A risk-averse expected-utility maximizer has initial wealth w0 and utility function u. She facesa risk of a financial loss of L dollars, which occurs with probability π. An insurance companyoffers to sell a policy that costs p dollars per dollar of coverage (per dollar paid back in theevent of a loss). Denote by x the number of dollars of coverage.(a) Give the formula for her expected utility V (x) as a function of x.(b) Suppose that u(z) = −e−zλ, π = 1/4, L = 100 and p = 1/3. Write V (x)using these values. There should be three variables, x, λ and w. Find the optimal value of x,as a function of λ and w, by solving the first-order condition (set the derivative of the expectedutility with respect to x equal to zero). (The second-order condition for this problem holds butyou do not need to check it.) Does the optimal amount of coverage increase or decrease in λ,where λ > 0?(c) Repeat exercise (b), but with p = 1/6.(d) You should find that for either (b) or (c), the optimal coverage is increasing in λ,and that in the other case it is decreasing in λ. Reconcile these two results.
A risk-averse expected-utility maximizer has initial wealth w0 and utility function u. She faces
a risk of a financial loss of L dollars, which occurs with probability π. An insurance company
offers to sell a policy that costs p dollars per dollar of coverage (per dollar paid back in the
event of a loss). Denote by x the number of dollars of coverage.
(a) Give the formula for her expected utility V (x) as a function of x.
(b) Suppose that u(z) = −e
−zλ
, π = 1/4, L = 100 and p = 1/3. Write V (x)
using these values. There should be three variables, x, λ and w. Find the optimal value of x,
as a function of λ and w, by solving the first-order condition (set the derivative of the expected
utility with respect to x equal to zero). (The second-order condition for this problem holds but
you do not need to check it.) Does the optimal amount of coverage increase or decrease in λ,
where λ > 0?
(c) Repeat exercise (b), but with p = 1/6.
(d) You should find that for either (b) or (c), the optimal coverage is increasing in λ,
and that in the other case it is decreasing in λ. Reconcile these two results.
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