Dr. Donna Feir

Economics 313

Problem Set 5

Uncertainty

1. Calculate the expected values of each of the gambles listed below:

a.  A dice roll that pays $1,200 if you roll a 1, $300 if you roll a 2 or a 3, and nothing if you roll anything else.

b.  A lottery ticket that pays $1,000,000 with a 1% chance, $50,000 with a 2% chance, and nothing with a 97% chance.

c.  You own a home, for which you paid $300,000. With a 5% chance the house will appreciate in value by 20%, with a 20% chance the house will appreciate in value by 10%, with a 65% chance the house value will remain constant, and with 10% chance the house depreciate in value by 10%.

2. Consider the following monetary payoffs for two investments, A and B, and the associated probabilities of earning these payoffs.

Investment A
Probability / Payoff
.1 / 0
.1 / 40
.2 / 50
.4 / 60
.2 / 100

Investment B gives a monetary payoff of $49 for sure.

a.  What are the mean and variance of monetary payoffs from these two investments?

b.  Which investment has the highest expected monetary payoff?

c.  Albert has the following utility function for money: u = 5x.25, where x is the amount of money he receives. Which of the two investments would he prefer?

d.  How high would a sure monetary payoff have to be for Albert to make him indifferent between this amount and investment A?

e.  Britney has the following utility function for money: u = x.5, where x is the amount of money she receives. Which of the two investments would she prefer?

f.  How high would a sure monetary payoff have to be for Britney to make her indifferent between this amount and investment A?

3. (Risk preferences) Suppose a person faces the following gamble: with probability .25 she receives $100 and with probability .75 she receives $200. Look/complete the graph below to answer the following questions:

a.  What in the graph below tells you about the person’s attitude towards risk?

b.  What is the person’s expected utility of the gamble?

c.  Indicate the person’s certainty equivalent in the graph!

d.  If the person would instead of the gamble receive an amount for sure equal to the expected monetary payoff of the gamble what would be the person’s expected utility?

4. Clarice is an expected utility maximizer and her utility function over money is given by u = x ½. Clarice’s friend, Hannibal, has offered to bet with her $1,000 on the outcome of the toss of a coin. That is, if the coin comes up heads, Clarice must pay Hannibal $1,000 and if the coin comes up tail, Hannibal must pay Clarice $1,000. The coin is a fair coin, so that the probability of heads and the probability of tails are both ½. Clarice has $10,000 and is trying to figure out whether she should take the bet. Note that if Clarice accepts the bet and heads comes up, she will have 10,000-1,000=9,000.

a.  If Clarice accepts the bet, then if tails comes up, she will have how much money?

b.  What is Clarice’s expected utility if she accepts the bet?

c.  What is Clarice’s expected utility if she does not accept the bet?

d.  Does Clarice take the bet? Explain why or why not Clarice takes the bet.

e.  Clarice later asks herself, “If I make a bet where I lose all my money, that is all my $10,000 if the coin comes up heads, what is the smallest amount that I would have to win in the event of tails in order to make the bet a good one for me to take?” Find the answer to Clarice’s question.

5. Suppose that a consumer has the utility of wealth function Uw=w2. This consumer faces a risky gamble that pays $100 with chance 3/5 and $200 with chance 2/5.

a.  Calculate the expected value of the gamble.

b.  Calculate the utility that this consumer would attain if he were to receive with certainty the amount you calculated in part a). That is, what is the utility of the expected value of the gamble?

c.  Now calculate the expected utility of the gamble.

d.  Compare your answer in part b) to your answer in part c) and use this comparison to draw a conclusion about this consumer’s risk preferences.

e.  Calculate the certainty equivalent of the gamble.

f.  Calculate the risk premium of the gamble.

g.  Draw a diagram illustrating your answers to part a) through f).

6. (Insurance) Jane owns a house worth $100,000. She cares only about her wealth, which consists entirely of the house. In any given year, there is a 20% chance that the house will burn down. If it does, its scrap value will be $30,000. Jane’s utility function is U=x ½.

a.  Draw Jane’s utility function.

b.  Is Jane risk-averse or risk preferring?

c.  What is the expected monetary value of Jane’s gamble?

d.  How much would Jane at most be willing to pay to fully insure her house against being destroyed by fire?

e.  Homer is the president of an insurance company. He is risk-neutral and has a utility function of the following type: U = x. Between what two prices could a beneficial insurance contract be made by Jane and Homer?

7. Suppose that a consumer owns two assets: a own a house valued at $200,000 and $100,000 in a money market fund. With probability 10% her house will be destroyed by a fire, leaving her only with the money market fund (assume the money market fund has zero risk). Her utility of wealth function is given by V(w) = ln w.

a.  Calculate the certainty equivalent and the risk premium of the gamble faced by this consumer.

Now suppose that this consumer can purchase as much or as little insurance as she wishes at a per dollar premium of $0.20.

b.  Draw a diagram illustrating this consumer’s insurance budget line. Write down an equation for this budget line.

c.  After this consumer maximizes her expected utility, what will be her contingent consumption bundle?

d.  Given your answer to part b), how much insurance coverage will she purchase?

e.  Draw a diagram illustrating your answers to parts b) and c).

Now suppose that the price of insurance rises; now each dollar’s worth of insurance costs $0.25.

f.  After the consumer maximizes her expected utility, what will be her new state-contingent consumption bundle after choosing her level of insurance coverage? How much coverage will she buy? Draw a new diagram illustrating this new choice.

8. Suppose that Joon and Yoosoon both have a good paying job that pays $ 120.000 a year to each of them. They both have a risk of getting laid off of 5%. Yoosoon’s utility function over money is U= x ½, and Joon’s utility function over money is U= x ¼.

a.  Without risk pooling, what is the mean and the variance of the “job gamble”?

b.  Find the expected utility of the “job gamble” for each of them without risk pooling.

Suppose Joon and Yoosoon pool their risk by agreeing to split their income equally no matter what happens.

c.  With risk pooling, what is the mean and the variance of the new “job gamble”?

d.  What is the expected utility for each of them with risk pooling?

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