Alice and Ben are both expected utility maximizers. Alice's vNM utility is u1 (x) = x ? while Ben's vNM utility is u2 (x) = x ? where both ? and ? are strictly between 0 and 2. Consider the following lotteries:

? Lottery P: (25%, $2; 75%, $0)

? Lottery Q: (q, $1; 1 ? q, $0)

For any lottery Q where q is between 0 and 1, if Alice prefers P to Q then Ben also prefers P to Q.

1. What is the relationship between ? and ?? Do we know which is larger?

2. Suppose Alice is risk-loving. Do we know if Ben is risk-averse or risk-loving?

3. Suppose Ben is risk-loving. Do we know if Alice is risk-averse or risk-loving?

Problem 2. (20 points total) Alice and Ben are both expected utility maximizers. Alice's vNM utility is un (x) = r" while Ben's vNM utility is uz (r) = x where both a and 3 are strictly between 0 and 2. Consider the following lotteries: . Lottery P: (25%, $2; 75%, $0) . Lottery Q: (q, $1; 1 - q, $0) For any lottery Q where q is between 0 and 1, if Alice prefers P to Q then Ben also prefers P to Q. 1. What is the relationship between o and ? Do we know which is larger? (10 pts) 2. Suppose Alice is risk-loving. Do we know if Ben is risk-averse or risk-loving? (5 pts) 3. Suppose Ben is risk-loving. Do we know if Alice is risk-averse or risk-loving? (5 pts)