Exam 12:Uncertainty-Part A

Sally Kink is an expected utility maximizer with utility function pu(c₁)+ (1- p)u(c₂),where for any x $\lt$ $2,000,u(x)=2x,and for x greater than or equal to $2,000,u(x)= 2,000 + x.

Buck Columbus is thinking of starting a pinball palace near a large Midwestern university.Buck is an expected utility maximizer with a von Neuman-Morgenstern utility function,U(W)=1 - (6,000/W),where W is his wealth.Buck's total wealth is $24,000.With probability .2 the palace will be a failure and he'll lose $18,000,so that his wealth will be just $6,000.With probability .8 it will succeed and his wealth will grow to $x.What is the smallest value of x that would be sufficient to make Buck want to invest in the pinball palace rather than have a wealth of $24,000 with certainty?

Yoram's expected utility function is pc^1/2₁ +(1 -p)c^1/2₂,where p is the probability that he consumes c₁ and 1 - p is the probability that he consumes c₂.Wilbur is offered a choice between getting a sure payment of $Z or a lottery in which he receives $2,500 with probability .30 and $3,600 with probability .70.Wilbur will choose the sure payment if

The certainty equivalent of a gamble is defined to be the amount of money which,if you were promised it with certainty,would be indifferent to the gamble. a.If an expected utility maximizer has a von Neuman-Morgenstern utility function U(W)=W^1/2 (where W is wealth)and if the probability of events 1 and 2 are both 1/2,write a formula for the certainty equivalent of a gamble that gives you x if event 1 happens and y if event 2 happens. b.Generalize your formula in part (a)to the case where the probability of event 1 is p and the probability of event 2 is 1 -p. c.Generalize the formula in part (a)to the case where U(W)= W ^a for a $\gt$ 0.

Gaston Gourmand loves good food.Due to an unusual ailment,he has a probability of 1/4 of losing his sense of smell,which would greatly reduce his enjoyment of food.Gaston finds an insurance company that will sell him insurance where Gaston gets $3x if he loses his smell and pays $x if he doesn't.He can also buy negative insurance,where Gaston pays $3x if he loses his sense of smell and gets $x if he doesn't.Gaston says,"Money will be only half as important to me if I lose my sense of smell." If we look at his expected utility function,we see what he means.Where c₁ is his consumption if he retains his sense of smell and c₂ is his income if he loses his sense of smell,Gaston has the expected utility function U(c₁,c₂)= 3/4 c^1/2₁ +1/8 c^1/2₂.What insurance should he buy?

Of any two gambles,no matter what their expected returns,a risk averter will choose the one with the smaller variance.

Joe's wealth is $100 and he is an expected utility maximizer with a von Neumann-Morgenstern utility function U(W)=W^1/2.Joe is afraid of oversleeping his economics exam.He figures there is only a 1 in 10 chance that he will,but if he does,it will cost him $100 in fees to the university for taking an exam late.Joe's neighbor,Mary,never oversleeps.She offers to wake him one hour before the test,but he must pay her for this service.What is the most that Joe would be willing to pay for this wake-up service?

Quincy's expected utility function is pc^1/2₁+ (1 -p)c^1/2₂,where p is the probability that he consumes c₁ and 1 - p is the probability that he consumes c₂.Wilbur is offered a choice between getting a sure payment of $Z or a lottery in which he receives $3,600 with probability .60 and $12,100 with probability .40.Wilbur will choose the sure payment if

Ronald has $18,000.But he is forced to bet it on the flip of a fair coin.If he wins he has $36,000.If he loses he has nothing.Ronald's expected utility function is .5x^.5 + .5y^.5,where x is his wealth if heads comes up and y is his wealth if tails comes up.Since he must make this bet,he is exactly as well off as if he had a perfectly safe income of

Timmy Qualm's uncle gave him a lottery ticket.With probability 1/2 the ticket will be worth $100 and with probability 1/2 it will be worthless.Let x be Timmy's wealth if the lottery ticket is a winner,and y his wealth if it is a loser.Timmy's preferences over alternative contingent commodity bundles are represented by the utility function U(x,y)=min{2x - y,2y -x}.He has no risks other than the ticket.

Every $1 invested in Safe Sox will yield $2 for sure.Each $1 invested in Wobbly Umbrellas will yield $8 with probability 1/2 and $0 with probability 1/2.An investor has $10,000 to invest in these two companies and her von Neumann-Morgenstern utility function is the expected value of the natural logarithm of the total yield on her investments.If S is the amount of money that she invests in Safe Sox and $10,000 2 S is the amount that she invests in Wobbly Umbrellas,what should S be to maximize her expected utility? (Pick the closest answer. )

Clancy has $5,000.He plans to bet on a boxing match between Sullivan and Flanagan.He finds that he can buy coupons for $5 that will pay off $10 each if Sullivan wins.He also finds in another store some coupons that will pay off $10 if Flanagan wins.The Flanagan tickets cost $5 each.Clancy believes that the two fighters each have a probability of 1/2 of winning.Clancy is a risk averter who tries to maximize the expected value of the natural log of his wealth.Which of the following strategies would maximize his expected utility?

Buck Columbus is thinking of starting a pinball palace near a large Midwestern university.Buck is an expected utility maximizer with a von Neuman-Morgenstern utility function,U(W)=1 -(3,000/W),where W is his wealth.Buck's total wealth is $12,000.With probability .2 the palace will be a failure and he'll lose $9,000,so that his wealth will be just $3,000.With probability .8 it will succeed and his wealth will grow to $x.What is the smallest value of x that would be sufficient to make Buck want to invest in the pinball palace rather than have a wealth of $12,000 with certainty?

Gary likes to gamble.Donna offers to bet him $70 on the outcome of a boat race.If Gary's boat wins,Donna would give him $70.If Gary's boat does not win,Gary would give her $70.Gary's utility function is U(c₁,c₂,p₁,p₂)=p₁c²₁ +p₂c²₁,where p₁ and p₂ are the probabilities of events 1 and 2 and where c₁ and c₂ are his consumption if events 1 and 2 occur respectively.Gary's total wealth is currently only $80 and he believes that the probability that he will win the race is .3.

Mabel and Emil were contemplating marriage.They got to talking.Mabel said that she always acted according to the expected utility hypothesis,where she tried to maximize the expected value of the log of her income.Emil said that he too was an expected utility maximizer,but he tried to maximize the expected value of the square of his income.Mabel said,"I fear we must part.Our attitudes toward risk are too different." Emil said,"Never fear,my dear,the square of income is a monotonic increasing function of the log of income,so we really have the same preferences." Who is right about whether their preferences toward risk are different?

Socrates owns just one ship.The ship is worth $200 million dollars.If the ship sinks,Socrates loses $200 million.The probability that it will sink is .02.Socrates' total wealth including the value of the ship is $225 million.He is an expected utility maximizer with von Neuman-Morgenstern utility U(W)equal to the square root of W.What is the maximum amount that Socrates would be willing to pay in order to be fully insured against the risk of losing his ship?

Prufrock is risk averse.He is offered a gamble in which with probability 1/4 he will lose $1,000 and with probability 3/4,he will win $500.

Diego has $6,400.He plans to bet on a soccer game.Team A is a favorite to win.Assume no ties can occur.For $.80 one can buy a ticket that will pay $1 if team A wins and nothing if B wins.For $.20 one can buy a ticket that pays $1 if team B wins and nothing if A wins.Diego thinks the two teams are equally likely to win.He buys tickets so as to maximize the expected value of lnW (the natural log of his wealth).After he buys his tickets,team A loses a star player and the ticket price moves to $.50 for either team.Diego buys some new tickets and sells some of his old ones.The game is then played and team A wins.How much wealth does he end up with?

If someone has strictly convex preferences between all contingent commodity bundles,then he or she must be risk averse.

Wilma is not risk averse.She is offered a chance to pay $10 for a lottery ticket that will give her a prize of $100 with probability .06,a prize of $50 with probability .1,and no prize with probability .85.If she understands the odds and makes no mistakes in calculation,she will buy the lottery ticket.