Question

1. It’s been estimated that if a truck collides with a car, the damage to the car one is A_C = $4,000. There is no damage to the truck. The probability of a collision is described by p(s_T, s_C) = (s_T²+ s_C²)/100,000, where s_T is the truck speed and s_C is the car speed, in km/h. The maximum speed the vehicles are capable of is 100 km/h.

Each driver can take precautions; assume the only feasible precaution is to reduce the speed. The cost of taking precautions by the truck driver is 4×(100 – s_T), that is the truck driver loses $4 value for every 1 km/h slowing down. The cost of taking precautions by the truck driver is 3×(100 – s_C), that is the car driver loses $3 value for every 1 km/h slowing down.

1. What are the efficient speeds by each driver? How often do collisions happen (i.e., what is the probability of a collision p) at the efficient speeds?

2. Suppose the rule is no liability = nobody compensates anyone else for the damages, in case of an accident. What speed would the truck driver choose? What speed would the car driver choose? How often do collisions happen (i.e., what is the probability of a collision p) at these chosen speeds?

3. Suppose the rule is strict liability = the truck driver must fully compensate the car owner for the damages no matter how they drive, in case of an accident. What speed would the truck driver choose? What speed would the car driver choose? How often do collisions happen (i.e., what is the probability of a collision p) at these chosen speeds?

4. Suppose the rule is shared liability = the truck driver must compensate the car owner for half of the damages no matter how they drive, in case of an accident. What speed would the truck driver choose? What speed would the car driver choose? How often do collisions happen (i.e., what is the probability of a collision p) at these chosen speeds?

5. If the rule were any other allocation of liability and damages where the truck driver must pay some compensation L_T and the car driver assumes the rest of the damages, A_C – L_T?, would the efficient speeds be chosen by both drivers? Explain.

6. What do you conclude from these examples? I.e., can the efficient outcome be achieved by any ex post allocation of liability and damages? Explain.

7. Now consider how this could be solved by introducing a negligence rule. Let us pick a simple negligence rule = if the truck driver does not exceed the legal speed limit of 50 km/h, the truck driver is not liable for any damages; if the truck driver exceeds the legal speed limit of 50 km/h, the truck driver is fully liable for the damages.

1. Draw the truck driver’s expected full cost (cost of taking precautions + expected liability) as a function of s_T. What speed would the truck driver choose?

2. Draw the car driver’s expected full losses (cost of taking precautions + expected damages) as a function of s_C if the truck driver is not speeding. What speed would the car driver choose?

3. Draw the car driver’s expected full losses (cost of taking precautions + expected damages + expected compensation [remember the damages and compensation have opposite signs for the car driver]) as a function of s_C if the truck driver is speeding. What speed would the car driver choose?

4. What are the equilibrium s_T and s_C? Explain. Is this outcome efficient?

Answer 1

a) efficient speed is the speed for which the car and the truck don't collide.

thus for p to be 0, then St ² + Sc² = 0

so the speed should be equal and opposite in direction.

b) if there is no liability then the cost of the precaution is zero. so applying it to the two equations we find the speed to be 100 km/h. the probability will be 0.2

c) since the cost of damage is given to be 4000 dollars. thus the equating it to the cost of precautions taken will be 4(100-St)

thus St= 900 km/h and Sc = 233.3km/h

d) if the cost is half of the cost of damage, then the cost of precautions = 2000 = 4(100- St)

thus, St=400 km/h and Sc = 566.6