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Question: 1 tom has an initial wealth of 100 his car...

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1. Tom has an initial wealth of 100. His car may be stolen so he runs a risk of a loss of 20 Hollars. The probability of loss ik 0.2. It is also possible that someone breaks in Toms home. In this case, he runs a risk of a loss of (80 dollars. The probability of loss is 0.1, These two probabilities (events) are independent. It is possible, however, for Tom to buy insurance. One unit of car insurance costs r dollars and pays 1 dollar if the loss occurs. On the the other hand, one unit of home insurance costs dollars and pays 1 dollar if the loss occurs. Thus if a units of car insurance are bought and a units of home insurance are bought, the wealth of Tom will be 100- Tra -ra if there is no loss. In the case that only his car is stolen, his wealth is 100-ra- 1-20to when only his home is broken in, his wealth is 100 ra-ra-80+ a. In the worstscenario that both his car is stolen and his home is broken in, his wealth is 100-rα_ra,--20-80 + α+ α,. Tom is an expected utility maximizer with Bernoulli utility function v()In() where x is his wealth in a state (a) (15 pts) Write down Toms lexpected utility if he buys a units of car insurance and α, units of home insurance. Differentiate it with respect to g and a to derive the first order conditions which will be useful later. (b) (10 pts) Suppose insurance is actuarially fair in the sense that both car and home insurance companies break even on average. What should r be? What should r be?

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