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

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Tom has an initial wealth of 100. His car may be stolen so he runs a risk of a loss of 40 dollars. The probability of loss is 0.2. It is possible, however, for Tom to buy insurance. One unit of insurance costs γ dollars and pays 1 dollar if the loss occurs. Thus, if α units of insurance are brought, the wealth of Tom will be 100-γα if there is no loss (a good state) and 100-γα-40+α if the loss occurs (a bad state). Tom is an expected utility maximizer with Bernoulli utility function u(x)=ln(x) where x is his wealth in a state.

(a)(10%) Write down Tom's expected utility if he buys α units of insurance. Different- iate it with respect to α to derive the first order condition which will be useful later.

(b)(10%) Suppose in 2016 insurance is actuarially fair in the sense that the insurance company breaks even on average. What should γ be? How much insurance (α) will Tom buy? What is his wealth in the good state? What is his wealth in the bad state? Does Tom face any risk after being insured?

(c)(10%) In 2017 insurance company decides to change the premium so now γ becomes 0.25 How much insurance (α) will Tom buy now? What is his wealth in good state? What is his wealth in the bad state?

(d)(10%) Facing the premium change from 2016 to 2017, what is the Slutsky substitution effect of Tom's wealth in the good state?

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