Question

(20 points) Clarice is an expected utility maximizer and her utility function over money is given by u = cả. Clarice's friend, Hannibal, has offered to bet with her $1,000 on the outcome of the toss of a coin. That is, if the coin comes up heads, Clarice must pay Hannibal $1,000 and if the coin comes up tail, Hannibal must pay Clarice $1,000. The coin is a fair coin, so that the probability of heads and the probability of tails are both. Clarice has $10,000 and is trying to figure out whether she should take the bet. Note that if Clarice accepts the bet and heads comes up, she will have $10,000- $1,000=$9,000. (a) (5 points) If Clarice accepts the bet, then if tails comes up, she will have how much money? (b) (5 points) What is Clarice's expected utility if she does not accept the bet? (c) (5 points) Does Clarice take the bet? Explain why or why not Clarice takes the bet. (d) (5 points) Clarice later asks herself, "If I make a bet where I lose all my money, that is all my $10,000 if the coin comes up heads, what is the smallest amount that I would have to win in the event of tails in order to make the bet a good one for me to take” ? Find the answer to Clarice's question.

Answer 1

Q)

(a) The total Money that Claire has when tails comes up = Intial money with Claire + $1000 paid by Hannibal as part of the bet

= 10000 + 1000 = $11000

(b) IN case of no betting, Claire will have a constant $10000. Her utility function over money is u = x^1/2

Thus, we get u = 10000^1/2 = 100

(c) Calire's expected utitliy from taking the bet = pr(winning)*utiltiy from winning + pr(losing)*utility from losing

= 0.5*(11000^1/2) + 0.5*(9000^1/2)

= 99.87

Now we can observe that the expected utility from accepting the bet = 99.87 is lesser than the utiltiy without betting = 100, and since Claire is an expected utility maximizer, she will not take the bet.

(d) The new bet good only if her expected utility from betting > utility from not betting =100

Let Y be the total amount she has if she wins

=> 0.5*(0^1/2) + 0.5*(y^1/2) > 100

=> y^1/2 > 200

=> y > 200²

=> y > 40000

Thus, the smallest amount that Claire has to win to make bet a good one is = Final amount with Claire - Initial amount with claire =  40000 - 10000 = $30000