Liability, Insurance and the Incentive to Obtain Information About Risk. Vickie Bajtelsmit * Colorado State University

Transcription

1 \ins\liab\liabinfo.v3d Liability, Insurance and the Incentive to Obtain Information About Risk Vickie Bajtelsmit * Colorado State University Paul Thistle University of Nevada Las Vegas December 2008 Thistle s research was supported by the Nevada Insurance Education Foundation. We retain the responsibility for any remaining errors. Bajtelsmit: Department of Finance and Real Estate, Colorado State University, Ft. Collins CO Phone: Fax: vickie.bajtelsmit@colostate.edu Thistle: Department of Finance, University of Nevada Las Vegas, Las Vegas NV Phone: Fax: paul.thistle@unlv.edu

2 Liability, Insurance and the Incentive to Obtain Information About Risk ABSTRACT Tort law provides incentives for individuals to obtain information about the risks they impose on others. Liability insurance interacts with tort law and modifies those incentives. We examine the incentives to obtain information about risk under strict liability and negligence rules when insurance is available. We show that, when the social value of information is positive, the availability of insurance implies that the private value of information may be either positive or negative. Keywords: information, tort, moral hazard JEL Classification: K13, G22, D82

3 Liability, Insurance and the Incentive to Obtain Information About Risk 1. Introduction and Background An important purpose of tort law is to provide incentives for individuals to exercise care to reduce the risk of injuries to others. Another, perhaps equally important, role of tort law is to provide incentives for individuals to become informed about the risks that they create. Clearly, people cannot exercise the appropriate amount of care if they are unaware or misinformed about these risks. Shavell (1992) analyzes the incentives that liability rules create for potential injurers to obtain information about the severity of potential risks and to take care. Kaplow and Shavell (1992) examine the value of ex ante legal advice (i.e., information) about the determination of liability. Both Kaplow-Shavell and Shavell show that strict liability leads to socially optimal acquisition of information and levels of care. Kaplow- Shavell argue that the negligence rule creates excessive private incentives to acquire information. Shavell considers several negligence rules, where liability may depend on both the decision to become informed and the level of care or may depend solely on the level of care. Negligence rules that require or assume optimal acquisition of information lead to optimal acquisition of information and optimal care. Other negligence rules may result in too much or too little information and consequently, in too much or too little care. To focus on incentive effects, Kaplow-Shavell and Shavell assume that potential injurers are risk neutral. As a result, they do not consider the effect of insurance on the incentives for people to acquire information and the incentives to take care. Insurance is widely used by individuals and businesses to manage liability risks and substantial

4 3 amounts of money are spent each year on liability insurance. Liability insurance premiums were $256 billion for the U.S. in 2006 (A. M. Best, 2007). Automobile liability insurance, including both private passenger and commercial automobile liability insurance, accounts for about half of the premiums. Workers compensation premiums were $51 billion and product liability premiums were $4.5 billion. The premiums on insurance against liability arising from negligence are also substantial. This includes $11 billion for medical malpractice insurance, $15 billion for the liability portion of commercial multiple peril insurance and $55 billion for other liability insurance. 1 Given the widespread use of insurance when both strict liability and negligence rules apply, it is important to take account of the effects of insurance on the incentives to obtain information and to take care. Prior research suggests that the availability of insurance affects the value of information about risks. Ligon and Thistle (1996a) examine the value of information under a strict liability rule when there is a monopoly insurer. They show that both precontract and post-contract information are privately valuable when adverse selection and moral hazard are present. Obtaining information allows potential injurers to obtain better contracts and to make better choices of the level of care. Crocker and Doherty (2000) analyze the value of information under a negligence rule when the insurance market is competitive. They argue that potential injurers ignorance about the severity of risks they face may provide a motive for the purchase of liability insurance under a negligence rule. In their analysis, potential injurers face a choice between becoming informed and taking appropriate precautions or remaining 1 Other liability includes coverages for liability resulting from negligence, carelessness or failure to act. This category includes, among others, professional liability (e.g., lawyers, accountants), directors and officers, errors and omissions and employment practices liability.

5 4 uninformed and purchasing liability insurance. Crocker and Doherty show that the private value of information may be positive or negative. Liability insurance may allow the potential injurers to hedge the classification risk, that is, the risk associated with the unknown cost of meeting the standard of due care. Bajtelsmit and Thistle (2007) examine the value of information about the frequency of accidents assuming that the courts apply a uniform ( reasonable person ) negligence standard. They show that the private value of information depends on insurance companies ability to observe potential injurers information (informed/uninformed) status. If insurers can observe policyholders information status, then the private value of information may be positive or negative. If insurers cannot observe information status, then they face an adverse selection problem. In the resulting screening equilibrium, the private value of information is positive. In this paper, we examine the effect that the availability of liability insurance has on individuals incentives to obtain information about risks and to take care to prevent injuring others. We consider the strict liability rule and several negligence rules. We consider negligence rules where negligence depends only on whether the standard of care is met. We also consider negligence rules that depend on both the injurer s information and whether the standard of care is met. Following Shavell and Crocker-Doherty, we initially assume individuals do not know a priori whether any accidents they cause will result in small or large damages. We also analyze the case where potential injurers know the severity of damages but do not know the cost of mitigating the risk. Individuals have the opportunity to obtain information about the severity of the risk or the cost of mitigating the risk. We show that, in general, the private value of information may be

6 5 either positive or negative when the social value of information is positive. In the cases where the private value of information may be positive or negative, we give sufficient conditions for it to be positive. The interaction of liability insurance with the tort law has important implication for potential injurers decisions to obtain information about risk. Since potential injurers can choose to remain rationally ignorant or can choose to learn about the risks they face, our work is related to previous research on endogenous information in insurance markets. (Hoy, 1982, Crocker and Snow, 1992, Doherty and Thistle, 1996, Ligon and Thistle, 1996b). 2 In these analyses, information is socially valuable if consumers have some prior hidden knowledge and the benefits of improved risk classification outweigh the costs of increased signaling. Information is privately valuable if insurers cannot determine consumers information status. This earlier work considered pure adverse selection economies, whereas we allow potential injurers to take actions to reduce the probability of loss. The next section describes the basic assumptions of the model, defines the liability rules and the value of information. The third and fourth sections analyze the incentives to obtain information and to take care insure under the strict liability and negligence rules when the severity of the risk is initially unknown. The fifth section considers the case where the severity of the risk is known but the cost of care is not. The final section provides brief concluding remarks. 2 Crocker and Snow (2000) provide a survey of the literature on risk classification, including risk classification with endogenous information. Our work is also related, although less directly, to the literature on information disclosure rules (e.g., Viscusi, 1978, Matthews and Postlewaite1985).

7 6 2. The Basic Model. 2.1 Assumptions. The analytical model employed is an extension of the standard model of accidents in the law and economics literature (e.g., Shavell 1982, 1987). For simplicity, we assume that only the potential injurer s care affects the probability of an accident and there is no contractual relationship that the potential victim can use to provide incentives for the potential injurer to take care. We let x denote the level of activity intended to reduce the risk of accidents or care. We assume that the cost of care is equal to one, although we relax this assumption later in the paper. The probability of an accident is π(x) when care is x and is the same for all potential injurers. The probability of an accident is a strictly decreasing, strictly convex function of care. Accidents are never entirely certain not entirely preventable, 0 < π(x) < 1. If an accident occurs, the damages inflicted may be large, d D, or small, d S, where d D > d S 0. We refer to potential injurers who can inflict high damages as dangerous and those can who inflict low damages as safe. We let θ D and θ S be the proportions of dangerous and safe potential injurers, where θ D, θ S > 0 and θ D + θ S =1. Average damages are d = θ S d S + θ D d D. An important aspect of the analysis here is that potential injurers do not necessarily know whether they are safe or dangerous. 3 However, they do know that they can be safe or dangerous with probabilities θ S and θ D. We use the subscript U to denote uninformed potential injurers. Potential injurers have an all-or-nothing choice between remaining uninformed or becoming informed. We assume throughout that information is 3 We can extend the models analyzed here to allow a portion of the potential injurers to be exogenously informed. This does not change any of the results in the paper.

8 7 costless. Potential injurers become informed if the private value of information (defined below) is non-negative. Potential injurers have initial wealth w > d D and the von Neuman-Morgenstern utility function u, which is strictly increasing and strictly concave. We assume potential victims are risk neutral, since this leads to a simple characterization of the optimal negligence rule. Individuals know whether they are a potential injurer or a potential victim. If an accident occurs, then whether the injurer is safe or dangerous is revealed ex post by the amount of damage. We assume that the courts work perfectly. This implies that victims, courts, and insurers can perfectly verify the injurer s level of care if an accident occurs. Since the injurer s care can be verified, we assume care is observable. This implies that the insurance premium can depend on the level of care. 4 An insurance policy consists of a premium, p, an indemnity, q S, paid in the event damages to a victim are low and an indemnity, q D, paid in the event damages to a victim are high. We assume throughout that the insurance market is competitive and that insurers earn zero expected profit, given the information they have about policyholders. The zero expected profit constraint is then p = π(x)(θ S q S + θ D q D ). Since the premium is actuarially fair, if potential injurers insure, they will choose full coverage. For informed individuals with risk severity d i, full coverage means they choose q i = d i and q j = 0. For uninformed individuals, full coverage means they choose q S = d S and q D = d D. Given their perceived potential damages, individuals who are insured choose the level of care 4 This is an assumption about timing, namely, that care is observed ex ante so that the premium can depend on care. However, under the alternative assumption that care is only observed ex post in the event of an accident, the indemnity can depend on the level of care.

9 8 that minimizes their expected total cost of accidents and care (henceforth, accident costs ). 2.2 Value of Information. The social value of information is the reduction in the social accident costs that results from individuals being informed. 5 We let a i (x) = x + π(x)d i denote the accident cost when potential damages are d i. The socially optimal level of care then minimizes accident costs, x i * = argmin a i (x). For uninformed individuals accident costs are a (x) = x + π(x) d and the socially optimal level of care, x *, minimizes a (x). Then the social value of information is S = a ( x *) θ S a S (x S *) θ D a D (x D *) (2.1) = θ S [a S ( x *) a S (x S *)] + θ D [a D ( x *) a D (x D *)] > 0. Information is socially valuable since it leads to better choices of the level of care. We measure the private value of information as the expected increase in expected utility from becoming informed. Let z = (p, q S, q D, x) be the arguments of the potential injurer s expected utility function. Let U denote expected utility and let ẑ i maximize expected utility for an individual of type i = D, S, U, subject to any relevant constraints. Then the value of information is the change in expected utility from becoming informed: I = θ S U S ( ẑ S ) + θ D U D ( ẑ D ) U U ( ẑ U ). (2.2) We assume that individuals choose to become informed if the private value of information is non-negative. 5 Since potential injurers are risk averse and potential victims are risk neutral, optimal risk sharing implies that all risk should be born by potential victims. Then social welfare is maximized by minimizing social accident costs. The increase in social welfare that results from individuals becoming informed is equal to the reduction in social accident costs. Note also that the social value of information is positive if, and only if, accident cost is concave in damages.

10 9 A possible objection to this definition is that individuals who wish to become informed may need to expend resources to become informed. An alternative approach to measuring the value of information is to use the decision-maker s willingness to pay, that is, the reduction in wealth that makes informed expected utility equal to uninformed expected utility. However, the two approaches are equivalent, that is, the value of information is positive if, and only if, willingness to pay is positive. 3. Strict Liability. Under a strict liability rule, injurers are responsible for the damages they cause, whether or not they are informed about the risks. Kaplow-Shavell and Shavell show that, if potential injurers are risk neutral, then the private and social value of information are the same, injurers choose to become informed and choose the optimal level of care given their potential damages. However, the situation is not quite the same if potential injurers are risk averse. First, suppose that insurance is not available. Then expected utility for an informed individual is U i (x) = (1 π(x))u(w x) + π(x)u(w x d i ), i = D, S. (3.1) Let xˆ i be the level of care that maximizes expected utility. A uninformed individual may be safe with probability θ S or dangerous with probability θ D. Then expected utility is U U (x) = θ S U S (x) + θ D U D (x); (3.2) let xˆ U maximize expected utility for the uniformed. The private value of information is then I = θ S U S ( xˆ S ) + θ D U D ( xˆ D ) U U ( xˆ U ) (3.3)

11 10 = θ S [U S ( xˆ S ) U S ( xˆ U )] + θ D [U D ( xˆ D ) U D ( xˆ U )] > 0. That is, under a strict liability rule, risk averse but uninsured individuals have an incentive to become informed. However, the levels of care that individual s choose once they become informed, namely, xˆ i, are not, in general, equal to the socially optimal levels of care, x i *. This proves the following result: Proposition 1: Assume a strict liability rule and that insurance is not available. Then (a) the private value of information is positive and (b) individuals choose levels of care that are not socially optimal. The socially optimal level of care, x i *, is the level of care that would be chosen by a risk neutral potential injurer. The level of care, xˆ i, chosen by risk averse potential injurers may be either greater or less than x i * (Jullien, Salanie and Salanie, 1999). Now suppose that liability insurance is available. Then expected utility for an informed individual is U i (p, q S, q D, x) = (1 π(x))u(w p x) + π(x)u(w p x d i + q i ) (3.4) for i = D, S. The actuarially fair premium is p i = π(x)q i. Then the individual fully insures, chooses care to minimize their accident cost given the potential damages and obtains expected utility u(w a i (x i *)). Again, a uninformed individual may be safe with probability θ S or dangerous with probability θ D, so that expected utility is U U (p, q S, q D, x) = θ S U S (p, q S, q D, x) + θ D U D (p, q S, q D, x) (3.5)

12 11 The actuarially fair premium is p U = π(x)(θ S q S + θ D q D ). The uninformed individual fully insures, q S = d D and q D = d D. They choose the level of care, x *, to minimize the average accident cost, a (x), and obtain expected utility u(w x * π( x *) d ) = u(w a ( x *)). The private value of information is I L = {θ S u(w a S (x S *)) + θ D u(w a D (x D *))} u(w a ( x *)) (3.6) In general, the private value of information may be either positive or negative. The term in braces can be written as u(w θ S a S (x S *) θ D a D (x D *) ρ L ), where ρ L is a risk premium. Then the private value of information is positive if, and only if, a ( x *) > θ S a S (x S *) + θ D a D (x D *) + ρ L. (3.7) This is equivalent to S > ρ L. The decision to become informed exposes the uninformed to classification risk, that is, the risk that arises from the ex ante randomness of accident costs. The risk premium ρ L provides a measure of the degree of classification risk, which is a private cost of becoming informed. If individuals remain uninformed, then safe and dangerous types are pooled and the classification risk is avoided. This proves the following result: Proposition 2: Assume a strict liability rule and that insurance is available. Then (a) the private value of information may be either positive or negative, (b) the private value of information if positive if, and only if, S > ρ L, and (c) individuals choose levels of care that are socially optimal given their information. Comparing Propositions 1 and 2, it is clear that the possibility that the private value of information may be negative under a strict liability rule is due to availability of liability insurance.

13 12 4. Negligence. Under a negligence rule, injurers are responsible for the damages they cause if they fail to exercise the appropriate amount of care. Under the complete negligence rule, injurers are also negligent if they fail to become informed, regardless of their actual level of care. This rule would seem to provide strong incentives for individuals to become informed. We assume that the negligence standard is set equal to the socially optimal level of care for each type. If potential injurers become informed, then they meet the negligence standard and obtain expected utility u(w x i *), i = D, S. If individuals remain uninformed, then they are effectively under a strictly liability rule, since they will be held responsible for any damages they cause. We initially assume that insurance is not available. Then expected utility for the uniformed is again given by (3.2). Then the value of information is I = θ S u(w x S *) + θ D u(w x D *) U U ( xˆ U ) (4.1) = θ S [u(w x S *) U S ( xˆ U )] + θ D [u(w x S *) U D ( xˆ U )] > 0. Then we have the following result, similar to the strict liability rule: Proposition 3: Assume a complete negligence rule and that insurance is not available. Then (a) the private value of information is positive and (b) individuals choose levels of care that are socially optimal. Now we assume that liability insurance is available. Uninformed individuals are still effectively under a strict liability rule. Then, with insurance, their expected utility is: U U (p, q S, q D, x) = θ S U S (p, q S, q D, x) + θ D U D (p, q S, q D, x) (3.5)

14 13 The actuarially fair premium is p U = π(x)(θ S q S + θ D q D ). Uninformed individuals fully insure, q S = d S and q D = d D, choose the level of care x *, and obtain expected utility u(w a ( x *). With the complete negligence rule, the private value of information is: I C = {θ S u(w x S *) + θ D u(w x D *)} u(w a ( x *)). (4.2) Again, the private value of information may be either positive or negative. The term in braces can be written as u(w θ S x S * + θ D x D * ρ N ), where ρ N is a risk premium. Then the necessary and sufficient condition for the private value of information to be positive is a ( x *) > θ S x S * + θ D x D * + ρ N, (4.3) This leads to the following result: Proposition 4: Assume a complete negligence rule and that insurance is available. Then (a) the private value of information may be either positive or negative, (b) the private value of information is positive if, and only if, the inequality in (4.3) holds and (c) individuals choose levels of care that are socially optimal given their information. As under the strict liability rule, the possibility that the value of information may be negative is due to the availability of liability insurance. However, informed individuals are better off under the complete negligence rule than under the strict liability rule. Consequently, the value of information is higher under the complete negligence rule than under strict liability, I C > I L. This is consistent with the results of Kaplow-Shavell that negligence provides stronger incentivesto become informed than strict liability. However, since the private value of information can be negative under the complete negligence rule, it does not follow that the incentives are socially excessive. As Shavell points out, there are other possible negligence rules depending on how potential injurer s decisions to obtain information are incorporated. One possibility is

15 14 that the negligence standard is based on the level of care that is socially optimal conditional on socially optimal acquisition of information. We refer to this as the conditional negligence rule, since negligence standards are conditional on the social value of information. Another possible rule is based on the optimal level of care assuming that information is always obtained (whether or not it is optimal). We refer to this as the unconditional negligence rule. 6 Since information is socially valuable and we have assumed information is costless, these rules lead to the same negligence standards, x S * and x D *, and the value of information is the same under both of these negligence rules. 7 If potential injurers become informed, then they meet the negligence standard and obtain expected utility u(w x i *), i = D, S. Now consider uninformed potential injurers. Since x S * < x D *, if the potential injurer chooses to spend less than x S * on care, the injurer is always negligent. If the potential injurer chooses to spend between x S * and x D *, the injurer is only negligent if damages are d D. If the potential injurer spends at least x D *, the injurer is never negligent. The ex ante probability that an uninformed potential injurer will be negligent is then Π(x) = π(x) if x < x S * (4.4) θ D π (x) if x S * x < x D * 0 if x D * x Crocker-Doherty and Bajtelsmit-Thistle show that an uninformed potential injurer will fully insure (q D = d D ) and choose a level of care x S * xˆ U < x *. Then expected 6 This is the negligence rule considered in Kaplow-Shavell and Crocker-Doherty. 7 Suppose that the social value of information is negative, net of the cost of information. Then under the conditional negligence rule the negligence standard is x *. Then the private value of information is zero.

16 15 utility for uninformed potential injurers is u(w xˆ U θ D π( xˆ U )d D ). The private value of information is then I N = {θ S u(w x S *) + θ D u(w x D *)} u(w xˆ U θ D π( xˆ U )d D ) (4.5) The value of information is then positive if xˆ U + θ D π( xˆ U )d D > θ S x S * + θ D x D * + ρ N (4.6) The expression of the left-hand side of the inequality is the accident cost for the uninformed, while the expression on the right-hand side is the same as under the complete negligence rule. This yields the following result: Proposition 5: Assume either a conditional or unconditional negligence rule and that insurance is available. Then (a) the private value of information may be either positive or negative, (b) the private value of information is positive if, and only if, the inequality in (4.6) holds, (c) informed individuals choose levels of care that are socially optimal given their information and (d) uninformed individuals choose levels of care that are too low given their information. Another possible negligence rule sets the standard of care to the level that is socially optimal given the information that the individual actually possesses. We refer to this as the actual information negligence rule. Under this negligence rule, informed individuals choose the level of care x i * and obtain expected utility u(w x i *). Uniformed individuals choose the level of care x * and obtain expected utility u(w x *). Since both informed and uninformed potential injurers meet the appropriate standard of care, the availability or unavailability of liability insurance is irrelevant. Then the private value of information is: I A = {θ S u(w x S *) + θ D u(w x D *)} u(w x *). (4.7)

17 16 Again, the private value of information may be either positive or negative. A sufficient condition for the value of information to be positive is x * θ S x S * + θ D x D * + ρ N (4.8) The following proposition summarizes these results: Proposition 6: Assume an actual information negligence rule. Then (a) the private value of information may be either positive or negative, (b) the inequality in (4.8) is a sufficient condition for the private value of information to be positive and (c) individuals choose levels of care that are socially optimal given their information. This is essentially the result obtained by Shavell for the actually information rule where potential injurers are risk neutral. The condition (4.8) requires that the optimal level of care is sufficiently concave in damages. If the optimal level of care is a convex function of damages then the value of information is negative. Both the complete and absolute information negligence rules require that the courts determine whether injurers are informed. However, the uninformed are better off under the absolute information rule, so the private value of information is lower than under the complete negligence rule. 5. Unknown Cost of Care In many cases, potential injurers know that a risk exists and know the magnitude of the risk. However, it may be difficult for potential injurers to know the costs of mitigating the risk. As we show in this section, what potential injurers know and don t know has an important effect on decisions to obtain information about risk Assumptions. In this section we assume all potential injurers know that they will cause damages d if an accident occurs. However, they do not initially know the cost of care. Individuals can have either a high cost of care, c H, or a low cost of care, c L, where 0

18 17 < c L < c H. Expenditure on care is then c i x, i = H, L. Potential injurers have high cost of care with probability θ H and low cost of care with probability θ L. The accident cost is then a i (x) = c i x + π(x)d, i = H, L and x i * minimizes accident cost. For uninformed individuals the expected accident cost is a (x) = cx + π(x)d, where c = θ H c H + θ L c L, and the socially optimal level of care, x *, minimizes a (x). Then the social value of information is again give by (2.1). Since there is only one level of damages, an insurance policy consists of a premium, p, and an indemnity q. Letting z = (p, q, x), then the private value of information is given by (2.2). All of the other assumptions of the basic model are retained Strict Liability and Complete Negligence. We begin by assuming that insurance is not available. Under the strict liability rule, informed potential injurers expected utility is given by U i (x) = (1 π(x))u(w c i x) + π(x)u(w c i x d), i = H, L. (5.1) Let xˆ i be the level of care that maximizes expected utility and observe that, again, xˆ i x i *. Under the complete negligence rule, informed potential injurers meet the standard of care and obtain expected utility u(w c i x i *). Under either rule, uninformed individual are strictly liable for damages, so their expected utility is U U (x) = θ S U S (x) + θ D U D (x); let xˆ U maximize expected utility for the uniformed. Then (abusing the notation slightly) the value of information is given by (3.2) or (4.1); in both cases the value of information is positive. Combining these results: Proposition 7: Assume either a strict liability or a complete negligence rule and that insurance is not available. Then (a) the private value of information is positive, (b) under the strict liability rule individuals choose levels of care that are not socially optimal and (c) under the complete negligence rule individuals choose levels of care that are socially optimal

19 18 Now suppose that liability insurance is available. Then under a strict liability rule, expected utility for an informed individual is U i (p, q, x) = (1 π(x))u(w p c i x) + π(x)u(w p c i x d + q) (5.2) for i = H, L. The actuarially fair premium, p = π(x)q, is the same for both high and low cost potential injurers. Then informed individuals fully insure, choose care to minimize their accident cost given their cost of care, and obtain expected utility u(w a i (x i *)). A uniformed individual may be low cost with probability θ L or high cost with probability θ H, so that expected utility is U U (p, q, x) = θ L U L (p, q, x) + θ H U H (p, q, x) (5.3) The actuarially fair premium is p = π(x)q so uniformed individuals fully insure and obtain expected utility U U (p, d, xˆ U ) = θ L u(w a L ( xˆ U )) + θ H u(w a H ( xˆ U )) (5.4) where xˆ U is the expected utility maximizing level of safety. The level of safely chosen by the uniformed, xˆ U, is less that the level of safety x *, that minimizes the accident cost given the average cost of care. The value of information is then I L = θ L u(w a L (x L *))+ θ H u(w a H (x H *)) (5.5) [θ L u(w a L ( xˆ U )) + θ H u(w a H ( xˆ U ))] > 0. Suppose instead of strict liability, we have the complete negligence rule. Informed potential injurers meet the negligence standards and obtain utility u(w c i x i *).

20 19 If individuals remain uninformed, then they are effectively under a strictly liability rule, and obtain expected utility U U (p, d, xˆ U ). Then the private value of information is: I N = θ L u(w c L x L *)+ θ H u(w c H x H *) (5.6) [θ L u(w a L ( xˆ U )) + θ H u(w a H ( xˆ U ))] > 0. The results are summarized in Proposition 8: Proposition 8: Assume either a strict liability or a complete negligence rule and that insurance is available. Then (a) the private value of information is positive (b) informed potential injurers choose the socially optimal levels of care. As in the case where the severity of the damage is unknown, the value of information is higher under the complete negligence rule than under the strict liability rule. The result in Proposition 8 that the value of information is positive contrasts with the potentially negative value of information in Propositions 2 and 4 where the cost of care is known but severity of damages is unknown. C. Conditional, Unconditional and Actual Information Negligence. Under the conditional, unconditional and actual information negligence rules, the results for the value of information and choice to care are the essentially same whether the severity of the risk or cost of care is initially unknown. The arguments leading to the conclusions are also essentially the same. Consequently, we simply report the results and note the differences. For the conditional and unconditional negligence rules, the value of information can be written as I N = θ H [u(w c H x H *) u(w c H xˆ U θ L π( xˆ U )d)] (5.7) + θ L [u(w c L x H *) u(w c L xˆ U θ L π( xˆ U )d)]

21 20 The first term in (5.7) is always positive while the second term may be positive or negative. If c L x L * c L xˆ U + θ L π( xˆ U )d, (5.8) then the private value of information is positive. This leads to the following result: Proposition 9: Assume either a conditional or unconditional negligence rule and that insurance is available. Then (a) the private value of information may be either positive or negative, (b) the inequality in (5.8) is a sufficient condition for the value of information to be positive, (c) informed potential injurers choose the socially optimal level of care and (d) uninformed potential injurers do not choose the socially optimal level of care given their information. Under the actual information negligence rule, informed and uninformed potential injurers meet the relevant standard of care. Then the value of information is I A = {θ L u(w c L x L *) + θ H u(w c H x H *)} (5.9) {θ L u(w c L x *) + θ H u(w c H x *)} The second term in braces can be written as u(w c x * ρ A ), where the risk premium ρ A reflects the uncertainty about the cost of care. A sufficient condition for the value of information to be positive is c x * + ρ A θ L c L x L * + θ H c H x H * + ρ N (5.10) The following proposition summarizes these results: Proposition 10: Assume an actual information negligence rule. Then (a) the private value of information may be either positive or negative, (b) the inequality in (5.10) is a sufficient condition for the private value of information to be positive and (c) individuals choose levels of care that are socially optimal given their information. The value of information under the actual information negligence rule may be either higher or lower than under the complete negligence rule.

22 21 6. Conclusions Tort law provides incentive for individuals to exercise care to avoid injuring others. Tort law also provides incentives for individuals to learn about the risks that they may impose on others. Since insurance is widely used to manage liability risks, it is important to understand how insurance interacts with tort law and how it affects the incentives to obtain information about risks. We show that insurance affects the incentives to learn about risks. We assume that information is socially valuable and the insurance market is perfectly competitive. While information is socially valuable, the private value of information about risks may, in general, be either positive or negative. In addition, potential injurers may choose levels of care that are not socially optimal, even given their information. The advantage of obtaining information is that it allows better choices of how much to invest in care. But the decision to obtain information exposes the uninformed to classification risk, that is, the ex ante random cost of investing in care. Remaining uniformed and purchasing insurance provides a method of hedging the classification risk, as well as the underlying liability risk. The private value of information depends on the liability rule, the availability of insurance, the information that potential injurers may obtain and potential injurers risk aversion. We show that, under the strict liability rule, the private value of information about the severity of a risk is positive when insurance is not available. However, potential injurers do not choose optimal levels of care. When insurance is available the value of information about risk severity may be positive or negative and potential injurers choose

23 22 levels of care that are socially optimal given their information. Put differently, the availability of insurance may change individual s decisions about whether to seek legal advice. When the severity of damages in known, but the cost of care is not known ex ante, the value of information is positive whether or not insurance is available. The complete negligence rule requires potential injurers to become informed and to meet the standard of care to avoid liability. This rule would seem to provide strong incentives for potential injurers to become informed. But, as under strict liability, the value of information about risk severity may be either positive or negative when insurance is available. Informed potential injurers meet the (optimally set) negligence standards, while uninformed potential injurers choose socially suboptimal levels of care. When the cost of care is not known ex ante, the value of information is positive whether or not insurance is available. The private value of information is higher under the complete negligence rule than under strict liability. The conditional negligence rule and the unconditional negligence rule set the negligence standard to the socially optimal level of care for each type of potential injurer. Under these rules, the private value of information may be either positive or negative when insurance is available. If the value of information is positive, then potential injurers choose the socially optimal levels of care. If the value of information is negative, then potential injurers do not choose the socially optimal level of care, even conditional on their information. The actual information liability rule sets the negligence standard to the socially optimal level of care, given the injurers actual information. Since potential injurers meet the appropriate liability standard, the availability of insurance is irrelevant under this rule. However, the value of information is still ambiguous.

24 23 There are a number of other factors that could affect the value of information that are not considered in our analysis. We assume that insurance is available at actuarially fair prices. If insurance premiums contain a loading and are actuarially unfair, then potential injurers buy less than full coverage. This would increase the value of information under some negligence rules. We assume that the courts and the insurance company can determine whether or not individuals are informed. If insurance companies cannot determine whether potential injurers are informed, then informed high risk individuals may have an incentive to mimic the uninformed. This may lead to adverse selection problems in the insurance market, changing the insurance contracts that are available. It may be the case that the courts can determine the level of care, but that insurance companies cannot condition policies on this information. If so, then moral hazard problems arise in the insurance market. We have assumed that the courts work perfectly, in particular, that the negligence rules are applied without error. Courts do in fact make errors and legal errors weaken the incentive effect of tort law. All of these factors are likely to reduce the private value of information. Finally, we assume that the information that potential injurers obtain is perfect. If the information is noisy, then the value of information will be lower and the choice of the investment in care is likely to be affected. Even if some or all of these additional considerations are taken into account, the availability of liability insurance will affect the incentives created by the tort law to obtain information about risk and to invest in care.

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