Socrates owns just one ship. The ship is worth $200 million dollars. If the ship sinks,
Socrates loses $200 million. The probability that it will sink is .02. Socrates' total wealth, including the value of the ship is $225 million. He is an expected utility maximizer with utility U(W) equal to the square root of W. What is the maximum amount that Socrates would be willing to pay in order to be fully insured against the risk of losing his ship?
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To find the maximum amount that Socrates would be willing to pay in order to be fully insured against the risk of losing his ship, we need to calculate the expected utility of his total wealth.
Let's break down the problem step by step:
Step 1: Calculate the expected utility of Socrates' total wealth if he doesn't lose the ship.
The expected utility is given by E(U(W)) = √(W), where W is the total wealth.
So if the ship doesn't sink, Socrates' total wealth remains $225 million. Therefore, the expected utility of his total wealth in this case is √(225) = 15.
Step 2: Calculate the expected utility of Socrates' total wealth if he loses the ship.
The expected utility is given by E(U(W)) = √(W), where W is the total wealth.
If the ship sinks, Socrates' total wealth would be $225 million - $200 million = $25 million. Therefore, the expected utility of his total wealth in this case is √(25) = 5.
Step 3: Calculate the expected utility of Socrates' total wealth taking into account the probability of the ship sinking.
The expected utility is calculated as the weighted average of the expected utilities in each scenario:
E(U) = (probability of ship not sinking) x (expected utility of total wealth if not sinking) + (probability of ship sinking) x (expected utility of total wealth if sinking)
E(U) = (0.98) x (15) + (0.02) x (5)
E(U) = 14.7
Step 4: Determine the maximum amount that Socrates would be willing to pay to be fully insured against the risk of losing his ship.
Socrates is an expected utility maximizer, which means he seeks to maximize his expected utility.
To find the maximum amount he would be willing to pay, we need to calculate the difference in expected utility with and without insurance.
If Socrates buys insurance, his total wealth will be reduced by the cost of insurance.
Let's say the cost of insurance is X million dollars.
Therefore, his total wealth would be $225 million - X million dollars.
The expected utility of his total wealth in this case would be √(225 - X).
So, the difference in expected utility is √(225 - X) - 14.7.
Socrates would be willing to pay an amount such that the difference in expected utility is zero, meaning the expected utility with and without insurance is the same.
Therefore, we solve the following equation for X:
√(225 - X) - 14.7 = 0
By solving this equation, we find that X ≈ 210.9 million dollars.
So, the maximum amount that Socrates would be willing to pay in order to be fully insured against the risk of losing his ship is approximately 210.9 million dollars.
To determine the maximum amount that Socrates would be willing to pay in order to be fully insured against the risk of losing his ship, we need to calculate the expected utility of losing the ship and compare it with his current utility.
First, let's calculate the expected utility of losing the ship. The expected utility is the probability of an outcome multiplied by the utility of that outcome. In this case, the outcome is losing the ship, which has a probability of 0.02 and a value of -200 million. Using the utility function U(W) = √W, the expected utility can be calculated as follows:
Expected utility of losing the ship = Probability of sinking * Utility of losing the ship
= 0.02 * √(-200 million)
= 0.02 * √(-200) * √(1 million)
= 0.02 * 14.142 * 1000
= 2.828 * 1000
= 2828
Now, let's calculate Socrates' current utility. His total wealth, including the value of the ship, is $225 million. Using the utility function U(W) = √W, his current utility can be calculated as:
Current utility = √(225 million)
= √(225) * √(1 million)
= 15 * 1000
= 15000
To find the maximum amount Socrates would be willing to pay for full insurance, we need to find the difference between the expected utility of losing the ship and his current utility. In this case:
Maximum amount Socrates would be willing to pay = Expected utility of losing the ship - Current utility
= 2828 - 15000
= -12172
Since the result is negative, it means Socrates is not willing to pay anything for full insurance against the risk of losing his ship. This implies that he values the ship more than the expected loss of sinking it, given his utility function.
Take a shot, what do you think?
Hint: Assuming the guy is risk-neutral, calculate an insurance value Z such that 225-Z = expected wealth without insurance.