Hjalmer Ingqvist’s son-in-law, Earl, has not worked out very well. It turns out that Earl likes to gamble. His preferences over contingent commodity bundles are represented by the expected utility function
U (c1, c2, π1, π2) = π1c21 + π2c22.
(a) Just the other day, some of the boys were down at Skoog’s tavern when Earl stopped in. They got to talking about just how bad a bet they could get him to take. At the time, Earl had $100. Kenny Olson shuffled a deck of cards and offered to bet Earl $20 that Earl would not cut a spade from the deck. Assuming that Earl believed that Kenny wouldn’t cheat, the probability that Earl would win the bet was 1/4 and the probability that Earl would lose the bet was 3/4. If he won the bet, Earl would have _______dollars and if he lost the bet, he would have _______ dollars. Earl’s expected utility if he took the bet would be _______, and his expected utility if he did not take the bet would be _______. Therefore he refused the bet.
(b) Just when they started to think Earl might have changed his ways, Kenny offered to make the same bet with Earl except that they would bet $100 instead of $20. What is Earl’s expected utility if he takes that bet? _______. Would Earl be willing to take this bet? _______.
(c) Let Event 1 be the event that a card drawn from a fair deck of cards is a spade. Let Event 2 be the event that the card is not a spade. Earl’s preferences between income contingent on Event 1, c1, and income contingent on Event 2, c2, can be represented by the equation u = 1/ 4c21 + 3 / 4c22 .
Use blue ink on the graph below to sketch Earl’s indifference curve passing through the point (100, 100).
(d) On the same graph, let us draw Hjalmer’s son-in-law Earl’s indifference curves between contingent commodities where the probabilities are different. Suppose that a card is drawn from a fair deck of cards. Let Event 1 be the event that the card is black. Let event 2 be the event that the card drawn is red. Suppose each event has probability 1/2. Then Earl’s preferences between income contingent on Event 1 and income contingent on Event 2 are represented by the formula u = 1 / 2c21 + 1 2c22 .
On the graph, use red ink to show two of Earl’s indifference curves, including the one that passes through (100, 100).