Question

i) Suppose that Mary’s utility function is $kOFZjQAAAABJRU5ErkJggg==$ where W is wealth. Is she risk averse? Suppose that Mary has initial wealth of $125,000. How much of a risk premium would she require to participate in a gamble that has a 50% probability of raising her wealth to $160,000 and a 50% probability of lowering her wealth to $90,000?

ii) Suppose that Irma’s utility function with respect to wealth is U(W) = 100 + 80W − W². Find her Arrow-Pratt risk aversion measure and show that for W < 40, it increases with her wealth. (Hint: First find the Arrow-Pratt measure, then differentiate it with respect to w)

Answer 1

UCW) = Wt LIM -2 She is Risk Avense Expected wealth E(W) = from Gamble (160,000) + / (90,000) + 45,800 = = 80,00 125.000

U (W = 125,000) = (125,000) = 50 ? ? ? ? Expected ility from Comble Ecucw)) = 1 U(W = 160,00) + + @ UCw=70,00) = (54.28) t I (44.8) - + 22.4. 27.14 = 4A. 54

Mary gets expected utility of She can get the same utility 48.54 from gamble. from a certain income W = (ur. 54) = 121, 664 Rish Premium = Expected Wealth - utility Certifying Utility wcolth - 125,00 - 121,664 = $333.6 ( Appear.)

UCW) = 100+ zow-W2 A = - UCW) Arrow Pratt Measure o'cW) = 80 - zw O = - (-2) 8o-2h 80-2W yo-W

youw JA _-(2) E) dw (ao-W) For W Lyo As wealth ince.ases smaller hence o denomivabe of A becomis A becomas larger.

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