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1 Demand for Health Insurance

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2 Which Investment will you pick Expected Value $2600 Choice 2 $5000 -$1000 0.6 0.4 Choice 1 $5000 $1000 0.4 0.6

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3 Attitude towards risk In the absence of any objective criteria, how an individual or organization deals with uncertainty depends ultimately on their attitude towards risk and whether they are risk averse, risk neutral or a risk taker.

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4 Attitude towards risk Someone who would prefer, for example, the certainty of $1,000 rather than a 50% probability of $3,000. Someone who is indifferent, for example, between the certainty of $1,000 rather than a 50% probability of $2,000. Someone who would prefer, for example, the 50% probability of $5,000 rather than the certainty of $3,000. Risk averse Risk neutral Risk taker

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5 Different Approaches to Risk: Expected Value Maximin Maximax Hurwicz alpha index rule

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6 Payoff Matrix First Choice Probability0.20.70.1 Payoff-1000100010,000 Second Choice Probability0.10.60.3 Payoff010003,000 2 Choices for investment:

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7 Expected Value: sum of probabilities Payoffs First Choice Probability0.20.70.1 Payoff-1000100010,000 Second Choice Probability0.10.60.3 Payoff010003,000 EV1= 0.2 (-1000) + 0.7 (1000) + 0.1 (10,000) = 1500 EV2= 0.1 (0) + 0.6 (1000) + 0.3 (3000) = 1500

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8 Maximin: Pessimistic/conservative risk attitude First Choice Probability0.20.70.1 Payoff-1000100010,000 Second Choice Probability0.10.60.3 Payoff010003,000 1.Minimum gain of each choice

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9 Maximin: Pessimistic/conservative risk attitude First Choice Probability0.20.70.1 Payoff-1000100010,000 Second Choice Probability0.10.60.3 Payoff010003,000 1.Minimum gain of each choice

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10 Maximin: Pessimistic/Conservative risk attitude First Choice Probability0.20.70.1 Payoff-1000100010,000 Second Choice Probability0.10.60.3 Payoff010003,000 1.Minimum gain of each choice 2.Which is Maximum

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11 Maximax: Optimistic Criterion First Choice Probability0.20.70.1 Payoff-1000100010,000 Second Choice Probability0.10.60.3 Payoff010003,000 1.Maximum gain of each choice

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12 Maximax: Optimistic Criterion First Choice Probability0.20.70.1 Payoff-1000100010,000 Second Choice Probability0.10.60.3 Payoff010003,000 1.Maximum gain of each choice

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13 Maximax: Optimistic Criterion First Choice Probability0.20.70.1 Payoff-1000100010,000 Second Choice Probability0.10.60.3 Payoff010003,000 1.Maximum gain of each choice 2.Which is Maximum

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14 Hurwicz alpha index rule: The Hurwicz alpha variable is a measure of attitude to risk. It can range from = 1 (optimist) to = 0 (pessimist). A value of = 0.5 would correspond to risk neutrality. The Hurwicz criterion = maximum value x + minimum value x (1 – )

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15 Hurwicz alpha index rule: First Choice Probability0.20.70.1 Payoff-1000100010,000 Second Choice Probability0.10.60.3 Payoff010003,000 1.Weighted average of min and max for each choice. For = 0.5 : The Hurwicz criterion for First Choice: 0.5 (10,000)+ 0.5 (-1000) = 4500 The Hurwicz criterion for Second Choice: 0.5 (3,000)+ 0.5 (0) = 1500

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16 Hurwicz alpha index rule: First Choice Probability0.20.70.1 Payoff-1000100010,000 Second Choice Probability0.10.60.3 Payoff010003,000 1.Weighted average of min and max for each choice. 2.Select the action with the maximum value For = 0.5 : The Hurwicz criterion for First Choice: 0.5 (10,000)+ 0.5 (-1000) = 4500 The Hurwicz criterion for Second Choice: 0.5 (3,000)+ 0.5 (0) = 1500

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17 Hurwicz alpha index rule: First Choice Probability0.20.70.1 Payoff-1000100010,000 Second Choice Probability0.10.60.3 Payoff010003,000 1.Weighted average of min and max for each choice. 2.Select the action with the maximum value For = 0.1 : The Hurwicz criterion for First Choice: 0.1 (10,000)+ 0.9 (-1000) = 100 The Hurwicz criterion for Second Choice: 0.1 (3,000)+ 0.9 (0) = 300

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18 Hurwicz alpha index rule: The maximin strategy equates to the Hurwicz approach with a value of = 0. The maximax strategy corresponds to = 1.

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Insurance Logic The consumer pays insurer a premium to cover medical expenses in coming year. –For any one consumer, the premium will be higher or lower than medical expenses. But the insurer can pool or spread risk among many insurees. –The sum of premiums will exceed the sum of medical expenses.

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Characterizing Risk Aversion Recall the consumer maximizes utility, with prices and income given. –Utility = U (health, other goods) –health = h (medical care) Insurance doesn’t guarantee health, but provides $ to purchase health care. We assumed diminishing marginal utility of “health” and “other goods.”

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Diminishing marginal utility of income Utility Income

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Utility of Different Income Levels Assume that we can assign a numerical “utility value” to each income level. Also, assume that a healthy individual earns $40,000 per year, but only $20,000 when ill. $20,000 $40,000 70 90 IncomeUtility Sick Healthy

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Utility Income$20,000$40,000 90 70 Utility when healthy Utility when sick A B Utility of Different Income Levels

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Probability of Being Healthy or Sick Individual doesn’t know whether she will be sick or healthy. But she has a subjective probability of each event. –She has an expected value of her utility in the coming year. Define: P 0 = prob. of being healthy P 1 = prob. of being sick P 0 + P 1 = 1

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Expected Utility as A Function of Probability An individual’s subjective probability of illness (P 1 ) will depend on her health stock, age, lifestyle, etc. Then without insurance, the individual’s expected utility for next year is: E(U) = P 0 U($40,000) + P 1 U($20,000) = P 0 90 + P 1 70

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Expected Utility & Income As A Point on AB Line For any given values of P 0 and P 1, E(U) will be a point on the chord between A and B. Utility Income$20,000$40,000 70 90 A B

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Expected Utility & Income As A Point on AB Line Assume the consumer sets P 1 =.20. Then if she does not purchase insurance: E(U) = 0.8 90 + 0.2 70 = 86 E(Y) = 0.8 40,000 + 0.2 20,000 = $36,000 Without insurance, the consumer has an expected loss of $4,000.

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Utility Income$20,000$40,000 90 70 A B $36,000 C 86 Expected Utility & Income As Point C on AB Line

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Certain Point on Income-Utility Curve The consumer’s expected utility for next year without insurance = 86 “utils.” Suppose that 86 “utils” also represents utility from a certain income of $35,000. –Then the consumer could pay an insurer $5,000 to insure against the probability of getting sick next year. –Paying $5,000 to insurer leaves consumer with 86 utils, which equals E(U) without insurance.

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Utility Income$20,000$40,000 90 70 A B $36,000 C 86 $35,000 D Certain Point D on Income-Utility Curve

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Price of Insurance and Loading Fee At most, the consumer is willing to pay $5,000 in insurance premiums to cover $4,000 in expected medical benefits. $1,000 loading fee price of insurance Covers –profits –administrative expenses –taxes

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To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna 3-1 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Prepared by.

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