Exam 12:Uncertainty-Part A

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

There are two events,1 and 2.The probability of event 1 is p and the probability of event 2 is 1 - p.Sally Kink is an expected utility maximizer with a utility function is pu(c₁)+ (1 - p)u(c₂),where for any number x,u(x)= 2x if x $\lt$ 1,000 and u(x)=1,000 + x if x is greater than or equal to 1,000.

Oliver takes his wealth of $1,000 to a casino.He can bet as much as he likes on the toss of a coin but the "house" takes a cut.If Oliver bets $x on heads,then if heads comes up,he gets $.8x and,if tails comes up,he pays $x.Similarly if he bets $x on tails and if tails comes up,he wins $.8x and,if heads comes up,he pays $x.Draw a graph with dollars contingent on heads and dollars contingent on tails on the two axes.Show Oliver's budget constraint.Oliver is an expected utility maximizer with the utility function U(h,t)=1/2h²+1/2t²,where h is his wealth if heads comes up and t is his wealth if tails comes up.Draw the highest indifference curve that Oliver can reach with his budget.What bets if any does he make?

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.

Tom Cruiser's car is worth $100,000.But Tom is careless and leaves the top down and the keys in the ignition.Consequently his car will be stolen with probability .5.If it is stolen,he will never get it back.Tom has $100,000 in other wealth and his von Neumann-Morgenstern utility function for wealth is u(w)= ln(w).Suppose that Tom can buy $K worth of insurance at a price of $.6K.How much insurance will Tom buy?

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

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?

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

Linus Piecewise is an expected utility maximizer.There are two events,H and T,which each have probability 1/2.Linus's preferences over lotteries in which his wealth is h if event H happens and t if event T happens are representable by the utility function U(h,t)= u(h)/2 + u(t)/2.The function u takes the following form.For any x,u(x)= x if x $\lt$ 100 and u(x)= 100 + x/2 if x is greater than or equal to 100.Draw a graph showing the indifference curves for Linus that pass through a.the point (50,0) b.the point (50,100) c.the point (100,100) d.the point (150,100)

An expected utility maximizer's preferences between two bundles contingent on event 1 happening must be independent of what he will get if event 2 happens.

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.

Portia has waited a long time for her ship to come in,and she has concluded that it will arrive today with probability 1/4.If it does come,she will receive $16.If it doesn't come in today,it never will and she will have zero wealth.She has a von Neumann-Morgenstern utility function equal to the square root of her total income.What is the minimum price at which she would sell the rights to her ship?

Sally Kink is an expected utility maximizer with utility function pu(c₁)+ (1-p)u(c₂),where for any x $\lt$ $3,000,u(x)= 2x,and for x greater than or equal to $3,000,u(x= 3,0001+ x.

If Paul is risk loving and his basketball team has a probability of .5 of winning,then Paul would rather bet $10 on his team than $100.(When Paul bets X,he wins X if his team wins and loses X if his team loses. )

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 = 1/4.The value of Willy's factory is $300,000 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 $5x/18 whether there is a flood or not but he gets back $x from the company if there is a flood.Willy should 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

Oskar's preferences over gambles in which the probability of events 1 and 2 are both 1/2 can be represented by the von Neuman-Morgenstern utility function .5y^.5₁ +.5y^.5₂,where y₁ is his consumption if event 1 happens and y₂ is his consumption if event 2 happens.A gamble that allows him a consumption of $9 if event 1 happens and $25 if event 2 happens is exactly as good for Oskar as being sure to have an income of

After graduating,Sallie Handshake's best job offer will either be with a Big-8 accounting firm for $160,000 a year or as a State Farm agent in Grand Rapids,Michigan,for $40,000 a year.She can increase the probability of the former outcome by studying more,but such studying has its costs.If S represents her amount of studying (where S= 0 is no study and S = 1 is all-out effort),her probability of getting the job with a Big-8 firm just equals S.Her utility depends on how hard she studies and her subsequent annual income Y.She tries to maximize the expected value of the von Neuman-Morgenstern utility function U(S,Y)= Y^1/2 - 400S².If she chooses S to maximize her expected utility,how much will she study?

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