Suppose that the agent is insuring his car and once a week he goes to La Banquise to have poutine. There is no drive-through. Moreover, if the agent actually stops the car and turns off the engine, his car must be pushed in order for him to restart it, which requires him some effort e. If he turns off the car, he has a probability of π(e) of his car being stolen while he is in the restaurant. If he does not spend the effort e, he has a probability of π(0) of his car being stolen, where π(0) > π(e).
(a) Suppose that he is able to get the amount of insurance as in part (b), above. Show that he will always choose to leave his car running.
(b) Let v(y) = √y. If the agent lets his car run while inside La Banquise, his utility is (1 − π(0))p(y − αq) + π(0)p(y − αq − D + α)