Question

Bob has a utility function over money v(x) = √ x. There are two possible states of the world 1 and 2. State 1 can occur with probability π1 and state 2 can occur with probability π2 where π2 = 1 − π1. Bob’s wealth levels in the states 1 and 2 will be x1 and x2 respectively. Therefore Bob’s expected utility over the state-contingent consumption bundle is, U((x1, x2); (π1, π2)) = π1 √ x1 + π2 √ x2

As in class, we will call the state-contingent consumption bundle ((x1, x2); (π1, π2)), a lottery.

(a) Let π1 = 3/8 and π2 = 5/8 . Calculate the expected utility from the lottery that pays nothing in state 1 and $ 100 in state 2. Calculate the expected level of wealth from this lottery and Bob’s utility from receiving this expected value with certainty, i.e., in both states of the world. Which one is larger? What does it tell you about Bob’s attitude towards risk?

(b) As we saw in class, the certainty equivalent of a lottery is a payoff which, if received with certainty, would make the person indifferent between the lottery and receiving the certainty equivalent in each state. Calculate the certainty equivalent of the lottery in (a).

(c) Find the formula for an indifference curve, giving x2 as a function of x1, identifying combinations of x1 and x2 that give the same expected utility to Bob.

Answer 1