Exam 11: Uncertainty-Part A

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?

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 > 0.

Billy Pigskin from your workbook has a von Neumann-Morgenstern utility function U(c)= c^1/2.If Billy is not injured this season, he will receive an income of $25 million.If he is injured, his income will be only $10,000.The probability that he will be injured is .1 and the probability that he will not be injured is .9.His expected utility is

Willy's only source of wealth is his chocolate factory.He has the utility function pc^1/2_f +(1 - p)c^1/2_nf, where p is the probability of a flood, 1 - p is the probability of no flood, and c_f and c_nf are his wealth contingent on a flood and on no flood, respectively.The probability of a flood is p = 1/6.The value of Willy's factory is $500,000 if there is no flood and $0 if there is a flood.Willy can buy insurance where if he buys $x worth of insurance, he must pay the insurance company $2x/17 whether there is a flood or not but he gets back $x from the company if there is a flood.Willy should buy

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?

If the price of insurance goes up, people will become less risk averse.

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

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

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

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?

A consumer has a von Neumann-Morgenstern utility function of the form U(c_A, c_B, p_A, p_B)= p_Av(c_A)+ p_Bv(c_B), where p_A and p_B are the probabilities of events A and B and where c_A and c_B are consumptions contingent on events A and B respectively.This consumer must be a risk lover if v is an increasing function.

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 $1,800.He plans to bet on a boxing match between Sullivan and Flanagan.He finds that he can buy coupons for $9 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 $1 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?

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?

Billy Pigskin from your workbook has a von Neumann-Morgenstern utility function U(c)= c^1/2.If Billy is not injured this season, he will receive an income of $16 million.If he is injured, his income will be only $10,000.The probability that he will be injured is .1 and the probability that he will not be injured is .9.His expected utility is

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?

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.

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?

Clancy has $1,200.He plans to bet on a boxing match between Sullivan and Flanagan.For $4, he can buy a coupon that pays $10 if Sullivan wins and nothing otherwise.For $6 he can buy a coupon that will pay $10 if Flanagan wins and nothing otherwise.Clancy doesn't agree with these odds.He thinks that the two fighters each have a probability of 1/2 of winning.If he is an expected utility maximizer who tries to maximize the expected value of lnW, where lnW is the natural log of his wealth, it would be rational for him to buy

Harley's current wealth is $600, but there is a .25 probability that he will lose $100.Harley is risk neutral.He has an opportunity to buy insurance that would restore his $100 if he lost it.