PS7_Sol - Intermediate Microeconomics 2008 Problem Set No 7...

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Intermediate Microeconomics, 2008 Problem Set No 7: Solutions Problems (5.20) Jerome moonlights; he holds down two jobs. The higher-paying job pays w , but he can work at most eight hours. The other job pays w * , but he can work as many hours as he wants. Show how Jerome determines how many hours to work. Jerome’s budget line is kinked at 8 hours of work. The amount he will work is determined by his utility function and where it is tangent to the kinked budget line. (5.30) Cynthia buys gasoline and other goods. The government considers imposing a lump- sum tax, L dollars per person, or a tax on gasoline of τ dollars per gallon. If L and τ are such that either tax will raise the same amount of tax revenue from Cynthia, which tax does she prefer and why? Show your answer using a graph or calculus. Without the tax Cynthia chooses point e1 on budget constraint B1. An ad valorem tax changes the budget line to B2 with an optimal bundle e2. A lump sum tax with equal revenue generation changes the budget line to B3 with optimal bundle e3. The lump sum tax leaves her on a higher indifference curve. (16.4) A risk-averse individual has to choose between $ 100 with certainty and a risky option with two equally likely outcomes, $ 100 - x and $ 100 + x . Use a graph(or math) to show that this 1
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person’s risk premium is smaller, the smaller x is (the less variable the gamble is). When x increases from x 0 to x 1 , the chord showing expected utility shifts downward. Initially the risk premium is $10. After the increase in x , the risk premium increases to $40. Using math, we know that risk-averse preferences mean that the indifference curves are concave, so we can look at what happens to the derivative of EU with respect to x when x decreases. ∂EU ∂x = - U 0 (100 - x ) + U 0 (100 + x ) By concavity U 0 (100 - x ) > U 0 (100 + x ) so ∂EU ∂x < 0, meaning that as x decreases, EU increases. And we know that the certainty equivalent is the value that makes utility equal to expected utility, so if EU increases, then we know CE increases as well. Finally, we know that the risk premium, P = EV - CE . Since EV is fixed, if x decreases, CE increases, and therefore the risk premium, P , will decrease. (16.8) Illustrate how a risk-neutral plaintiff in a lawsuit decides whether to settle a claim or go to trial. The defendants offer $ 50,000 to settle now. If the plaintiff does not settle, the plaintiff believes that the probability of winning at trial is 60%. If the plaintiff wins, the amount awarded will be X . How large can X be before the plaintiff refuses to settle? How does the plaintiff’s attitude toward risk affect this decision? In order not to settle, the plaintiff must believe that the expected value of going to trial is greater than the amount offered as a settlement, or . 6 X > 50 , 000, or X > 50000 . 6 , or X > $83 , 333 . 33. If the plaintiff is risk-averse, he would accept a smaller settlement offer in order to avoid the risk of going to trial.
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