hw2

ECON2003Microeconomic Theory

Problem Set 2

1.James is risk averse.He is offered a gamble in which with probability 1/4he will lose $1,000and with probability 3/4,he will win $500.Will he take the gamble if it is free?If he is risk lover?How much would he be willing to pay for that gamble?

2.Socrates owns just one ship.The ship is worth $200million dollars.If the ship sinks,Socrates loses $200million.The probability that it will sink is .02.Socrates total wealth including the value of the ship is $225million.He is an expected utility maximizer with utility of wealth u (w )=√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?

3.Oskar’s preferences over gambles in which the probability of events 1and 2are both 1/2can be represented by the utility function U =0.5√y 1+0.5√y 2,where y 1is his consumption if event 1happens and y 2is his consumption if event 2happens.Consider a gamble that allows him a consumption of $9if event 1happens and $25if event 2happens.This gamble is exactly as good for Oskar as being sure to have an income of how much?

4.Wilbur’s expected utility function is pc 1/21+(1−p )c 1/2

2,where p is the probability that he consumes c 1and 1−p is the probability that he consumes c 2.Wilbur is offered a choice between getting a sure payment of Z or a lottery in which he receives $2,500with probability .30and $3,600with probability .70.Wilbur prefers the gamble for which values of Z ?

5.Oliver takes his wealth of $1,000to 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 on the x axis and dollars contingent on tails on the y axis.Draw Oliver’s budget constraint.Oliver is an expected utility maximizer with the utility function U (h,t )=1/2h 2+1/2t 2,where h is his wealth if heads comes up and t is his wealth if tails comes up.Is Oliver risk averse or risk lover?Draw the indifference curves (hint:look for the equation of a circle).Which bet,if any,maximizes his utility?Answer the same questions if U (h,t )=1/2h +1/2t .1

6.Suppose that you are an expected utility maximizer and you are indifferent between

gambles g=(π,w1,1−π,w2)and g =(π1,w

1,1−π1,w

2

).Suppose now that you are

offered a choice between the following gambles:thefirst gives you g with probability

π2and g =(π3,w

1,1−π3,w

2

)with probability1−π2.The second one gives you g

with probabilityπ2and g with probability1−π2.Which one do you prefer?

2