13.21 A decision maker faces a risky gamble in which she may obtain one of five outcomes. Label the outcomes A, B, C, D, and E. A is the most preferred, and E is least preferred. She has made the following three assessments. She is indifferent between having C for sure or a lottery in which she wins A with probability 0.5 or E with probability 0.5 She is indifferent between having B for sure or lottery in which she wins A with probability 0.4 or C with probability 0.6 She is indifferent between these two lotteries: A 50% chance at B and a 50% chance at D A 50% chance at A and a 50% chance at E What are U(A), U(B), U(C), U(D), and U(E)?

First, because A and E are best and worst outcomes, respectively, we can set U(A) = 1, and U(E) = 0. Then, based on the indifference assessment, we can set up equations First assessment: U(C) = 0.5*U(A) + 0.5*U(E) = 0.5(1) + 0.5(0) = 0.5 Second assessment: U(B) = 0.4*U(A) + 0.6*U(C) = 0.4(1) + 0.6(0.5) = 0.7 Third assessment: 0.5*U(B) + 0.5*U(D) = 0.5U(A) + 0.5U(E)  0.5(0.7) + 0.5U(D) = 0.5(1) + 0.5(0) U(D) = 0.30

15.14 A friend of yours is in a market for a new computer. Four different machines are under consideration. The four computers are essentially the same, but they vary in price and reliability. The least expensive model is also the least reliable, the most expensive is the most reliable, and the other two are in between. The computers are described as follows: Price: $998.95 Expected number of days in the shop per year: 4 Price: $1300.00 Expected number of days in the shop per year: 2 Price: $1350.00 Expected number of days in the shop per year: 2.5 Price: $1750.00 Expected number of days in the shop per year: 0.5 The computer will be an important part of your friend’s livelihood for the next two years. The magnitude of the losses are uncertain but are estimated to be approximately $180 per day that the computer is down. Can you give your friend any advice without doing calculations? Use the information given to determine weights kP and kR. What assumptions are you making? Calculate overall utilities for the computers. What do you conclude? Sketch three indifference curves that reflect your friend’s trade-off between reliability and price.

Let P = Price of computers ($) R = Reliability (the number of days in the shop per year) a. Compared to machine B, machine C has a higher price yet more number of days in the shop per year. Therefore, machine C is dominated by B and thus can be eliminated from the analysis b. Because A is the best on price but the worst on reliability, UP (A) = 1 and UR (A) =0 Because D is the worst on price but the best on reliability, UP (D) = 0 and UR (D) =1 UP (B) = (1300-1750)/(998.95-1750) = 0.60 and UR (B) =(2-4)/(0.5-4) = 0.57 Utility Alternative A B D UP 1 0.6 UR 0.57

An extra day of reliability is worth $180, and also assume proportionality for the utilities, then Machine A, with price $998.95 and expected down time of 4 days per year would be equivalent to hypothetical Machine E for (998.95+180=$1178.95) and expected down time of (4-1=3) days per year. Because A is the best on price but the worst on reliability, U(A) = kP(1) + kR (0) = kP UP (E) = (1178.95-1750)/(998.95-1750) = 0.76 and UR (E) =(3-4)/(0.5-4) = 0.29 U(E) = kP (0.76) + kR (0.29) U(A) = U(E) 0.76kP +0.29kR = kP (Eq. 1) kP + kR = 1 (Eq. 2) Solving Equations 1 and 2 for kP and kR , we can get kP = 0.547 and kR = 0.453 c. U(A) = 0.547(1) + 0.453(0) = 0.547 U(B) = 0.547(0.60) + 0.453(0.57) = 0.586 U(D) = 0.547(0) + 0.453(1) = 0.453

d. Utility  Days in shop/per year 0.586 0.547 0.453 A E B D Price ($)