Economics 401 Problem Set 8 Answer Key|everything inclusive

Economics 401 Problem Set 8 Answer Key Prof. David A. Miller This version: October 26, 2020 1 Choice under uncertainty 1.1 Expected utility practice Consider the following three expected utility functions, and the following three scenarios involving uncertainty. Using a calculator or computer, compute the expected utility for each utility function in each scenario. Expected utility functions: a) ua(c1 , c2) = π1 ln(c1) + π2 ln(c2) b) ub(c1 , c2) = π1 √ c1 + π2 √ c2 c) uc(c1 , c2) = 10( 1 − π1e −c1/100 − π2e −c2/100) Uncertainty scenarios: d) π1 = π2 = 1 2 , c1 = 10, and c2 = 150 e) π1 = π2 = 1 2 , c1 = 40, and c2 = 50 f ) π1 = π2 = 1 2 , c1 = 110, and c2 = 200 Answer: (a) (b) (c) (d) 3.66 7.70 4.36 (e) 3.80 6.70 3.62 (f ) 5.00 12.32 7.66 1 1.2 Expected utility of lotteries You may be familiar with common lotteries, where you pay a small price to have a small chance to win a large amount of money. This is a special case of what economists call “lotteries,” which are probability distributions over various levels of consumption. Consider a consumer with an expected utility preference, with the same state utility function v(c) = √ c in all states. (The consumer cannot buy any insurance.) Consider two lotteries that the consumer may face: • Lottery A: Lose 50 with probability 1 4 , win 20 with probability 3 4 . • Lottery B: Lose 40 with probability 1 10 , lose 10 with probability 6 10 , win 30 with probability 3 10 . Use a calculator or a computer for this problem, and report your answers to at least 3 decimal places. a) If the consumer starts with income 100, what is the consumer’s expected utility for each lottery? Answer: u(A, 100) = 1 4 · √ 100 − 50 + 3 4 · √ 100 + 20 ≈ 9.9836 u(B, 100) = 1 10 · √ 100 − 40