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ch09_solutions_solved edit - Solutions to Chapter 9...

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Solutions to Chapter 9 Exercises SOLVED EXERCISES S1. (a) Your neighbor has a sure income of $100,000. In addition, under the insurance contract, he will receive x when you have a good year and pay you $60,000 when you have a bad year. The lowest value of x such that your neighbor prefers to enter the contract will be the x for which his expected utility for entering the contract is equal to his utility for not entering the contract: 0.6 * √(100,000 + x ) + 0.4 * √40,000 = √100,000 √(100,000 + x ) = (√100,000 – 0.4 * √40,000) / 0.6 x = [(√100,000 – 0.4 * √40,000) / 0.6] 2 – 100,000 ≈ 55,009.8818 Rounding down to $55,009.88 would make your neighbor very slightly prefer not entering the contract, so the minimum x that your neighbor will agree to is $55,009.89. (b) Here we are looking for the level of x where you are indifferent between getting insurance (where you pay x in a good year and receive 60,000 in a bad year) and not getting insurance. That is, we’re looking for the x for which your expected utility with the insurance is equal to your expected utility without the insurance: 0.6 * √(160,000 – x ) + 0.4 * √100,000 = 0.6 * √160,000 + 0.4 * √40,000 0.6 * √(160,000 – x ) + 0.4 * √100,000 = 320 √(160,000 – x ) = (320 – 0.4 * √100,000) / 0.6 x = 160,000 – [(320 – 0.4 * √100,000) / 0.6] 2 ≈ 55,984.1891 Rounding up and paying $55,984.19 would make you very slightly prefer not having insurance. The highest x you would be willing to pay in a good year and still (barely) prefer to have insurance is $55,984.18. Solutions to Chapter 9 Exercises 1 of 18
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S2. (a) To achieve separation, t must be such that: t 2 / 160 > 10 and t 2 / 320 < 10 t 2 > 1,600 and t 2 < 3,200 t > 40 and t < 56.57 The minimum wait time that achives separation is the smallest moment longer than forty minutes. (b) The expected benefit has changed for the college students, so t must now satisfy: t 2 / 160 > 0.5*(10) + 0.5*(–5) = 2.5 and t 2 / 320 < 10 t 2 > 400 and t 2 < 3200 t > 20 and t < 56.57 The minimum wait time that achives separation is now a shade more than twenty minutes. The partial identification of college students reduces the minimum wait time required to achieve separation. When the charity has more information about the patrons—even if only partial information—this allows it to distinguish between the two types by means of a less stringent test. S3. (a) Buyers expect a random Citrus to be an orange with probability f = 0.6 and a lemon with probability 0.4. Risk-neutral buyers are then willing to pay up to: 0.6 * $18,000 + 0.4 * $8,000 = $14,000 (b) The willingness to accept (WTA) of owners of oranges is $12,500. Since the willingness to pay (WTP) of buyers is $14,000 for a random Citrus, sellers will be willing to sell and buyers will be willing to buy for any price in the range [$12,500, $14,500]. Thus, there will indeed be a market for oranges. (c) If f = 0.2, risk-neutral buyers will be willing to pay up to: Solutions to Chapter 9 Exercises 2 of 18
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0.2 * $18,000 + 0.8 * $8,000 = $10,000 (d) There will not be a market for oranges when f = 0.2. The probability of a random Citrus being an orange is so low that a buyer would only be willing to pay up to $10,000, but owners of oranges require at least $12,500. No oranges will be sold.
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ch09_solutions_solved edit - Solutions to Chapter 9...

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