Risk-averse case. Risk-averse parties bear less risk under act-based liability beca sanctions equal the expected harm rather than the realized harm 1.5 Actual use of rules. In fact. we often observe use of harm-based sanctions both on a strict basis and according to fault. Penalties may be imposed by the state for spills of toxic materials, for failure to pay proper taxes, and for many other harmful events. Perhaps more often however, we see that public law enforcement involves act-based sanctions. This is typically the case with violation of safety, environmental, and many financial regulations, where sanctioned behavior is that which creates a positive expected harm but need not do actual harm 2. Law Enforcement with a probability: The optimal probability and magnitude of Sanctions 2.1 Introduction. Here it will be assumed that it is costly to identify and penalize liabl parties, so that society has to choose a level of enforcement effort, which will determine the probability of applying sanctions, as well as the magnitude of sanctions In determining the social-welfare-maximizing choice of the probability and magnitude of sanctions, I will for simplicity assume that liability is strict and based on harm, for the major points to be made do not depend on the nature of the rule of liability(except as remarked in section 2.6 on fault-based liability) 2.2 Behavior given the probability and magnitude of sanctions. How will a person behave who will face a sanction only with a probability if he commits an act? If the person is risk neutral, he will evaluate the sanction in terms of its expected value. Hence, the person will commit an act if and only if his benefit exceeds the expected sanction Risk-averse case. If the individual is risk averse, he will commit an act if and only if his expected utility is raised by so doing, and in general he will not be equally deterred by different combinations of sanction and probability with the same expected value, he will be more deterred the higher the magnitude of the potential sanction in the combination, the expected sanction held constant. For example, a risk-averse person will be more deterred by a sanction of $1,000 borne with probability 20 percent than by a sanction of $500 borne with probability 40 percent even though their expected values, $200, are equal. The reason is that, for a risk-averse party, the disutility of sanctions rises more than in proportion to their size, when the sanction rises from $500 to $1,000, its disutility more than doubles. Comments.(a) Probability versus magnitude of sanction. It is sometimes asked whether an increase in the probability or an increase in the magnitude of sanctions would make a greater difference in deterrence. But this question is incomplete as stated, for it is not explicit about the degree of change of these two factors. Obviously, if the magnitude of the sanction rises by much more than the probability, an increase in the magnitude would exert a greater effect on More generally, if U is the utility of income function of a risk-averse person, y is income, g is the gain from the act, p is the probability of a sanction, and s is the magnitude of the sanction(this notation will be used in many later note as well), the person' s expected utility if he commits the act will be eU=(l! p)U +g)+pUC +g! s). If p falls to kp where k< 1, and s rises to s/k(so that the expected sanction is still ps), the person s expected utility becomes(1 l kp)U(+ 8)+ kpU+gI s/k). Differentiating the latter expression with respect to k yields P((s/k)UN+g! sk)! [U(+g)! U0+g! s/k)l:>0 because UMs decreasing. Hence, the lower is k, the lower is expected utility, and therefore the greaterChapter 20 – Page 5 Risk-averse case. Risk-averse parties bear less risk under act-based liability because sanctions equal the expected harm rather than the realized harm. 1.5 Actual use of rules. In fact, we often observe use of harm-based sanctions, both on a strict basis and according to fault. Penalties may be imposed by the state for spills of toxic materials, for failure to pay proper taxes, and for many other harmful events. Perhaps more often, however, we see that public law enforcement involves act-based sanctions. This is typically the case with violation of safety, environmental, and many financial regulations, where sanctioned behavior is that which creates a positive expected harm but need not do actual harm. 2. Law Enforcement with a Probability: The Optimal Probability and Magnitude of Sanctions 2.1 Introduction. Here it will be assumed that it is costly to identify and penalize liable parties, so that society has to choose a level of enforcement effort, which will determine the probability of applying sanctions, as well as the magnitude of sanctions. In determining the social-welfare-maximizing choice of the probability and magnitude of sanctions, I will for simplicity assume that liability is strict and based on harm, for the major points to be made do not depend on the nature of the rule of liability (except as remarked in section 2.6 on fault-based liability). 2.2 Behavior given the probability and magnitude of sanctions. How will a person behave who will face a sanction only with a probability if he commits an act? If the person is risk neutral, he will evaluate the sanction in terms of its expected value. Hence, the person will commit an act if and only if his benefit exceeds the expected sanction. Risk-averse case. If the individual is risk averse, he will commit an act if and only if his expected utility is raised by so doing, and in general he will not be equally deterred by different combinations of sanction and probability with the same expected value; he will be more deterred the higher the magnitude of the potential sanction in the combination, the expected sanction held constant. For example, a risk-averse person will be more deterred by a sanction of $1,000 borne with probability 20 percent than by a sanction of $500 borne with probability 40 percent even though their expected values, $200, are equal. The reason is that, for a risk-averse party, the disutility of sanctions rises more than in proportion to their size; when the sanction rises from $500 to $1,000, its disutility more than doubles.8 Comments. (a) Probability versus magnitude of sanction. It is sometimes asked whether an increase in the probability or an increase in the magnitude of sanctions would make a greater difference in deterrence. But this question is incomplete as stated, for it is not explicit about the degree of change of these two factors. Obviously, if the magnitude of the sanction rises by much more than the probability, an increase in the magnitude would exert a greater effect on 8 More generally, if U is the utility of income function of a risk-averse person, y is income, g is the gain from the act, p is the probability of a sanction, and s is the magnitude of the sanction (this notation will be used in many later notes as well), the person’s expected utility if he commits the act will be EU = (1 ! p)U(y + g) + pU(y + g ! s). If p falls to kp, where k < 1, and s rises to s/k (so that the expected sanction is still ps), the person=s expected utility becomes (1 ! kp)U(y + g) + kpU(y + g ! s/k). Differentiating the latter expression with respect to k yields p{(s/k)UN(y + g ! s/k) ! [U(y + g) ! U(y + g ! s/k)]} > 0 because UN is decreasing. Hence, the lower is k, the lower is expected utility, and therefore the greater is deterrence