (2006). Detection Avoidance and Deterrence: Some Paradoxical Arithmetics. Munich Personal RePEc Archive: Paper 1148. Malik, A.S. (1990). Avoidance, Screening, and Optimum Enforcement. RAND Journal of Economics 21: 341-353. Nussim, J. and A.D. Tabbach (2007). Controlling Avoidance: Ex-Ante Regulation Versus Ex-Post Punishment. Paper presented at the 2007 Annual Conference of the European Association for Law and Economics in Copenhagen. Polinsky, A.M. and S. Shavell (1998). Punitive Damages: An Economic Analysis. Harvard Law Review 111: 869-962. Polinsky, A.M. and S. Shavell (2000). Punitive Damages. In: B. Bouckaert and G. De Geest (eds.), Encyclopedia of Law and Economics, Vol. II, Cheltenham: Edward Elgar: 764-781. Rubin, P.H. (2005). Public Choice and Tort Reform. Public Choice 124: 223-236. Sanchirico, C.W. (2006). Detection Avoidance. New York University Law Review 81: 1331-1399. Shavell, S. (2004). Foundations of Economic Analysis of Law. Cambridge: Harvard University Press. Young, R., Faure, M. and P. Fenn (2006). Defences in Negligence: Implications for Tortfeasor Care. International Review of Law and Economics 26: 67-87.

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Chapter 9 Conclusion The literature on the economic analysis of tort law is rich in relation to that on many other areas of law and the study of precautionary incentives is in most instances at its heart. The already large number of contributions in this realm is hinted at in Chapter 2. This literature is relatively extensive - inter alia - due to the societal importance of the activities affected by tort law. However, from an absolute level in relation to other areas of interest in economics, the investigations presented are still relatively few in number and leave many interesting topics uncovered. Our own analyses fill some of these gaps left by previous literature. The main thread of the studies presented was the attainment of optimal care-taking by designing individual incentives using the instruments available in tort law as tools. Let us briefly recapitulate in which respects we contribute to the literature surveyed in Chapter 2. Individuals involved in accidents are usually very heterogeneous. The underlying aspect causing this heterogeneity is easily observed by others in some circumstances, but not all. The separation of heterogeneous individuals is often considered desirable for efficient care incentives. To disentangle individuals from one another in the circumstances in which characteristics are not easily observed usually demands the use of resources. For instance, a continuum of harm levels is possible and courts usually spend a great amount of effort on approximating the level of harm actually suffered. This implies incurring administrative costs of non-negligible magnitude. Although separating types is often desirable without positive separation costs, this is no longer true in all circumstances if separation is costly. In pursuit of this question, we touch upon different aspects. In all analyses related to this line of inquiry, we consider a setting in which victims vary in the level of harm suffered consequent to an accident and in which injurers cannot foresee the victim type. Obviously, one way to save administrative costs associated with the accurate assessment of harm given the knowledge on possible magnitudes of harm and their relative frequency is to compensate only average harm. From a distributional perspective, this 184

proceeding implies that some victims are overcompensated while others obtain a compensation below the level of harm suffered. Another important aspect is the way in which care-taking is changed. The first study (Chapter 3) is interested in the incentive effects of damage averaging if both parties to the accident can take care. It is shown that depending upon the formation of the measure used to compensate all victim types, optimal care may be induced. The measure which allows for efficient care is particular in that it incorporates the derivative of the accident probability function with respect to care to reflect that the productivity of injurer care in a bilateral-care model is usually dependent on the level of victim care taken. Apart from damage averaging as a method to reduce administrative costs, the question arises whether there are circumstances in which it is not only inconsequential but in fact desirable to use damage averaging with regard to incentives of individuals. In the analysis presented in Chapter 4, we precisely do this. In the framework used, injurers choose precaution and victims choose the value of the object they put at risk. The object, for instance, can be the bicycle in a car-bicycle accident. It is shown that victims will choose a value that is socially excessive, should they be compensated fully in the event of an accident. In contrast, victim incentives are first best if they obtain only the average value of the object put at risk. Victims will internalize the possibility that the object is destroyed with a positive probability. Consequently, this analysis conveys the idea that the widely followed practice of making the victim whole may have even higher economic costs than the administrative costs associated with the assessment of harm. In another study (Chapter 5), we test whether courts need to invest in the assessment of the victim type at all. The difference between victims might be sufficiently great so that victim type can be deduced from their conduct. Take the case in which there are two different levels of harm. If the victims with high losses obtain the same level of compensation that victims with low losses receive, the former will always value an increase in the amount of care more than the latter. Suppose a schedule of care and compensation levels equal to the efficient care level for and the harm suffered by the respective types were erected. It is proven that there are instances in which low-harm (high-harm) victims do not

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increase their care (put up with a lower compensation) in order to obtain the compensation above the level of harm suffered (to save on precaution costs). Stated alternatively, compensating victims fully if and only if they take first-best care for their harm level can be incentive compatible. In such instances, care-taking behavior provides all the information required by the court. In consequence, there certainly are circumstances in which there is no need to spend positive resources on the accurate assessment of harm in court while compensation makes respective victim types whole. In summary, the studies of chapters 3 to 5 all have a somewhat common thread. The remaining chapters are more diverse in topic. Yet, there is a factor linking the analysis in Chapter 4 to the one in Chapter 8. In Chapter 4, a new type of behavior, the choice over the value put at risk, is introduced and possible consequences are laid out. Similarly, there is the consideration of a new type of activity in Chapter 8. In that analysis, we allow injurers to choose avoidance activities besides precautionary measures. Avoidance activities have costs but entail the benefit for the injurer of reducing the probability of successful compensation requests. In this way, avoidance proves to be an alternative means from the injurer’s perspective to reduce expected liability. The consequences of this additional behavioral dimension are manifold. Accordingly, our analysis explores these effects in different fields. It is, for instance, established that the possibility of avoidance activities can argue for a damage level below harm, i.e., can rationalize doing the inverse of imposing punitive damages, which is usually recommended in similar situations. The advantageousness of punitive damages has also been questioned in a setting in which there is potential insolvency of the injurer who has the asset level at his discretion. We contribute to the literature on judgment proofness in Chapter 6. The literature has derived that risk-neutral and potentially judgment-proof injurers take less than individuals with sufficient funds if care is nonmonetary and affects the accident probability. We establish that this does not necessarily hold if the injurers considered are risk averse. In fact, it may turn out that some potentially judgment-proof injurers take more care than injurers who have sufficient assets to cover the harm resulting in the event of an accident. For individuals with sufficient funds, an increase in assets decreases both the

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marginal costs and the marginal benefit of care, whereas the second effect is not at work for potentially judgment-proof injurers. This follows from the fact that the change in assets has no effect on the utility of the accident state for these latter individuals. The result obtained is also present if care is monetary and can appease people worried about care incentives as judgment proofness is a potential problem in many areas in which society relies on liability rules to induce care-taking. The final contribution hints at possibly decisive parallels between two distinct ways of setting up the framework. The literature subsequent to the early contributions partly consisted of studies which tested the robustness of the results by complicating the setting. For instance, Arlen (1990) shows that contrary to claims by others, standard liability rules applied to a context in which both parties to the accident can suffer harm yield the firstbest outcome. Kim and Feldman (2006) extend the standard framework by considering that there are circumstances in which the role in an eventual accident is uncertain, i.e., in which there is role-type uncertainty. In Chapter 7, we contribute to the literature by establishing that incentives in the framework with role-type uncertainty are similar to those in the bilateral-harm framework. The underlying rationale is the fact that roletype uncertainty in a unilateral-harm model de facto creates a bilateral-harm context in expectation terms. However, in the analysis, we highlight that the analogy is not generally present but depends on factors such as the consistency of role-type probability.

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