(2) A doctor commutes 30 miles each morning to her job at Baptist Hospital. In order to avoid causing a car accident, she must exert a degree of precaution, which we denote x. Let's assume that x can be between 0 and 1. We can think of x as being the percent of effort the doctor gives to precaution: 0 = 0% (she's not paying attention at all), 1 = 100% (she is focused on the road and driving 30 MPH). Because she likes driving fast and because being stuck in traffic stops her from seeing patients, the cost of total precaution equals $600. The probability of an accident is (1-x)². So, for example, if she chooses precaution of x = 0.50, then the probability of an accident, p(x), is (1 - 0.5)² = (0.5)² = 0.25. If an accident occurs, the damage would be $1,000.

a. Find the socially optimal/efficient amount of precaution. [Hint: the derivative of (1 - x)² with respect to x is -2(1 - x).]

• (Another hint: Be careful with how you handle negative signs in your calculation. If you get an x that'snot between 0 and 1, then you made a mistake.)

b. How much precaution would the doctor choose under no liability? Strict liability? Negligence?

c. What is the probability of an accident under strict liability? Under negligence?

d. What formula do we use to calculatetotal social costs? What are the total social costs with the efficient level of care taken? What are the total social costs with full precaution (x = 1) taken?

Suppose that a tort reform bill is passed that caps all automobile accident damage awards at $600 but leaves the negligence due-care standard at its efficient level (which you solved for in part (a)).

e. What would be the doctor's total costs if she takes due care?

f. What would be the doctor's total costs if she minimizes her total costs without consideration of due care? (You must first find the level of care that minimizes the total cost function with A = 600.)

Note: For all parts of this problem that ask you what will happen under a negligence rule, assume that the negligence standard is set at its efficient level.

g. Based on your answers to e) and f), can you say whether the doctor will take due care with the underestimated harm of 600?