John is a professional football player and next year will be able to sign a $25 million contract if he does not get injured this year. If he gets injured this year, his value will be significantly reduced and he will only be able to sign a contract for $5 million. Suppose that the probability he gets injured is 5%. His utility function is of the form U = W^(1/2), where W is his wealth. (therefore he is risk adverse).

Suppose that an insurface company offers to insure him at a premium of $0.10 per $1 of coverage. Would John choose to FULLY insurance himself? (hint: think of the idea of actuarilly fair/unfair game) Explain, and show graphically.
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I am very confused by this question and I don't even know where to begin. Can someone kindly explain the answer to this question. If at all possible, please explain the graph as well.

Your help is very much appreciated!:)

typo...

The above should read
"Suppose that an INSURANCE company offers to insure him at a premium of..."

I believe what you want do do is calculate John's Expected Utility. Incorporate into the utility function the insurance option.

Let x be the number dollars of insurance coverage, each unit of x costs 0.10. So, the expected utility E(U) is as follows.
E(U) = .95*(25000000 - 0.1*x)^(.5) + .05*(5000000 + 0.9*x)^(.5)

Where the .95 and the .05 are the probabilities of not getting/getting injured, .1 is the insurance policy premium and .9 is the insurance policy payoff less the premium.

From the beginning, you could see the insurance policy was actuarily unfair. If the probability of getting injured was 10% instead of 5%, then the policy would be actuarily fair.

You can graph expected the expected utility function with respect to x. You should get a hump-shaped curve. From the graph, you can tell that John will not fully insure himself. Maximum expected utility occurs when x is between 600,000 and 700,000. (Use calculas to determine the exact amount) From the graph, you should be able to tell that John will NOT fully insure himself. Expected utility at x=20million is actually less than expected utility at x=zero.

To determine whether John would choose to fully insure himself, we need to compare the expected utility of his wealth with and without insurance.

First, let's calculate the expected utility of his wealth without insurance. We can calculate the expected utility by taking the sum of the expected utility in each scenario and weighting it by the probability of that scenario occurring.

Without insurance, there are two possible outcomes: either John gets injured and his wealth is reduced to $5 million with a probability of 5% (0.05), or he does not get injured and his wealth remains $25 million with a probability of 95% (0.95).

Expected utility without insurance:
E(U) = U($5 million) * P(Injured) + U($25 million) * P(Not Injured)
E(U) = ($5 million)^(1/2) * 0.05 + ($25 million)^(1/2) * 0.95

Now let's calculate the expected utility of his wealth with full insurance. With full insurance, he will pay a premium, but in case of injury, he will receive the insured amount. In case of no injury, he still has to pay the premium.

The premium is $0.10 per $1 of coverage, so for full insurance, he will pay a premium of 10% of the insured amount. Let's assume he insures the full $25 million.

Expected utility with full insurance:
E(U_insured) = [U($25 million - premium)] * P(Not Injured) - premium * P(Injured)
E(U_insured) = [($25 million - 0.10 * $25 million)^(1/2)] * 0.95 - (0.10 * $25 million) * 0.05

Now let's compare the expected utility without insurance and with full insurance.

If E(U) > E(U_insured), John would choose not to fully insure himself. If E(U) < E(U_insured), John would choose to fully insure himself. And if E(U) = E(U_insured), he would be indifferent.

Now let's graphically analyze the situation.

On the x-axis, we can plot the probability of being injured (ranging from 0 to 1), and on the y-axis, the expected utility.

- Plot a line representing the expected utility without insurance (E(U)) as a function of the probability of injury.
- Plot a line representing the expected utility with full insurance (E(U_insured)) as a function of the probability of injury.

The point where these two lines intersect represents the probability of injury at which John would be indifferent between full insurance and no insurance. If the point where the lines intersect lies to the left of the intersection, it means E(U) > E(U_insured), and John would choose not to fully insure himself. If it lies to the right, E(U) < E(U_insured), and John would choose to fully insure himself.

Based on this analysis, you can determine whether John would choose to fully insure himself by comparing the expected utilities and visualizing them on the graph.