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I. Answer all pans {aHe) of (A« question.
A risk-averse expected-utility maximizer bus initial wealth n unit utility function u\ She faces a risk of a financial loss of L dollars, which occurs with probability si. An insurance company offers to sell a policy that costs w dollars per dollar of coverage (|«r dollar paid hack in the event of a loss). Denote by ;r the number of dollars of coverage.
(a)15 marks! Give the formula for her expected utility V’t>: as a function of*.
(b)| 111 marks! Suppose that u(*J o t>m. a 1/4, L 100 and p 1/3. Write I’fx) using these values. There should be three variables, x. A and a . Find the optimal value of a-, as a function of A ami by solving the tlisi-oidei condition (set the derivative of the expected utility with respect to a equal to zero). (The second-order condition lur this problem holds but you do not need to check it. ’ Does the optimal amount of coverage increase or decrease in A?
(c)[10 marks] Repeat exercise (b). but with p = 1/(5.
id: |7 ntarksl You should find tiiat for either (b> orfcl. the optimal coverage is increasing in A. ami that m the nilicr case it is decreasing in A. Reconcile these two results.
I. /tower of,1 parti (a)fe/ of this question
A lisk-aveisc expected-utility maximizer has initial wealth >.- ami utility function a. She faces a nsk of a financial loss of L dollars. which occurs with probability x. An insurance company offers to sell a policy that costs p dollars per dollar of coverage fper dollar paid back in the event of a losst. Denote hy .c the numhei of dnltais of eoveiage.
(a)15 ituuksl Give the formula lur her expected utility l‘(z) as a function of jr.
(b)[10 inaiks) Suppose that i.”z! = r~’x . a = 1/4, f. = .fifiamlp = 1 /A. Wide V(y) using these values. There shuuld be three variables, x. A and to. Find the optimal value ol’x, as a function of A and a. by solving the first-order condition (set the derivative of the expected utility with respect to -.v equal to zerot. iTlie sccond-oidei condition for this pnihlemhnldshui you do nut need to check it,) Does the optimal amount of coverage increase or decrease in A1′ (ci 110 marks | Repeat exercise fl». but with y = 1/6.
id) 17 marks| You shuuld find that for either ib) or (c), the optimal coverage is increasing in A, and that in the other ease it is decreasing in A. Reconcile these two results.
I. Anstutrall jranx (ai-ftrj of ih’s> r/sufsiion.
A risk-averse expected-utility maximizer has initial wealth a- and utility function u. She faces a risk of a financial Iocs of J. dollars. which occurs with pmhability v. An insurance company oilers to sell a policy that costs p dollars per dollar of coverage (per dollar paid back in the event of a loss). Denote by z the number of dollars of coverage.
(a) [5 marks) Give the formula for her expected utility V’fx) as a function of x.
•lb) [10 marksl Suppose that &(*) = e-*4 . .r = 1/1. L – 10(1 and p = 1/3. Write V(x] using these values. There should he ihice variables, A and Fiivd the optimal value of as a function oi A anti v, by solving the first-order condition (vet 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 cheek it.i Does the optimal amount nf coverage increase nr decrease in A ? (c> 110 marksl Repeat exercise (b;. but with y 1/6.
[7 mark-] You should find that foi cither (hi or (e), the optimal eoveiag