Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Slide 58Slide 59Slide 60Slide 61Slide 62Slide 63Slide 64Slide 65Slide 66Slide 67Slide 68Slide 69Slide 70Slide 71Slide 72Slide 73Slide 74Slide 75Slide 76Slide 77Slide 78Slide 79Slide 80Slide 81Group-Clicker Question (P): What does the decision-maker want?Group-Clicker Question (P): What can the decision-maker have?Slide 84Slide 85Slide 86Slide 87Slide 88Slide 89Slide 90Slide 91Slide 92Slide 93Slide 94Slide 95Slide 96Slide 97Group-Clicker Question (1 pt): An individual’s vNM utility over money is u(x)=x1/2 where x is the individual’s change in wealth. Which lottery will she choose?Lottery 1: With a 20% chance, receive $100. With a 80% chance, receive $25.Lottery 2: With a 90% chance, receive $36. With a 10% chance, receive $16. 0%0%ECON 410 Preferences Over Risk“History has not dealt kindly with the aftermath of protracted periods of low risk premiums.” –Alan Greenspan1.Lottery 12.Lottery 22Th 3/20: HW5F 3/21 (7:20pm): NCAA Round 1Th 3/27: Midterm 2Th 3/20: HW5F 3/21 (7:20pm): NCAA Round 1Th 3/27: Midterm 2Class 17 - Preferences Over Risk3Class 17 - Preferences Over RiskExpected Utility TheoremIf an individual’s preferences over lotteries are complete, transitive, continuous, and satisfy the independence axiom, then(p1, p2,…pn) (q1, q2,…qn) if and only ifp1u1+p2u2+…+pnun >= q1u1+q2u2+…+qnun where the ui’s are known as von Neumann-Morgenstern (vNM) utilities and can be interpreted as the cardinal utility received from good i.�4Class 17 - Preferences Over RiskExpected Utility Theorem1. Given probabilities and vNM utilities, which lottery is preferred?2. Given probabilities and a given lottery preference, how high/low must vNM utilities be to be consistent with that preference?3. Given vNM utilities and a given lottery preference, how high/low must probabilities be to be consistent with that preference?5Class 17 - Preferences Over RiskWorking with the Expected Utility TheoremFinancial UncertaintyDecision-Making Under Uncertainty6Class 17 - Preferences Over RiskvNM Utility:2Things:vNM Utility:147Class 17 - Preferences Over RiskvNM Utility:2Things:vNM Utility:14$1$108Class 17 - Preferences Over RiskvNM Utility FunctionA function that gives the vNM utility for every amount of a specific (oftentimes cash) prize.9Class 17 - Preferences Over Risku(x)=2x1/2Example vNM Utility Function:Group-Clicker Question (P): An individual is considering purchasing car insurance. He currently has $100. If he purchases insurance, with a 100% probability he will lose $36 (and any accidents will not cost him extra). If he does not purchase insurance, with a 10% probability he will get in an accident and lose $64. With a 90% probability he will not get in an accident and he will not lose anything. His vNM utility function over final wealth is u(x)=x1/2Assuming it is not required by law, should he purchase insurance?1. Yes indeed2. No11Things:Lottery: .10 .90vNM Utility:(36)1/2=6 (100)1/2=10Expected Utility of the Non-Insurance Lottery = Class 17 - Preferences Over Risk$36 $100u(x)=x1/21 1 2 2( ) ( )EU pu x p u x= +1/2 1/20.1( ) 036 100.9( ) 9.6EU = =+12Things:Lottery: 1.0vNM Utility:(64)1/2=8Expected Utility of the Insurance Lottery = Class 17 - Preferences Over Risk$64u(x)=x1/21 1 2 2( ) ( )EU pu x p u x= +1/261.0( )4 8EU = =13Expected Utility of the Non-Insurance Lottery = Class 17 - Preferences Over RiskExpected Utility of the Insurance Lottery = �1/2 1/20.1( ) 036 100.9( ) 9.6EU = =+1/261.0( )4 8EU = =14Class 17 - Preferences Over RiskWorking with the Expected Utility TheoremFinancial UncertaintyDecision-Making Under Uncertainty15Class 17 - Preferences Over Risk0Group-Clicker Question (P): As a baseline scenario, assume an individual does not have insurance and currently has $10000. With a 20% probability she will get sick and face medical bills of $2000. With an 80% probability she will not get sick. Her vNM utility function over final wealth is u(x)=ln(x)What is her expected utility in this scenario?1. 2. 3. 4. 0.2(10000) 0.8(8000)EU = +0.2ln(10000) 0.8ln(8000)EU = +0.8ln(10000) 0.2ln(8000)EU = +0.2ln(10000) 0.8ln(9000)EU = +Group-Clicker Question (P): Now let’s explore the decision to purchase health insurance. Again assume an individual currently has $10000. If she purchases $H of partial insurance, then if she gets sick she’ll lose $2000 due to medical bills, but will get paid back $H<$2000 from her insurance company. $H of insurance costs her 0.2*H whether she is sick or healthy (i.e. her insurance premium is 0.2*H). Assume she gets sick with a 20% probability. Her vNM utility function over final wealth is u(x)=ln(x)What is her expected utility if she purchases $H of insurance?1. 2. 3. 4. 0.2ln(10000 2000 ) 0.8ln(10000)EU H= - + +0.2ln(10000 2000 0.2 ) 0.8ln(10000)EU H H= - + - +0.2ln(10000 2000 0.2 ) 0.8ln(10000 0.2 )EU H H H= - + - + -0.2ln(10000 2000 ) 0.8ln(10000 0.2 )EU H H= - + + -18Things:Lottery: .20 .80vNM Utility:ln(10000-2000+H-0.2H)ln(10000-0.2H)Expected Utility of the Insurance Lottery = Class 17 - Preferences Over Risk10000u(x)=ln(x)Sick: Healthy:+H-2000 -0.2H10000 -0.2H1 1 2 2( ) ( )EU pu x p u x= +0.2ln(10000 2000 0.2 ) 0.8ln(10000 0.2 )EU H H H= - + - + -Group-Clicker Question (P): An individual is considering purchasing health insurance. She currently has $10000. If she purchases $H of partial insurance, then if she gets sick she’ll lose $2000 due to medical bills, but will get paid back $H<$2000 from her insurance company. $H of insurance costs her 0.2*H whether she is sick or healthy (i.e. her insurance premium is 0.2*H). Assume she gets sick with a 20% probability. Her vNM utility function over final wealth is u(x)=ln(x)We determined her Expected Utility equalsIf we wanted to figure out how much health insurance she will purchase, what should we do?1. Maximize her expected utility with respect to H.2. Maximize her expected utility subject to a budget constraint.3. Maximize her utility if she’s sick. Then maximize her utility if she’s healthy. Compare the results. 4. None of the