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solmidII10fa - Your name Econ 420K Midterm Exam II Fall...

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Your name: Econ 420K Midterm Exam II Fall 2009 Svetlana Boyarchenko Solutions 1. (25 points) Alice’s utility of wealth is u ( w ) = w . Alice’s only asset is a valuable violin which is worth $400. There is a 48 percent chance that the violin will be damaged during the next year. If the violin is damaged, it will only be worth $100. Alice can buy insurance for her violin. If she pays $ γK to an insurance company, the insurance company will pay Alice back $ K if the violin is damaged. (a) (5 points) Determine the value of the premium rate, γ , if the in- surance company makes zero profit on average. The insurance company gets γK and with probability 0.48 it pays Alice back K . Therefore the expected profit is γK - 0 . 48 K = 0 γ = 0 . 48 . (b) (5 points) What is Alice’s expected wealth if she buys the insur- ance? E [ w ] = 0 . 48 · (100 - . 48 K + K ) + 0 . 52 · (400 - 0 . 48 K ) = 0 . 48 · (100 + 0 . 52 K ) + 0 . 52 · (400 - 0 . 48 K ) . (c) (5 points) What is Alice’s expected utility of wealth if she buys the insurance? E [ u ( w )] = 0 . 48 100 + 0 . 52 K + 0 . 52 400 - 0 . 48 K. (d) (10 points) What is the value of coverage, K , that maximizes Alice’s expected utility? We need to find K which solves the following problem: max K 0 . 48 100 + 0 . 52 K + 0 . 52 400 - 0 . 48 K. The first order condition is 0 . 48 · 0 . 52 2 100 + 0 . 52 K - 0 . 48 · 0 . 52 2 400 - 0 . 48 K = 0 ,
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2 equivalently, 1 100 + 0 . 52 K - 1 400 - 0 . 48 K = 0 , equivalently, 400 - 0 . 48 K = 100 + 0 . 52 K 400 - 0 . 48 K = 100 + 0 . 52 K, whence, K =$300. Hence Alice completely covers her potential loss.
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