Clarence Bunsen is an expected utility maximizer. His preferences among contingent commodity bundles are represented by the expected utility function
U (c1, c2, π1, π2) = π1√c1 + π2√c2.
Clarence’s friend, Hjalmer Ingqvist, has offered to bet him $1,000 on the outcome of the toss of a coin. That is, if the coin comes up heads, Clarence must pay Hjalmer $1,000 and if the coin comes up tails, Hjalmer must pay Clarence $1,000. The coin is a fair coin, so that the probability of heads and the probability of tails are both 1/2. If he doesn’t accept the bet, Clarence will have $10,000 with certainty. In the privacy of his car dealership office over at Bunsen Motors, Clarence is making his decision. (Clarence uses the pocket calculator that his son, Elmer, gave him last Christmas. You will find that it will be helpful for you to use a calculator too.) Let Event 1 be “coin comes up heads” and let Event 2 be “coin comes up tails.”
(a) If Clarence accepts the bet, then in Event 1, he will have ______ dollars and in Event 2, he will have ______dollars.
(b) Since the probability of each event is 1/2, Clarence’s expected utility for a gamble in which he gets c1 in Event 1 and c2 in Event 2 can be described by the formula ______. Therefore Clarence’s expected utility if he accepts the bet with Hjalmer will be ______. (Use that calculator.)
(c) If Clarence decides not to bet, then in Event 1, he will have ______dollars and in Event 2, he will have ______dollars. Therefore if he doesn’t bet, his expected utility will be 100.
(d) Having calculated his expected utility if he bets and if he does not bet, Clarence determines which is higher and makes his decision accordingly. Does Clarence take the bet?