Slide 1

Choice Under Uncertainty Introduction to uncertainty Law of large Numbers Expected Value Fair Gamble Von-Neumann Morgenstern Utility Expected Utility Model Risk Averse Risk Lover Risk Neutral Applications Gambles Insurance paying to avoid uncertainty Adverse Selection Full disclosure/Unraveling Slide 2 Introduction to uncertainty What is the probability that if I toss a coin in the air that it will come up heads? 50% Does that mean that if I toss it up 2 times, one will be heads and one will be tails? Slide 3 Introduction to uncertainty Law of large numbers - a statistical law that says that if an event happens independently (one event is not related to the next) with probability p every time the event occurs, the proportion of cases in which the event occurs approaches p as the number of events increases. Slide 4 Which of the following gambles will you take? Gamble 1: H: $150 T: -$1 Gamble 2: H: $300 T: -$150 Gamble 3: H: $25,000 T: -$10,000 Takers EV Expected value = EV =(probability of event 1)*(payoff of event 1)+ (probability of event 2)*(payoff of event2) What influences your decision to take the gamble? *150+*-1 =75-0.5=$74.50 *300+*-150 =150-75=$75 * 25000 +*- 10000 =12500-5000= $7500 Slide 5 Fair Gamble a gamble whose expected value is 0 or, a gamble where the expected income from gamble = expected income without the gamble Ex: Heads you win $7, tails you lose $7 EV = 1/2*$7+1/2*(-$7) = $3.5+-$3.5 = $0 Slide 6 Von-Neumann Morgenstern Utility Expected Utility Model Utility and Marginal Utility Relates your income to your utility/satisfaction Utility cardinal or numerical representation of the amount of satisfaction - each indifference curve represented a different level of utility or satisfaction Marginal Utility - additional satisfaction from one more unit of income Slide 7 Von-Neumann Morgenstern Utility Expected Utility Model: Prediction we will take a gamble only if the expected utility of the gamble exceeds the expected utility without the gamble. EU = Expected Utility = (probability of event 1)*U(M 0 +payoff of event) +(probability of event 2)* U(M 0 +payoff of event 2) M is income M 0 is your initial income! Slide 8 Risk Averse Defining Characteristic Prefers certain income over uncertain income Slide 9 Risk Averse Example: Peter with U=M could be many different formulas, this is one representation MUMU 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0=0 1=1 1-0=1 2 =1.41 1.41-1=0.41 9 =3 16=4 What is happening to U? Increasing What is happening to MU? Decreasing Each dollar gives less satisfaction than the one before it. Slide 10 Risk Averse Defining Characteristic Prefers certain income over uncertain income Decreasing MU In other words, U increases at a decreasing rate Slide 11 Risk Averse Example: MUMU 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0=0 1=1 1-0=1 2 =1.41 1.41-1=0.41 9 =3 16=4 What is Peters U at M=9? 3 By how much does Peters utility increase if M increases by 7?4-3=1 By how much does Peters utility decrease if M decreases by 7? 3-1.41=1.59 How would you describe Peters feelings about winning vs. losing? He hates losing more than he loves winning. Slide 12 Risk Seeker Defining Characteristic Prefers uncertain income over certain income Slide 13 Risk Seeker Example: Spidey with U=M 2 could be many different formulas, this is one representation MUMU 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0202 =0 1212 =1 1-0=1 2 =4 4-1=3 9292 =81 16 2 =256 What is happening to U? Increasing What is happening to MU? Increasing Each dollar gives more satisfaction than the one before it. Slide 14 Risk Seeker Defining Characteristic Prefers certain income over uncertain income Increasing MU In other words, U increases at an increasing rate Slide 15 Risk Seeker Example: MUMU 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0202 =0 1212 =1 1-0=1 2 =4 4-1=3 9292 =81 16 2 =256 What is Spideys U at M=9?81 By how much does Spideys utility increase if M increases by 7? 256-81= 175 By how much does Spideys utility decrease if M decreases by 7? 81-4=77 How would you describe Spideys feelings about winning vs. losing? He loves winning more than he hates losing. Slide 16 Risk Neutral Defining Characteristic Indifferent between uncertain income and certain income Slide 17 Risk Neutral Example: Jane with U=M could be many different formulas, this is one representation MUMU 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0=0 1=1 1-0=1 2 =2 2-1=1 9 =9 16=16 What is happening to U? Increasing What is happening to MU? Constant Each dollar gives the same additional satisfaction as the one before it. Slide 18 Risk Neutral Defining Characteristic Indifferent between uncertain income and certain income Constant MU In other words, U increases at a constant rate Slide 19 Risk Neutral Example: MUMU 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0=0 1=1 1-0=1 2 =2 2-1=1 9 =9 16=16 What is Janes U at M=9?9 By how much does Janes utility increase if M increases by 7? 16-9= 7 By how much does Janes utility decrease if M decreases by 7? 9-2=7 How would you describe Janes feelings about winning vs. losing? She loves winning as much as she hates losing. Slide 20 Summary Risk AverseRisk Seeker Risk Neutral MU Shape of U Fair Gamble decreasing increasingconstant Slide 21 Shape of U Chord line connecting two points on U Below = concaveAbove = convexOn = linear Slide 22 Summary Risk AverseRisk Seeker Risk Neutral MU Shape of U Fair Gamble decreasing increasingconstant concave convexlinear M 0 =$9 Coin toss to win or lose $7 (.5)16+ (.5)2 =2.7 81, Yes (.5)16+ (.5)2 =9 =9, indifferent EU gamble U no gamble Slide 23 Intuition check Why wont Peter take a gamble that, on average, his income is no different than without the gamble? Dislikes losing more than likes winning. The loss in utility from the possibility of losing is greater than the increase in utility from the possibility of winning. Slide 24 Gambles Suppose a fair coin is flipped twice and the following payoffs are assigned to each of the 4 possible outcomes: H-H: win 20; H-T: win 9; T-H: lose 7; T-T: lose 16 What is the expected value of the gamble? First, what is the probability of each event? H 1/2T H T H T The probability of 2 independent events is the product of the probabilities of each event. * = =.251/4 Slide 25 Problem 1: Suppose a fair coin is flipped twice and the following payoffs are assigned to each of the 4 possible outcomes: H-H: win 20; H-T: win 9; T-H: lose 7; T-T: lose 16 What is the expected value of the gamble? *(20)+ *(9) + *(-7)+ *(-16)= 5+2.25-1.75-4= 1.5 Fair? No, more than fair! Would a risk seeker take this gamble? Yes! Would a risk neutral take this gamble? Yes! Would a risk averse take this gamble? Slide 26 Gambles Suppose a fair coin is flipped twice and the following payoffs are assigned to each of the 4 possible outcomes: H-H: win 20; H-T: win 9; T-H: lose 7; T-T: lose 16 If your initial income is $16 and your VNM utility function is U= M, will you take the gamble? What is your utility without the gamble? U no gamble = M = 16 = 4 Slide 27 Gambles Suppose a fair coin is flipped twice and the following payoffs are assigned to each of the 4 possible outcomes: H-H: win 20; H-T: win 9; T-H: lose 7; T-T: lose 16 If your initial income is $16 and your VNM utility function is U= M, will you take the gamble? What is your EXPECTED utility with the gamble? EU = *(16+20)+ *(16+9)+ *(16-7)+*(16-16) EU = *(36)+ *(25)+ *(9)+*(0) EU = *6+ *5+ *3+*0 EU = 1.5+1.25+0.75+0 EU = 3.5 Slide 28 Von-Neumann Morgenstern Utility Expected Utility Prediction - we will take a gamble only if the expected utility of the gamble exceeds the expected utility without the gamble. U no gamble =4 EU gamble = 3.5 What do you do? U no gamble >EU gamble Therefore, dont take the gamble! Slide 29 What is insurance? Pay a premium in order to avoid risk and Smooth consumption over all possible outcomes Magahee Slide 30 Example: Mia Dribble has a utility function of U=M. In addition, Mia is a basketball star starting her senior year. If she makes it through her senior year without a serious injury, she will receive a $1,000,000 contract for playing in the new professional womens basketball league (the $1,000,000 includes endorsements). If she injures herself, she will receive a $10,000 contract for selling concessions at the basketball arena. There is a 10 percent chance that Mia will injure herself badly enough to end her career. Slide 31 Mias utility If M=0, U= 0=0 If M=10000, U= 10000=100 If M=1000000, U= 1000000=1000 10000 Slide 32 Mias utility If M=250000, U= 250000=500 If M=640000, U= 640000=800 If M=810000, U= 810000=900 If M=1210000, U= 1210000=1100 10000 Slide 33 Mias utility Utility if income is certain! U=M Risk averse? Yes Slide 34 Mias utility U if not injured? 1000000=1000 Label her income and utility if she is not injured. U=M Label her income and utility if she is injured. 10000=100 M not injured Unot injured 10000 M injured Uinjured Slide 35 What is Mias expected Utility? No injury: M = $1,000,000 Injury: M = $10,000 Probability of injury = 10 percent = 1/10=0.1 E(U) = 9/10*(1000000)+1/10* (10000)= 9/10*1000+1/10*100= 900+10 = 910 Probability of NO injury = 90 percent = 9/10=0.9 Slide 36 What is Mias expected Income? No injury: M = $1,000,000 Injury: M = $10,000 Probability of injury = 10% = 1/10=0.1 E(M) = 9/10*(1000000)+1/10* (10000)= 900000+1000 = 901,000 Probability of NO injury = 90% = 9/10=0.9 Slide 37 Mias utility Label her E(M) and E(U). Is her E(U) certain? No, therefore, not on U=M line U=M M not injured Unot injured 10000 M injured Uinjured E(U) E(M)=901000 E(U)=910 Slide 38 Remember prediction: will take a gamble only if the expected utility of the gamble exceeds the utility without the gamble. If Mia pays $p for an insurance policy that would give her $1,000,000 if she suffered a career- ending injury while in college, then she would be sure to have an income of $1,000,000-p, not matter what happened to her. What is the largest price Mia would pay for this insurance policy? What is the E(U) without insurance? 910 Slide 39 Remember prediction: will take a gamble only if the expected utility of the gamble exceeds the utility without the gamble. If Mia pays $p for an insurance policy that would give her $1,000,000 if she suffered a career- ending injury while in college, then she would be sure to have an income of $1,000,000-p, not matter what happened to her. What is the largest price Mia would pay for this insurance policy? What is the U with insurance? U = (1,000,000-p) Slide 40 Remember prediction: will take a gamble only if the expected utility of the gamble exceeds the utility without the gamble. Buy insurance if U=(1,000,000-p) > 910 = E(U) Solve Square both sides Slide 41 Remember prediction: will take a gamble only if the expected utility of the gamble exceeds the utility without the gamble. Buy insurance if U=(1,000,000-p) > 910 = E(U) Solve Square both sides Solve for p Interpret: If the premium is less than $171,000, Mia will purchase insurance Slide 42 Mias utility What certain income gives her the same U as the risky income? 1,000,000- 171,900 $828,100 U=M M not injured Unot injured 10000 M injured Uinjured E(U) E(M)=901000 E(U)=910 828,100 U = 910 Slide 43 Leah Shooter also has a utility function of U=M. Lea is also starting college and she has the same options as Mia after college. However, Leah is notoriously clumsy and knows that there is a 50 percent chance that she will injure herself badly enough to end her career. Slide 44 Leahs utility If M=0, U= 0=0 If M=10000, U= 10000=100 If M=1000000, U= 1000000=1000 10000 Slide 45 Leahs utility If M=250000, U= 250000=500 If M=640000, U= 640000=800 If M=810000, U= 810000=900 If M=1210000, U= 1210000=1100 10000 Slide 46 Leahs utility U if not injured? 1000000=1000 Label her income and utility if she is not injured. U=M Label her income and utility if she is injured. 10000=100 M not injured Unot injured 10000 M injured Uinjured Slide 47 What is Leahs expected Utility? No injury: M = $1,000,000 Injury: M = $10,000 Probability of injury = 50 % =0.5 E(U) = 1/2*(1000000)+1/2*(10000)= 550 Probability of NO injury = 0.5 Slide 48 What is Leahs expected income? No injury: M = $1,000,000 Injury: M = $10,000 Probability of injury = 50% = 0.5 E(M) = 1/2*(1000000)+1/2* (10000)= 500000+5000 = 55,000 Probability of NO injury = 0.5 Slide 49 Leahs utility Label her E(M) and E(U). U=M M not injured Unot injured 10000 M injured Uinjured E(U) E(M)=550,000 E(U)=550 Slide 50 Remember prediction: will take a gamble only if the expected utility of the gamble exceeds the utility without the gamble. What is the largest price Leah would pay for the above insurance policy? Intuition check: Will Leah be willing to pay more or less? Slide 51 Remember prediction: will take a gamble only if the expected utility of the gamble exceeds the utility without the gamble. What is the largest price Leah would pay for the above insurance policy? What is the E(U) without insurance? 550 What is the U with insurance? U = (1,000,000-p) Buy insurance if U=(1,000,000-p) > 550 = E(U) Slide 52 Remember prediction: will take a gamble only if the expected utility of the gamble exceeds the utility without the gamble. Buy insurance if U=(1,000,000-p) > 550 = E(U) Solve p < 697,500 Slide 53 Leahs utility What certain income gives her the same U as the risky income? 1,000,000- 697,500= $302,500 U=M M not injured Unot injured 10000 M injured Uinjured E(U) E(M)=550,000 E(U)=550 302,500 U = 550 Slide 54 Thea Thorough runs an insurance agency. Unfortunately, she is unable to distinguish between coordinated players and clumsy players, but she knows that half of all players are clumsy. If she insures both Lea and Mia, what is her expected value of claims/payouts (remember, she has to pay whenever either player gets injured)? Slide 55 Theas expected value of claims/payouts What does Thea have to pay if the basketball player gets injured? Difference in incomes w/ and w/o injury 1,000,000-10,000 = 990,000 Expected claim from Mia = 0.1*990000= $99,000 Expected claim from Leah= 0.5*990000= $495,000 Slide 56 Theas expected value of claims/payouts Expected claim from Mia = $99,000 Expected claim from Leah= $495,000 Theas expected value of claims = 0.5*99,000 + 0.5*495,000 =$297,000 Probability of risky player Probability of non-risky player Slide 57 Premium=$297,000 Willingness to pay: Mia: $171,900, Leah: $697,500 Suppose Thea is unable to distinguish among clutzy and non-clutzy basketball players and therefore has to change the same premium to everyone. If she sets her premium equal to the expected value of claims, will both Lea and Mia buy insurance from Thea? Only Leah will buy insurance. Mia will not because she is only willing to pay $171,900 Adverse Selection - undesirable members of a group are more likely to participate in a voluntary exchange Slide 58 What do you expect to happen in this market? Only the risky players will buy insurance. Premiums will increase The low-risk players will not be able to buy insurance. Slide 59 What is the source of the problem? Asymmetric information cannot tell how risky Is all information asymmetric? No, sex, age, health all observable (and cannot fake) Therefore, insurance companies can charge higher risk people higher rates Illegal to use certain characteristics, like race and religion Slide 60 How do insurance companies mitigate this problem? Offer different packages: 1. Deductibles the amount of medical expenditures the person has to pay before the plan starts paying benefit risky people reveal themselves by choosing low deductibles 2. Do not cover preexisting condition Slide 61 Other examples of adverse selection Slide 62 Another Adverse Selection Example Used Cars Why does your new car drop in value the minute you drive it off the lot? Slide 63 Another Adverse Selection Example used Cars First assume that there are two kinds of used cars - lemons and peaches. Lemons are worth $5,000 to consumers and peaches are worth $10,000. Assume also that demand is perfectly elastic and consumers are risk neutral. There is a demand for both kinds of cars and a supply of both kinds of cars. Is the supply of lemons or peaches higher? Peaches Lemons PP Q of Peaches Q of Lemons D10,000 5,000 D S S Q * (perfect info) Slide 64 Another Adverse Selection Example Used Cars Assume there is perfect information Buyers are willing to pay ___________ for a lemon and ___________ for a peach. Peaches Lemons PP Q of Peaches Q of Lemons D10,000 5,000 D S S 10,000 Q * (perfect info) Slide 65 Another Adverse Selection Example Used Cars Case 1: Assume that buyers think that there is a 50% chance that the car is a peach. What is their expected value of any car they see? 0.50*$10000+0.50*$5000 =$7500 If they are risk neutral, how much are they willing to pay for the car? $7500, indifferent between certain and uncertain income Slide 66 Another Adverse Selection Example Used Cars Case 2: Will the ratio of peaches to lemons stay at 50/50? If not, what will happen to the expected value? Demand for peaches falls, demand for lemons rises Peaches Lemons PP Q of Peaches Q of Lemons D10,000 5,000 D S S 7,500 D(50/50) 7,500 D(50/50) Q * (p.i.) Ratio shifts to fewer peaches and more lemons Expected value falls as beliefs about # of lemons increases More peaches drop out. Q * (new) Slide 67 Another Adverse Selection Example Used Cars Ultimately In the extreme case, no peaches, all lemons Peaches Lemons PP Q of Peaches Q of Lemons D10,000 5,000 D S S 7,500 D(50/50) 7,500 D(50/50) Q * (p.i.) Q * (new) Slide 68 What could you do to signal to someone that your car is not a lemon? Pay for a mechanic to inspect it. Offer a warranty on the car. Generally, offer something that is costly to fake. Slide 69 Role for the Government? Does the asymmetric info mean the govt can/should be involved? http://www.oag.state.ny.us/consumer/cars/ qa.html http://www.oag.state.ny.us/consumer/cars/ qa.html (look up the Lemon Law for MI) Slide 70 Other examples of signaling Brand names company advertising Dividends versus Capital gains Football players How can you signal how good of an employee you will be? Slide 71 III. Full disclosure/Unraveling Youre on a job interview and the interviewer knows what the distribution of GPAs are for MSU graduates: Expected/Average grade for everyone: 0.2*1+0.3*2+0.3*3+0.2*4 =2.5 The job counselor at MSU advises anyone who had a B average to volunteer their GPA. Is this a stable outcome? Per- cent 0.20.3 0.2 GPA 1.02.03.04.0 3.0 or better What does the potential employer believe about the people who stay quiet? They know their GPA is below a 3.0, but how far below? Slide 72 III. Full disclosure/Unraveling Employers know their GPA is below a 3.0, but how far below? Expected/Average grade for those who dont reveal: Percent GPA0.10.2 0.4*1+0.6*2 =1.6 Therefore, those w/ a 2.0 should revealunravels so that there is full disclosure. Those who dont reveal: Original percent divided by what share of students remain 0.20/.50 =0.40 0.30/.50 =0.60 Intuitively, those who are above the expected average dont want employers to think they are average, so they disclose! Slide 73 Intuition check What does this full disclosure principle say about whether only peaches will provide a signal of their value? Slide 74 Voluntary disclosure and SAT scores Institutional Details Voluntary disclosure question Data Results Slide 75 Institutional Details Increasing # of schools are adopting policies where submitting your SAT scores are optional I.e., students can submit high school G.P.A., extracurricular activities etc, and exclude standardized test score on their application School will judge based on submitted material Slide 76 Voluntary disclosure question If it is fairly costless to reveal your scores, all by the students with the lowest scores should reveal to avoid being considered the average of those who dont reveal. Is it only the students with very low SAT scores that dont reveal? Slide 77 Data Liberal arts college 1800 students Mean SAT score > 1300 (out of 1600) 1020 is the mean SAT score of those who take it