Question

You are a competitive athlete, and will earn $1,000,000 if you play for the entire
season. If you are injured before the season starts, you will earn only $40,000. Your
utility function is U = sqrt(Income) . There is a 5% probability that you will be injured,
and a 95% probability that you will not be injured.
(a) What is your expected income for the year?
(b) What is your expected utility for the year?
(c) An insurance company offers you a policy that will fully insure your income. What
is the maximum price you are willing to pay for this policy?
(d) A second insurance company offers you a policy that will partially compensate
you, but not fully insure you. This second company offers a $900,000 payout in the
event of an injury, and charges a fee of $40,000. If the first insurance company
charges a fee of $70,000 for full insurance, which insurance policy do you buy? Why?

Answer 1

Let H denotes being Healthy(not injured) and I denotes injured

P(H) = 95% = .95

P(I) =5% = .05

Income(M) when not injured:- $1000000

Income(M) when injured:- $40000

Utility, U = M1/2

Part (a)

Expected Income:- P(H) x M(H) + P(I) x M(I) = .95 x 1000000 + .05 x 40000 = $952000

Part (b)

Expected Utility:- P(H) x U(H) + P(I) x U(I) = .95 x 10000001/2 + .05 x 400001/2 = 950 + 10 = 960

Part (c)

My expected utility is 960

U = M1/2

960 = M1/2

M = 921600

For an income of $921600, I can maintain my expected utility of 960.

Total amount I will get from the insurance after injury with full coverage :- 1000000 = $100000

So, my maximum willingness to pay for the policy is:- 1000000 - 921600 = $78400

Part (d)

Net benefit with the second insurance plan:- 900000 - 40000 = $860000

Utility:- 8600001/2 = 927.36

Net benefit with the first plan:- 1000000 - 70000 = $930000

Utility:- 9300001/2 = 964.36

So, I will go with the first insurance company as the utility provided is higher than the second one.