The Social Cost of Accidents

11/10/09 Tort_B2 * The Social Cost of Accidents The model: p is the probability of an accident (bad event) occurring x is the amount of precaution taken p = p(x) where ?p / ?x < 0 (p decreases as x increases - the more precaution taken, the lower the probability of an accident) 11/10/09 Tort_B2 * A picture of the relationship between the probability of an accident and the amount of precaution - p(x) Probability of an accident p(x) x Precaution 0 1.0 p(x) Note that this is a probability and must be between 0 and 1 on the vertical axis 11/10/09 Tort_B2 * A is the monetary harm from an accident (medical costs, lost income, property damage, etc.) so that, p(x)A is the expected cost of harm – dollar value of expected harm note that the expected cost of harm, p(x)A, decreases as x increases even though A is fixed (since ?p(x)A / ?x < 0) because ?p(x) / ?x < 0 11/10/09 Tort_B2 * A picture of the relationship between the expected cost of harm and the amount of precaution - p(x)A Expected cost of harm p(x)A x Precaution 0 $ p(x)A Same basic shape as p(x) only shifted by the amount of A Note the $ on the vertical axis 11/10/09 Tort_B2 * w is the per unit cost of precaution (time, money, convenience, etc.) measured in $’s note we assume that w is fixed, independently of the amount of precaution taken x – we get a straight line as a result (not a necessary assumption) wx the total expenditure on precaution 11/10/09 Tort_B2 * A picture of the relationship between the amount of precaution taken and the total cost of precaution - wx Total cost of precaution wx x Precaution 0 $ wx Note the $ on the vertical axis Straight line because w is assumed fixed 11/10/09 Tort_B2 * In this model there are two types of costs to be considered: Cost of precaution [wx] and Expected cost of harm [p(x)A] The Social Costs of accidents is given by the sum of the two SC = wx + p(x)A is the expected social cost of accidents considering both the cost of an accident and the cost of trying to avoid the accident Why SOCIAL COSTS? Because the costs to all members of society are included –potential victims and potential injurers 11/10/09 Tort_B2 * A picture of the relationship between the social cost of accidents and the amount of precaution SC = wx + p(x)A x Precaution 0 $ p(x)A wx SC = wx +p(x)A 11/10/09 Tort_B2 * The social cost curve is the vertical sum of the wx and p(x)A curves This curve must be U-shaped because wx is positively sloped and p(x)A is negatively sloped and they cross Therefore some minimum level of social costs must exist, denoted x*. x* is the level of precaution that minimizes the expected social costs. x* is the efficient level of precaution 11/10/09 Tort_B2 * A picture of the relationship between the social cost of accidents and the amount of precaution SC = wx + p(x)A x Precaution 0 $ p(x)A wx x* SC = wx +p(x)A x* is the ‘efficient’ level of precaution 11/10/09 Tort_B2 * Some Math – an aside Knowing that SC = wx + p(x)A, how do we determine x*? Taking the derivative of SC = wx + p(x)A with respect to x and setting it equal to zero yields: w + p’A = 0 or w = -p’(x*)A where p’(x*) is the derivative of p(x) at the SC minimizing level of x, [x*]. OPTIONAL STUFF! 11/10/09 Tort_B2 * w is the marginal social costs of precaution - the cost of one more, or one less, unit of precaution (x) -p’(x*)A is the marginal social benefit associated with one more, or one less, unit of precaution evaluated at x* The marginal social benefit being the decrease in the expected costs of harm resulting from the last unit of precaution purchased Go back to the numerical example (Ms. Jones) In plain English 11/10/09 Tort_B2 * The social cost of the last unit of precaution (the marginal social cost) should just equal the increase in social benefit (marginal social benefit) that it produces. Why? If social cost of the last unit of precaution < social benefit it produces, then it has a net benefit to society. If social cost of the last unit of precaution > social benefit it produces, then it is a waste. 11/10/09 Tort_B2 * A change in the underlying riskiness (probability function) of an accident p(…) x Precaution 0 $ p(x)A wx x* SC = wx +p(x)A x* the ‘efficient’ level of precaution increases as the underlying riskiness increases x* p(x)A SC = wx +p(x)A 11/10/09 Tort_B2 * A change in the cost of an accident A x Precaution 0 $ p(x)A wx x* SC = wx +p(x)A x* the ‘efficient’ level of precaution increase as A increase x* p(x)A SC = wx +p(x)A 11/10/09 Tort_B2 * A change in the cost of an accident cost per unit of precaution w x Precaution 0 $ p(x)A wx x* SC = wx +p(x)A x* the ‘efficient’ level of precaution decreases as the per unit cost of precaution increases x* SC = wx +p(x)A wx 11/10/09 Tort_B2 * Potential Victim – Potential Injurer So far we have not indicated who is taking the precaution, the potential injurer or the potential victim. It could be either, or more likely, both (ie. driving safely/wearing a seat belt). We say potential because we are considering the possibility of a future accident – you can only take precaution in advance. 11/10/09 Tort_B2 * Internalizing externalities Imposing risks on others is like blowing smoke in their face – it is an external cost to those around you. We will use the above model to assess the incentive effects of various aspects of tort law. An incentive will be ‘efficient’ if the decision maker internalizes the marginal costs and marginal benefits of her actions. We will want the decision makers to act so as ensure that x* is the amount of precaution taken. 11/10/09 Tort_B2 * A Generalization of the Efficient Precaution Model We can generalize the above story and allow for precaution by both the potential injurer and the potential victim. Consider, xv the amount of precaution taken by the potential victim wv the cost per unit of precaution taken by the potential victim xi the amount of precaution taken by the potential injurer wi the cost per unit of precaution taken by the potential injurer 11/10/09 Tort_B2 * Now p(xv, xi) is the probability of an accident where p’ < 0 (p is a decreasing function of both xv and xi) p(xv, xi)A is the expected dollar value of harm faced by the potential victim. Note that we are now considering the case in which both the potential injurer and the potential victim can take precautions that reduce the probability of accidents – xv , xi So that the social costs of accidents is given by: SC = wvxv + wixi + p(xv, xi)A Where’s the picture? It is three dimensions