Suppose, after undergoing genetic testing, you discover that you have a health condition that could result in the emergence of a disability which would make it impossible for you to continue to work. The probability of this happening is 50%. Currently your expected lifetime earnings are $5,000,000, but if the disability hits, your expected lifetime earnings will consist primarily of income earned from government support programs -- and will not add up to more than $1 million.
a. Suppose your tastes are state-independent and the function 
can be used to
represent your tastes in the expected utility form. Are you risk averse?
b. What is the highest premium you would pay to get fully insured?
c. What is the equation (in terms of 
-- consumption in the bad state -- and 
-- consumption in the good state) that defines the full menu of actuarily fair insurance contracts?
d. Set up the optimization problem that you would solve as you choose among actuarily fair insurance contracts.
e. Solve the optimization problem. What does this imply will be the insurance contract (b,p) that you buy -- where b is the benefit level and p is the insurance premium?
f. Finally, suppose you had state dependent tastes and that the functions 
and 
allowed us to use the expected utility form to represent your tastes. How does your answer to (e) change?
What will be an ideal response?
b. The expected utility without insurance is
.
The highest amount you would pay to get consumption level x that is independent of the state is an amount that would result in 
Solving for x, this gives us approximately x=2,236,000. Thus, the most you would be willing to give up in the good state to fully insure is 5,000,000-2,236,000=$2,764,000.
c. 
d. 
e. The problem solves to 
which implies b=4,000,000 and p=2,000,000.
f. The optimization problem becomes 
which solves to 
and 
This implies b=2,000,000 and p=4,000,000.
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In the short-run
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