On the Judgment Proof Problem

Transcription

1 The Geneva Papers on Risk and Insurance Theory, 27: , 2002 c 2003 The Geneva Association On the Judgment Proof Problem RICHARD MACMINN Illinois State University, College of Business, Normal, IL 61790, USA macminn@exchange.cob.ilsta.edu Received November 3, 1999; Revised September 22, 2001 Abstract A party who causes harm to others and is found legally liable but cannot fully pay is said to be judgment proof. When the party who causes the harm is judgment proof, the incentives provided by the negligence and strict liability rules diverge. The payment probabilities implied by the two rules also differ. If the cost of care is non-monetary, as in Shavell s analysis, then the different probabilities generated by the two rules and the injurer s risk aversion combine to show that greater care is optimal under the negligence rule than the strict liability rule. If, however, the cost of care is monetary then the difference in probabilities generated by the two rules suffices to show greater care under the strict liability rule than under the negligence rule. The latter case holds for either a risk averse injurer or a corporate injurer. Key words: strict liability, negligence, limited liability, judgment proof JEL Classification No.: D000, D630, G00, K00 Introduction As Shavell noted Parties who cause harm to others may sometimes turn out to be judgment proof, that is, unable to pay fully the amount for which they have been found legally liable. 1 Using strict liability and negligence rules, Shavell showed that the both rules would yield less than the efficient 2 level of care for sufficiently small asset values but that the negligence rule would yield greater care than the strict liability rule for some asset values sufficiently close to but less than the loss. The care choice is a function of the agent s asset value and the jump from taking less than the efficient care to taking that level of care is due to a perceived discontinuity in the expected utility function. Kahan showed that the negligence rule unambiguously yields a level of care less than the efficient level if the injurer is judgment proof. 3 Kahan did not, however, compare the care choices for the two rules. The analysis here resolves the disparity between the two results and provides a comparison across rules. The model is reconstructed in the next section for the case of a risk-averse injurer. The basic structure of the model is the same as the Shavell model but according to Kahan: As a matter of legal doctrine, injurers are liable for accident damages if two conditions are satisfied. First, of course, the injurer must have acted negligently that is, he must have exercised less than due care. Secondly, the injurer s negligence must have cause the accident that is, the accident would not have occurred had there been no negligence. 4

2 144 MACMINN Following Kahan in using both causal requirements for liability under the negligence rule, the analysis shows that the expected utility function is continuous. Then two cases are considered for the risk-averse injurer. The first case allows for a non-monetary cost of care and the analysis shows that the judgment proof injurer takes more care under the negligence rule than the strict liability rule for some income levels. The second case allows for only a monetary cost of care and the analysis shows that the judgment proof injurer always takes more care under the strict liability rule than under the negligence rule. The model is modified in the penultimate section to allow for a judgment proof injurer who is also a corporation; that analysis shows that limited liability yields greater care under the strict liability rule. The final section provides some concluding remarks. Risk-averse behaviour Shavell considers a case in which the injurer s assets are not sufficient to cover the loss in an accident event [Shavell, 1986]. He also considers a risk-averse injurer. The notation is as follows: x Monetary equivalent of care; x 0 y Asset value; 0 < y l l Loss; l 0 p(x) Accident probability; p(x) (0, 1), p < 0, p > 0 u Utility function; u > 0, u < 0 Strict liability Let H(x, y) denote the expected utility in the strict liability case and let p(x) denote the accident probability as a function of the care level. Then H(x, y) = (1 p(x))u(y x) + p(x)u(max{0, y l} x) = (1 p(x))u(y x) + p(x)u( x) (1) This form of the expected utility is due to Shavell who finesses the cost of care by assuming that care is non-monetary but that x is the monetary equivalent. The second equality in (1) follows if the agent is judgment proof. Due to limited liability the injurer s income given an accident is max{0, y l} =0 by assumption. Hence, the loss in the accident event is y l. Let x s (y) denote the optimal care level as a function of asset value given the strict liability rule. Let x e denote the efficient level of care. Then the following proposition follows easily. Proposition 1: In the strict liability case, the optimal care level is increasing in asset value i.e., x s (y) > 0 for 0 < y <l.

3 ON THE JUDGMENT PROOF PROBLEM 145 Proof. To see that x s (y) is an increasing function note that H x D 1 H(x, y) = (1 p(x))u (y x) pu ( x) p (x)u(y x) + p (x)u( x) 5 (2) 2 H x y D 12 H(x, y) = (1 p(x))u (y x) p (x)u (y x) > 0 (3) The inequality in (3) follows since p < 0, u > 0 and u < 0. Since the utility function is concave, 2 H/ 2 x D 11 H < 0 and it follows that 2 H x s (y) = x y 2 H > 0 (4) x 2 The strict liability result is stated separately here for emphasis. Next, we turn our attention to the negligence rule. Negligence Shavell specifies the expected utility in the negligence case as H(x, y)ifx < x e and u(y x) otherwise; this specification of expected utility, however, is discontinuous at the efficient care level as long as the accident probability is positive there. It is not surprising then that there is also a discontinuity in the injurer s optimal choice of care x n (y). The claim made here is that this result is a consequence of using the probability of being found liable given negligent behaviour. Under the negligence rule the injurer is held liable for the losses only if two conditions are satisfied [Kahan, 1989]. First, the injurer must have acted negligently, i.e., must have exercised less than due care, where due care is specified by the courts. Second, the injurer s negligence must have caused the accident, i.e., the accident would not have occurred in the absence of the negligence. Both conditions play a role in developing the probability of being found liable. Although Kahan did not develop the probability of being found liable, he did tell a story which distinguishes between the two conditions and which will be used here to motivate the probability. Consider accidents in which a cricket ball flies over a fence surrounding the playing field and causes damage. Suppose the fence must be ten feet tall for the owner of the cricket field to avoid negligence. The cricket field owner is only liable for accidents that are caused by her negligence. Hence, if the fence is built nine feet high then the owner is only liable for damages caused by balls that fly over the fence between nine and ten feet. Let t denote the event that a ball flies over the fence at a height of t feet and causes damage; let F(t) be the distribution function. Similarly, interpret x as the fence height selected by the field owner. Then the accident probability is p(x) where p(x) = x df(t) (5)

4 146 MACMINN This, however, is not the probability of being found liable in the negligence case. Let x e denote the height required by the due care standard. Then the probability of being found liable is λ(x) where xe λ(x) = df(t) df(t) = df(t) 6 (6) x x e x Note that this probability is the probability of an accident minus the probability that the negligence did not cause the accident. Let G(x, y) denote the expected utility in the negligence case. Assume, as in the previous section, that the expected utility is concave. Now it is possible to specify the expected utility G(x, y), in a fashion similar to Shavell, as { (1 λ(x))u(y x) + λ(x)u( x) x < xe G(x, y) = (7) u(y x) x x e This is not the expected utility function that Shavell specifies; the probability of being found liable given negligent behaviour is used here rather than the accident probability. Since the probability of being found liable is zero at the due care level, the expected utility is continuous and may be stated succinctly as G(x, y) = (1 λ(x))u(y x) + λ(x)u( x). Given the structure of G and H, it should come as no surprise that the optimal care choice x n (y) in the negligence case is non-decreasing in the interval (0,l) and equal to the efficient care level in the interval [l, ]. What may be less apparent is that the optimal care level is greater in the negligence case than the strict liability case for some asset values where limited liability is important, i.e., x n (y) > x s (y) for all y (0, y 0 ). 7 Proposition 2: In the negligence case, if the expected utility is strictly concave in care then the optimal care level is non-decreasing in asset value for 0 < y <l,i.e., x n 0 for 0 < y <l,and the optimal care is greater than that in the strict liability case, i.e., x n (y) > x s (y), for 0 < y < min{y 0,l}. Proof. Using (7) the marginal expected utility is G x D 1G(x, y) [(1 λ(x))u (y x) + λ(x)u ( x)] = λ u(y x) + λ u( x) for x < x e (8) u (y x) for x x e u (y x) + λ(x)[u (y x)] u ( x)] = λ [u(y x) u( x)] for x < x e u (y x) for x x e Note that (8) implicitly defines the optimal care as a function of asset value, i.e., x n (y). While G is continuous, G/ x D 1 G has a discontinuity at x e. Since λ(x e ) = 0 and the inequality u (y x) λ [u(y x) u( x)] > u (y x)

5 ON THE JUDGMENT PROOF PROBLEM 147 Figure 1. Marginal expected utility. 8 clearly holds, it follows that the marginal expected utility, i.e., D 1 G, is as shown in figure 1. Now, observe that the marginal expected utility is increasing in asset value since 2 G y x D 12G(x, y) { (1 λ(x))u (y x) λ u (y x) for x < x e = u (9) (y x) for x x e > 0 This is also depicted in figure 1. For sufficiently small asset values the optimal care is determined by the first order condition D 1 G(x, y) = 0. There is, however, an asset value y 0 such that [(1 λ(x e ))u (y 0 x e ) + λ(x e )u ( x e )] λ (x e )u(y o x e ) + λ (x e )u( x e )) = 0 (10) and for all asset values y y 0 the optimal care is the efficient care x e since expected marginal utility is positive for less care and negative for more care, e.g., for y 2 > y 0, D 1 G(x, y 2 ) > 0 for x < x e and D 1 G(x, y 2 ) < 0 for x > x e ; this is depicted in figure 1. For asset values y < y 0, it follows by the implicit Function Theorem that the fuction x n (y) exists, is continuous, and has the derivative property dx n dy = 2 G x y 2 G x 2 = D 12G D 11 G > 0 (11) The last inequality in (11) follows due to the inequality in (9) and the concavity of the expected utility. This shows that x n (y) is non-decreasing in the interval (0,l). The function x n (y) is depicted in figure 2.

6 148 MACMINN Figure 2. Optimal care comparison. Next, consider the comparison of x n and x s for asset values y < y 0. To demonstrate that x n (y) > x s (y) for y < y 0,itsuffices to show that G x H x = [(1 λ(x))u (y x) + λ(x)u ( x)] λ u(y x) + λ u( x) { [(1 p(x))u (y x) + p(x)u ( x)] p u(y x) + p u( x)} > 0 (12) Since λ = p, the inequality in (12) holds if and only if (1 λ(x))u (y x) + λ(x)u ( x) < (1 p(x))u (y x) + p(x)u ( x) (13) but (13) obviously holds since it is equivalent to u (y x) < u ( x). Therefore x n (y) > x s (y) for all y (0, y 0 ). Risk aversion obviously drives the result that the negligence rule generally yields more care than the strict liability rule; the risk-averse individual puts less value, at the margin, on high income, i.e., low care, and so the individual increases the investment in care farther under the negligence rule than under the strict liability rule, i.e., at least for y < y 0 and, by continuity, a neighbourhood above y 0. The results in the last proposition, however, are not only driven by risk aversion but also by the finesse in handling the cost of care. That finesse yielded an income of max{0, y l} x. An artless use of the limited liability assumption suggests an income of max{0, y l x} and such an assumption is employed in the next proposition. Proposition 3: If the income is max{0, y l x} then each rule yields care that is nondecreasing in asset value. In addition, the negligence rule motivates an optimal care level that is less than that in the strict liability case, i.e., x n (y) < x s (y), for 0 < y <l.

7 ON THE JUDGMENT PROOF PROBLEM 149 Proof. Suppose that the individual s income is specified as max{0, y l x} rather than max{0, y l} x. Then the expected utilities in the strict liability and negligence rule cases are H(x, y) = (1 p(x))u(y x) + p(x)u(0) and { (1 λ(x))u(y x) + λ(x)u(0) for x < xe G(x, y) = u(y x) for x x e respectively. Given a strict liability rule, the optimal care as a function of asset value is determined by the first order condition D 1 H = (1 p(x))u (y x) p u(y x) + p u(0) = 0 and the function x s (y) has the following derivative property dx s dy = D 12 H D 11 H = (1 p(x))u (y x) p u (y x) D 11 H > 0 (14) Given a negligence rule, the marginal expected utility is { (1 λ(x))u (y x) λ u(y x) + λ u(0) for x < x e D 1 G(x, y) = u (15) (y x) for x x e Since λ(x)u (y x) λ u(y x) + λ u(0) > 0, there is a discontinuity in D 1 G at x e. The analysis generates an asset value y 0 such that x n (y) = x e for all y y 0 in a manner analogous to that in Proposition 2. 9 For asset values y < y 0 the negligence rule yields an optimal care that is increasing in asset value and has the following derivative property dx n dy = D 12G D 11 G = (1 λ(x))u (y x) λ u (y x) D 11 G > 0 (16) Next, consider the comparison of x n and x s for asset values y < y 0. To demonstrate that x n (y) < x s (y) for y < y 0,itsuffices to show that D 1 G D 1 H = (1 λ(x))u (y x) λ u(y x) + λ u(0) { (1 p(x))u (y x) p u(y x) + p u(0)} = (p(x) λ(x))u (y x) (17) < 0 It follows that x n < x s.

8 150 MACMINN Figure 3. Optimal care comparison too. The comparison of care levels with the alternate specification of limited liability is provided in figure 3. Corporate behaviour The time dimension implicit in the model must be made explicit here. Suppose the care decision is made at date zero and all costs and benefits occur at date one; refer to dates zero and one as now and then, respectively. 10 Let δ denote a risk adjusted discount factor so a dollar then is equivalent to δ dollars now. A corporate manager making decisions on behalf of shareholders will make the care decision to maximize corporate value V. In the strict liability case that value is V s (x, y) = (1 p(x))δ(y x) (18) In the negligence case the value is { (1 λ(x))δ(y x) for x < xe V n (x, y) = (19) δ(y x) for x x e These values follow because the corporate net asset value is y x in the no accident event and max{0, y l x} =0 in the accident event. Proposition 4: In the corporate case, with either the strict liability rule or the negligence rule, the optimal care level is non-decreasing in asset value for 0 < y <l. The optimal care level in the negligence case is less than that in the strict liability case, i.e., x s (y) > x n (y), for all asset values 0 < y <l.

9 ON THE JUDGMENT PROOF PROBLEM 151 Proof. The marginal corporate value in the strict liability case is V s x = δ( (1 p(x)) (y x)p (x)) 11 (20) Note that x s (y) is an increasing function if dx n d y = 2 V s x y 2 V s > 0 (21) x 2 Since p < 0 and p > 0 makes the denominator negative, the inequality in (21) holds if and only if 2 V s x y = δ( p (x)) > 0 (22) (22) holds since p (x) < 0 and hence x s (y) is increasing in y. Similarly, the marginal corporate value in the negligence case is { V n (1 λ(x))δ λ x = (x)δ(y x) for x < x e (23) δ for x x e The same method used in proposition one may be used here to establish a critical asset value y 0 such that for y < y 0 the function x n (y) exists and is increasing while for y > y 0, the function x n (y) is a constant equal to the efficient care level. 12 Next, consider the relationship between x s and x n, in this corporate case. Note that x s (y) > x n (y) for all y < y 0 if V s x V n x = [ (1 p(x))δ p (x)δ(y x)] [ (1 λ(x))δ λ (x)δ(y x)] = (p(x) λ(x))δ (24) > 0 The inequality in (24) follows since λ = p and p >λ. Finally, since x s is increasing at y 0 and x n becomes a constant, it follows that x s (y) > x n (y) for all y (0,l). While risk aversion drove the difference in care choices in proposition two, the difference in the probabilities drives the difference in care choices made by the corporation; the smaller probability in the negligence case does not alter the marginal benefit of care but it does increase the marginal cost of care. 13 Risk aversion does play a role but it is indirect. The risk aversion of the agents determines the discount factor δ and so has an impact on the quantity of care choice but not on the qualitative comparison.

10 152 MACMINN Concluding remarks This note resolves the disparity between the results in the literature on the judgment proof problem. Being judgment proof does distort the incentives for care even in the presence of risk aversion. If the cost of care is non-monetary, as in Shavell s analysis, then the difference in probabilities and risk aversion combine to show greater care under the negligence rule. If the cost of care is monetary then the difference in probabilities suffices to show greater care under the strict liability rule; the latter case holds for the risk-averse or corporate injurer. The model provides some simple and intuitive results but those results are not clearly very robust. The size of the loss does not depend on care. Here the injurer is judgment proof but that may also be unknown at the time decisions are made. The probability of a loss and the size of the loss may also depend on conditions in product and financial markets. Acknowledgment I appreciate the comments provided by the referee. The points made were constructive and helped improve the paper. Notes 1. See Shavell [1986], p As in Shavell, the efficient level of care is that level at which the marginal cost of care equals the marginal benefit. In subsequent notation the efficient care decision is implicitly defined by the condition 1+ p (x)l = See Kahan [1989], pp See Kahan, p The D 1 H represents the partial derivative of H with respect to its first argument. 6. Note that the marginal probability of being found negligent is λ (x) = f (x) = p (x). Similarly, λ (x) = f (x) = p (x) > The asset value y 0 will be defined subsequently; it does depend on the injurer s measure of risk aversion. 8. The marginal expected utility is not generally linear as depicted. The figure is intended only to demonstrate the discontinuity. 9. The asset value y 0 is implicitly defined by the condition. (1 λ(x e ))u (y 0 x e ) λ (x e )u(y 0 x e ) + λ (x e )u(0) = We could have assumed that the expenditure on care occurred now rather than then but have not so that the comparison with the previous results is as close as possible. 11. The marginal corporate value may be rewritten as marginal benefit minus marginal cost as follows: δ( p (x))y δ( (1 p(x)) xp (x)). 12. In this case, the condition (1 λ(x e ))δ λ (x e )δ(y 0 x e ) = 0 implicitly defines y Simply note that δ((1 p(x)) xp (x)) <δ((1 λ(x)) xλ (x)) since λ = p and λ<p. References KAHAN, M. [1989]: Causation and Incentives to Take Care under the Negligence Rule, Journal of Legal Studies, 18, SHAVELL, S. [1986]: The Judgment Proof Problem, International Review of Law and Economics,6,45 58.