The conventional model of the demand for insurance assumes utility is a function of income. Based on this model, use the information provided to determine the answers to the following questions: maximum income if Andy Dwyer does not get sick = $90,000; Cost of Illness = $15,000; Probability of Illness = 25%, Utility with respect to income is: U = 2*Y^1/2

1) What is the expected value of Andy's income loss?

2) How much is Andy willing to pay to avoid risk in relation to an income associated with an illness?

3) What is the maximum amount Andy is willing to pay for insurance?

(U) + Utility as a function of income with risk aversion. Utility A U=90 U Expected Income = - = = = = = 1 U,=E(U)=86 $36,000 B Expected Utility: 86 U,=70 0 Y1 Y2 E(Y) Yo Income (Y) $20,000 $36,000 $40,000 $35,000 Chord AB represents the expected utility associated with different probability values of illness (nt) with no risk aversion.What We Know: Utility as a function of income: U 2Y1\" Probability of illness 2 0.5 Income without getting sick: 575,000 Expected income loss from getting sick: 530,000 I. What is income and utility coordinate pair for Pete's maximum income? A. Income = $75,000 (GIVEN) B. Utility.- U = Y'"1 = {75,000)l-"1 = 2.73.86 2. What is the minimum income along the utility curve? 0 Y1 Y\" Income m A. Min income = max income income loss = 75.000 30,000 = 45,000 B. Utility: U = Y\"1 = (45,000)\"! = 212.13 What We Know: - Utility as a function of income: U =1"1er [U] - Probability of illness = 0.5 mm\" u - Income without getting sick: $75,000 U0: n ' ' ' . 273.85 Expected income loss from getting suck. $30,000 U1= 1. What is Pctc's expected incomc'.' 212.13 A. Expected income = (Prob of Illness x Max Income} + {Prob of Illness x Max Income) = 0. 595,000) + 0.5{45,000) = 60000 ' ' 2. What is Pete's expected utility? 0 Y1 Em Yo Income m $45,000 550,000 $15,000 A. 0.5{273 .86) + 0.5{212.13} 242.995 3. How much does Pete expect his medical expenses in the coming year to cost'? A. Max income expected income = 75,000 60,000 = 15,000 What We Know: Utility as a function of income: U =y1/2 (U) Probability of illness = 0.5 Utility A Income without getting sick: $75,000 UO= U 273.86 Expected income loss from getting sick: $30,000 D U.= 1. What is Pete willing to pay to avoid risk? 212.13 A. Expected income - risk averse pay B. Use expected utility to solve for risk averse pay A. Expected utility (from previous calculation) = 242.995 0 Y1 E(Y) Yo Income (Y) $45,000 $60,000 $75,000 B. U = Y1/2 $59,046 C. 242.995 = Y1/2 D. Solve for Y which gives you 59046.57 C. 60,000 - 59,046.57 = $953.43 What We Know: Utility as a function of income: U =Y1/2 (U) Probability of illness = 0.5 Utility A . Income without getting sick: $75,000 U 273.86 D Expected income loss from getting sick: $30,000 U,= 1. If he is offered insurance for $20,000, will he 212.13 buy it? A. Expected Medical expenses + willingness to pay to avoid risk B. 15,000 + 953.43 = 15,953.43 0 Y1 Y2 E(Y) Yo Income (Y) C. No, he will not buy the $20,000 insurance $45,000 $60,000 $75,000 $59,046 because it exceeds his willingness to pay to avoid risk plus his expected medical expensesFIGURE 6-1 Expected Utility (Conventional) Model (U) + Utility as a function of income with risk aversion. Utility A U=90 U If we trace from Point U2=E(U)=86 C, towards the y axis, C U,=70 we cross the indifference curve at Point D where utility is at the same level as at Point C. Y1 Y2 E(Y) Yo Income (Y) $20,000 $36,000 $40,000 $35,000 Use the utility value from Point D and the utility function and solve for Y. Here assume the utility function is ~U = 0.45*Y1/2. Solve for Y = $35,000. utility function mas willing to pay for insur: 5K 22 FIGURE 6-1 Expected Utility (Conventional) Model (U) + Utility as a function of income with risk aversion. Utility A Point D = ($35K, 86) U=90 U U2=E(U)=86 Point D shows that a U,=70 risk-averse person is indifferent in terms of utility between losing a known amount of $5,000 at point D and 0 Y2 E(Y) Yo Income ( Y) $20,000 $36,000 $40,000 an expected amount of $35,000 $4,000 at point C because expected involves uncertainty