Question

1. Suppose that David has the following utility function over consumption level c:

U(c) = √c (that’s the square root of c)

Also, suppose David has wealth $1000, but faces the risk of a financial loss $500 with probability 0.2 (20% chance). He does not save anything (he consumes all the wealth he can).

Now suppose that David has a choice of insuring his potential losses. The insurance policy pays him $0 if he does not have financial loss and pays him $500 if he does have loss.

1. Describe intuitively why David will value insurance.
2. What would the actuarially fair price for this insurance policy be?
3. Suppose the price of the insurance policy is $100. What is David’s expected utility if he buys insurance?
4. Will David buy the insurance at a price of $100?
5. Now suppose the price of insurance is $300. What is David’s expected utility if he buys insurance now?
6. Will David buy the insurance at a price of $300?

Answer 1

Ans a)

U(c)=sqrt(c)

U'(c)=0.5*c^(-0.5)>0 for all c>0

U"(c)=-0.25c^(-1.5)<0 for all c>0 then

Arrow Pratt Risk measure is positive if(-U"(c)/U'(c)) when agent is Risk averse and in our case (-U"(c)/U'(c))>0

David is Risk averse and will always choose to value insurance

Ans b)

Lets say "p" is the value of actuarily fair insurance such that

Expected utility after insurance is equals to Utility from the expected income

0.8sqrt(1000-p)+0.2sqrt(500-p+500)=sqrt(0.8(1000)+0.2(500))

sqrt(1000-p)=30

Ans c)

sqrt(1000-p)=30

p=100..(actuarily fair price of insurance)

EU=30...Expected Utility if he buys insurance

Ans d)

Yes if he buys insurnace worth $100 then he is indifferent between buying insurance and not buying the insurance

Ans e)

When price of insurance is $300

EU=sqrt(1000-300)=sqrt(700)=26.46