1. Let x be the wealth of an individual. Suppose that his initial wealth is $100 and he is offered a bet: Pay $10 if a fair coin comes up Heads and get $10 if the coin comes up Tails. He is also offered the chance to enter two such bets simultaneously. So he is choosing between three lotteries; one when he rejects the bet altogether, one when he accepts the bet on one coin and one when he accepts the bet on two coins. Write the probability function and distribution functions of the three lotteries and find their means and variances. Explain whether the distributions can be ranked by First Order, or Second,Order Stochastic Dominance.
2. Consider a game where a person is given the choice between two envelopes both containing money but one envelope with double the amount in the other envelope. Upon choosing the person is given the chance to switch envelopes. The person reasons as follows: with probability half the other envelope has double the amount of my envelope and with probability half it has half the amount of my envelope. If he bases his decision on this argument will he switch his envelope? Is his argument correct?
3. (Problem thanks to Alex) Suppose that the game master chooses randomly to roll either six faced or a four faced dice (the first has faces with numbers 1 to 6 and the second faces with numbers 1 to 4). Assume that the numbers on both are equally likely and you are told that the number that came up is 2. What is the probability that the dice that was rolled was the six-faced one?
4. Consider two lotteries
L ={0.3, 0.7: $0, $200}
L’ ={0.9, 0.1: $0, $1200}
Grether and Plott (1979) found that in experiments when asked which lottery they would prefer to play many people prefer L to L’ but when asked what is the sure $ value of each lottery (the sure amount that they value the same as the lottery – the certainty equivalent) they give a higher value for L’ than for L. Explain why this is a problem. What assumptions about preferences are violated?
5. Consider an expected utility maximizer with vNM (Bernoulli) utility function u(x )= -0.1x2 + 100x
(a) The consumer evaluates lotteries on the outcome space {0,100,200}:
L ={p1 , p2 , 1-p1 - p2 : 10, 100, 200}
Determine how this consumer evaluates such a lotteries.
(b) Construct a pair of lotteries that is comparable by First Order Stochastic Dominance and one that is comparable by Increase in Risk on the outcome space {10,100,200} and determine how the consumer’s preferences rank each pair.
(c) Construct a pair of lotteries that is not comparable by either First or Second
Order Stochastic Dominance and explain how this consumer compares the two.