This appendix summarizes the results reported in the body of the Article. We consider an individual i engaged in an activity that imposes monetary harm h on a third party j with probability p(x), where x is the amount of precaution taken by i. The probability function has properties p’(x) < 0 < p’’(x). We assume that the marginal cost of precaution is 1, and that if the harm occurs then i is strictly liable for monetary sanction D, which is transferred to j. The socially optimal amount of precaution x* is chosen if the costs to i from the creation of harm are equal to the social costs of the harm. We assume that the cost of imposing a sanction D is increasing in the amount of the sanction.^[45]

We consider first the case of a utilitarian social cost function, in which social costs are equal to the sum of the costs borne by i and j from the activity.

• Selfish. In the baseline case, i is selfish, so she chooses x to minimize x + p(x)D. Social costs are therefore x + p(x)h. Private costs are equal to social costs only if D = h.

• Sympathy. Suppose that i cares about the utility of j, and assigns weight α to j’s well-being. In that case, i chooses x to minimize x + p(x)(D + αh − αD). Social costs are equal to x + p(x)(h + αh − αD). Again, private costs are equal to social costs only if D = h.

• Harm aversion. Suppose instead that i cares about the harm that she imposes on j, but either does not value the benefit that j gets from D, or the monetary sanction is a fine that is paid to the government (not to j). In that case, i chooses x to minimize x + p(x)(D + αh). Social costs are equal to x + p(x)(h + αh). Again, private costs are equal to social costs only if D = h.

In all three cases, the amount of damages required to induce efficient precaution is the same and equal to the harm caused. Next, we consider what happens if i can be induced to act as if she had preferences other than those that she has, perhaps out of habit or a moral commitment.

• If i has selfish preferences, then social costs are x + p(x)h.

1. If i can be induced to act as if she is sympathetic, then she chooses x to minimize x + p(x)(D + αh − αD). Private costs are equal to social costs only if D = h.

2. If i can be induced to act as if she is harm averse, then she chooses x to minimize x + p(x)(D + αh). Private costs are equal to social costs only if D = h(1 − α).

• If i is sympathetic, then social costs are x + p(x)(h + αh − αD).

1. If i can be induced to act as if she is harm averse, then she chooses x to minimize x + p(x)(D + αh). Private costs are equal to social costs only if D = h 1 + α .

2. If i can be induced to act as if she is selfish, then she chooses x to minimize x + p(x)D. Private costs are equal to social costs only if D = h.

• If i is harm averse, then social costs are x + p(x)(h + αh ).

1. If i can be induced to act as if she is sympathetic, then she chooses x to minimize x + p(x)(D + αh − αD). Private costs are equal to social costs only if D = h 1 − α .

2. If i can be induced to act as if she is selfish, then she chooses x to minimize x + p(x)D. Private costs are equal to social costs only if D = h(1 + α).

We consider next the case of an atomistic social cost function that ignores other-regarding preferences, which implies that the social cost function is equal to x + p(x)h.

• If i is sympathetic, she will choose x to minimize x + p(x)(D + αh −αD). Social costs are equal to private costs only if D = h.

• If i is harm averse (or the monetary sanction is a fine), she will choose x to minimize x + p(x)(D + αh). Social costs are equal to private costs only if D = h(1 − α).