Question 7 Emily is trying to decide if she should buy a brand new electric bicycle so that she can commute to the office faster. The net value of owning the bike (after purchasing it) is $1000. But if she buys the bike, she will face some uncertainty... With probability p she gets in a wreck and damages her bike, and with probability 1 - p she has a safe ride. In the event of a wreck, she has to pay $100 in repairs to have her bike fixed. However, the bicycle company offers a warranty that costs $W and will completely cover the cost of her repairs. If she wrecks her bike, with probability q she also injures herself, and with probability 1 - q only the bike is damaged and Emily is okay. Going to the hospital for an injury costs her $899. However if she purchases health insurance for $//, an amount she can choose, then she receives a premium of $3/1 to cover some of the cost. Emily needs to decide what she should buy along with her bike. Her utility function is u(x) = In(x). (a) What is Emily's expected utility function if she does not purchase the warranty or the insurance, in other words, she only buys the bike? [2 points] (b) What is Emily's expected utility function if she purchases both the warranty and some level of health insurance? [2 points] (c) What is the optimal amount of insurance for Emily to purchase? Describe the relationship between the optimal I/ and the price of the warranty W. [5 points] (d) Suppose W = 100, which basically negates the warranty program since that is equal to the cost of her repairs. What amount of health insurance should she purchase? [1 point] (e) Suppose the probabilities are p = 0.3 and q - 0.9, and the warranty costs $25. Should Emily purchase health insurance, the warranty, both, or neither? [10 points] (f) BONUS (not graded): Suppose the bicycle company is trying to decide what price W to set for the warranty. They want to set the highest price such that Emily will still purchase the warranty, but they incorrectly believe that Emily will not purchase any health insurance. What equation determines W? Do not solve