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## Choice Under Uncertainty

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**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**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?**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.**Which of the following gambles will you take?**½*150+½*-1 ½*300+½*-150 ½*25000+½*-10000 =150-75=$75 =12500-5000= $7500 =75-0.5=$74.50 What influences your decision to take the gamble? Expected value = EV =(probability of event 1)*(payoff of event 1)+ (probability of event 2)*(payoff of event2)**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**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**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(M0+payoff of event) • +(probability of event 2)* U(M0+payoff of event 2) • M is income • M0 is your initial income!**Risk Averse**• Defining Characteristic • Prefers certain income over uncertain income**Risk Averse Example:**√0 =0 1-0=1 √1 =1 1.41-1=0.41 √2 =1.41 • Peter with U=√M could be many different formulas, this is one representation • What is happening to U? • Increasing • What is happening to MU? • Decreasing • Each dollar gives less satisfaction than the one before it. √9 =3 √16 =4**Risk Averse**• Defining Characteristic • Prefers certain income over uncertain income • Decreasing MU • In other words, U increases at a decreasing rate**Risk Averse Example:**√0 =0 1-0=1 √1 =1 1.41-1=0.41 √2 =1.41 How would you describe Peter’s feelings about winning vs. losing? He hates losing more than he loves winning. √9 =3 What is Peter’s U at M=9? 3 By how much does Peter’s utility increase if M increases by 7? 4-3=1 By how much does Peter’s utility decrease if M decreases by 7? 3-1.41=1.59 √16 =4**Risk Seeker**• Defining Characteristic • Prefers uncertain income over certain income**Risk Seeker Example:**02 =0 1-0=1 12 =1 4-1=3 22 =4 • Spidey with U=M2 could be many different formulas, this is one representation • What is happening to U? • Increasing • What is happening to MU? • Increasing • Each dollar gives more satisfaction than the one before it. 92 =81 162 =256**Risk Seeker**• Defining Characteristic • Prefers certain income over uncertain income • Increasing MU • In other words, U increases at an increasing rate**Risk Seeker Example:**02 =0 1-0=1 12 =1 4-1=3 22 =4 How would you describe Spidey’s feelings about winning vs. losing? He loves winning more than he hates losing. 92 =81 What is Spidey’s U at M=9? 81 256-81= 175 By how much does Spidey’s utility increase if M increases by 7? 81-4=77 By how much does Spidey’s utility decrease if M decreases by 7? 162 =256**Risk Neutral**• Defining Characteristic • Indifferent between uncertain income and certain income**Risk Neutral Example:**0 =0 1-0=1 1 =1 2-1=1 2 =2 • Jane with U=M could be many different formulas, this is one representation • What is happening to U? • Increasing • What is happening to MU? • Constant • Each dollar gives the same additional satisfaction as the one before it. 9 =9 16 =16**Risk Neutral**• Defining Characteristic • Indifferent between uncertain income and certain income • Constant MU • In other words, U increases at a constant rate**Risk Neutral Example:**0 =0 1-0=1 1 =1 2-1=1 2 =2 How would you describe Jane’s feelings about winning vs. losing? She loves winning as much as she hates losing. 9 =9 What is Jane’s U at M=9? 9 16-9= 7 By how much does Jane’s utility increase if M increases by 7? 9-2=7 By how much does Jane’s utility decrease if M decreases by 7? 16 =16**Summary**increasing constant decreasing**Shape of U**Below = concave Above = convex On = linear Chord – line connecting two points on U**Summary**increasing constant decreasing concave convex linear (.5)162+ (.5)22 =130 (.5)√16+ (.5)√2 =2.7 (.5)16+ (.5)2 =9 >81, Yes <3, NO =9, indifferent EUgamble Uno gamble M0=$9 Coin toss to win or lose $7**Intuition check…**• Why won’t 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.**Gambles**1/4 ½* ½ = ¼=.25 1/4 1/4 • 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? The probability of 2 independent events is the product of the probabilities of each event. 1/2 T H 1/2 T H H 1/2 1/2 T 1/2 1/2**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? • ¼ *(20)+ ¼ *(9) + ¼ *(-7)+ ¼*(-16)= • 5+2.25-1.75-4= • 1.5 • Fair? • No, more than fair! Yes! Would a risk seeker take this gamble? Yes! Would a risk neutral take this gamble? Would a risk averse take this gamble?**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? • Uno gamble = √M • = √16 • = 4**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 = ¼*√(16+20)+ ¼*√(16+9)+ ¼*√(16-7)+¼*√(16-16) • EU = ¼*√(36)+ ¼*√(25)+ ¼*√(9)+¼*√(0) • EU = ¼*6+ ¼*5+ ¼*3+¼*0 • EU = 1.5+1.25+0.75+0 • EU = 3.5**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. • Uno gamble=4 • EUgamble = 3.5 • What do you do? • Uno gamble>EUgamble • Therefore, don’t take the gamble!**What is insurance?**• Pay a premium in order to avoid risk and • Smooth consumption over all possible outcomes • Magahee**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 women’s 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.**Mia’s utility**• If M=0, U= • √0=0 • If M=10000, U= • √10000=100 • If M=1000000, U= • √1000000=1000 10000**Mia’s 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**Mia’s utility**U=√M • Utility if income is certain! • Risk averse? • Yes**Mia’s utility**U=√M Unot injured • U if not injured? • √1000000=1000 • Label her income and utility if she is not injured. • Label her income and utility if she is injured. • √10000=100 Uinjured 10000 Minjured M not injured**What is Mia’s expected Utility?**• No injury: M = $1,000,000 • Injury: M = $10,000 • Probability of injury = 10 percent = 1/10=0.1 • Probability of NO injury = • 90 percent = 9/10=0.9 • E(U) = • 9/10*√(1000000)+1/10* √(10000)= • 9/10*1000+1/10*100= • 900+10 = 910**What is Mia’s expected Income?**• No injury: M = $1,000,000 • Injury: M = $10,000 • Probability of injury = 10% = 1/10=0.1 • Probability of NO injury = • 90% = 9/10=0.9 • E(M) = • 9/10*(1000000)+1/10* (10000)= • 900000+1000 = 901,000**Mia’s utility**U=√M Unot injured • Label her E(M) and E(U). • Is her E(U) certain? • No, therefore, not on U=√M line E(U)=910 E(U) Uinjured 10000 E(M)=901000 Minjured Mnot injured**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**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)**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**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**Mia’s utility**U=√M U = 910 Unot injured • What certain income gives her the same U as the risky income? • 1,000,000-171,900 • $828,100 E(U)=910 E(U) Uinjured 10000 E(M)=901000 828,100 Minjured Mnot injured**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.**Leah’s utility**• If M=0, U= • √0=0 • If M=10000, U= • √10000=100 • If M=1000000, U= • √1000000=1000 10000**Leah’s 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**Leah’s utility**U=√M Unot injured • U if not injured? • √1000000=1000 • Label her income and utility if she is not injured. • Label her income and utility if she is injured. • √10000=100 Uinjured 10000 Minjured M not injured**What is Leah’s expected Utility?**• No injury: M = $1,000,000 • Injury: M = $10,000 • Probability of injury = 50 % =0.5 • Probability of NO injury = • 0.5 • E(U) = • 1/2*√(1000000)+1/2*√(10000)= • 550**What is Leah’s expected income?**• No injury: M = $1,000,000 • Injury: M = $10,000 • Probability of injury = 50% = 0.5 • Probability of NO injury = 0.5 • E(M) = • 1/2*(1000000)+1/2* (10000)= • 500000+5000 = 55,000**Leah’s utility**U=√M Unot injured • Label her E(M) and E(U). E(U) E(U)=550 Uinjured 10000 E(M)=550,000 Minjured Mnot injured**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?