Problem 4: Consider a risk-averse individual with an initial wealth of 100 and a utility function of the form u(W) = W0:

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Problem 4: Consider a risk-averse individual with an initial
wealth of 100 and a utility function of the form u(W) = W0:5 where
W represents the payo  from a particular outcome. Suppose that this
individual faces a loss of L with probability 0.5. To reduce her
risk, she would like to purchase an insurance policy that gives her
a payment of B if she su ers the loss. Suppose that the cost of the
policy is equal to the the premium, M.
(a) Assuming that the insurance policy is actuarially fair, what
is the premium that insurance companies will charge?
(b) Use all of the information above to write down an expression
for this individual's expected utility with the actuarially fair
insurance.
(c) Let the individual's objective be to choose the optimal
insurance pay- ment, B. What is the optimal B?
(d) Suppose that for each dollar of bene t paid out, an
insurance com- pany incurs an additional fraction   > 0 as an
administrative cost. The insurance company passes this cost on to
the buyer by reducing the bene t paid by a fraction  . What is the
individual's optimal B in this case? How does it compare to your
answer in part (c)? You can assume that the the insurance industry
is competitive and the premium will be actuarially fair.
(e) Now let L = 40. Will this individual prefer having an
insurance policy where   is greater than zero to not having
insurance at all? Explain.