Suppose you have $30,000 in wealth and have the choice of two possible gambles:
Gamble 1: a single flip of a coin would pay you $16,000 if it is heads, cost you $16,000 if it is tails.
Gamble2: two coin flips, each payuing you $8,000 if it is head, each costing you $8,000 if it is tails.
Suppose you utility function for money is U=$1/2. How much less costly is the second gamble than the first.
I presume that U=($)^(1/2) = sqrt($).
Calculate the expected utility under both choices.
U1 = .50*sqrt(46000) + .50*sqrt(14000)
U2 = .25*sqrt(46000) + .50*sqrt(30000) + .25*sqrt(14000)
Take it from here. BTW, not playing at all gives the highest expected utility.
To determine how much less costly the second gamble is compared to the first, we need to calculate the expected utility for each gamble.
First, let's calculate the expected utility for Gamble 1:
The probability of getting heads on a single flip of a fair coin is 1/2, so the expected outcome for Gamble 1 is:
(1/2) * $16,000 + (1/2) * (-$16,000) = $8,000 - $8,000 = $0
Now let's calculate the expected utility for Gamble 2:
The probability of getting heads on each flip of a fair coin is (1/2) * (1/2) = 1/4. Since the coin flips are independent, we can multiply the probabilities to calculate the overall probability of a particular outcome.
The four possible outcomes for Gamble 2 are:
1. Heads - Heads: (1/4) * $8,000 + (1/4) * $8,000 = $2,000 + $2,000 = $4,000
2. Heads - Tails: (1/4) * $8,000 + (1/4) * (-$8,000) = $2,000 - $2,000 = $0
3. Tails - Heads: (1/4) * (-$8,000) + (1/4) * $8,000 = -$2,000 + $2,000 = $0
4. Tails - Tails: (1/4) * (-$8,000) + (1/4) * (-$8,000) = -$2,000 - $2,000 = -$4,000
The expected outcome for Gamble 2 is the sum of all possible outcomes weighted by their respective probabilities:
(1/4) * $4,000 + (1/4) * $0 + (1/4) * $0 + (1/4) * (-$4,000) = $1,000 - $1,000 = $0
Now that we have calculated the expected utility for both gambles, let's evaluate the difference in utility:
The utility for Gamble 1 is $0, and the utility for Gamble 2 is also $0.
Therefore, the second gamble is not less costly than the first gamble in terms of utility, as both have the same expected utility of $0.