1. (a) Driver's probability of having a traffic accident: 0.05- 0.04 x 0.5 = 0.03 Magnitude of loss : 1,000 If driver is risk-neutral, maximum amount she is willing to pay : 0. 03 x 1, 000 = 30 ( b) if driver is risk- averse , maximum amount she is willing to pay will be different. let's assume the maximum amount in this case is y . Then, 1 1,200 - 4 = 0.03 x 1,200-1,000 + (1-0.03) XJ1, 200 :. y = 42. 23 (c) The amount policy covers : ( 1,000 - 200 ) x 0.7 = 560 If driver is risk-neutral , maximum amount she is willing to pay : 5,box 0.03 = 16.8 (d) For Insurance A, If driver is risk- averse , maximum amount she is willing to pay will be larger than 16.8 for Insurance B , If driver is risk-averse , maximum amount ( let's assume z ) she is willing to pay will be : 200 + ( 1, 000 - 200 ) x 0.2 = 360 (1- 0.03 ) x 1,200-2 + 0.03 x 1,200- 2 - 360 = 0.03 x 1,200-1,000 + (1-0.03) x 1,200 . . 7 13 greater than 16.8 ( 2 7 16.8 )1 Adverse Selection Suppose the quality of a driver is characterized by a random variable z. A higher value of x indicates a better driver and the probability of accident is lower. x is uniformly distributed between 0 and 1. Suppose the probability of having a traffic accident is 0.05 0.04z. For example, if a driver has = = 0.2, the probability of having a traffic accident is 0.05 0.04 x 0.2 = 0.042. If the traffic accident happens, the loss is 1000. All people have an intiail wealth of 1200. 1. Now we consider one single driver with x = 0.5. (15") (a) (b) What is the maximum amount she is willing to pay for an auto insurance policy that covers the cost of the traffic accident in full, if she is risk-neutral? (3') How will your answer to question (a) change if she is risk averse with utility function u(w) = w? (3") How will your answer to question (a) change if the policy requires a coinsurance of 30% with a deductible of 200 (i.e., the buyer should pay 200 deductible plus 30% of the total loss in excess of the deductible)? (3") If buyers are risk averse, consider two different insurance. Insurance A: coinsurance of 30% with a deductible of 200; Insurance B: coninsurance of 20% with a deductible of 400. Lay out the equations that solves for the maximum willingness to pay for the risk-averse buyers with utility function u(w) = /w. (3') Consider the risk-averse buyer with utility function u(w) = /w. Denote maximum willingness to pay for every unit of coverage for the insurance with full coverage as Weun, the maximum willingness to pay for every unit coverage for Insurance A as Wy, the maximum willingness to pay for every unit coverage for Insurance B as Wg. Rank Wiun, Wa, Wg and explain. (3')