Dr. Donna Feir

Economics 313

Problem Set 5: Solutions

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.

EV = (1/6 × 1,200) + (2/6 × 300) + (3/6 × 0) = 200 + 100 + 0 = $300

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.

EV = (1/100 × 1,000,000) + (2/100 × 50,000) + (97/100 × 0) = 10,000 + 1,000 = $11,000

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%.

EV = (5/100 × 360,000) + (20/100 × 330,000) + (65/100 × 300,000) + (10/100 × 270,000)

= 18,000 + 66,000 + 195,000 + $27,000 = $306,000

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.

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

a)  Investment A: mean=.1(40)+.2(50)+.4(60)+.2(100)=58

Variance=.1(0-58)2+.1(40-58)2+.2(50-58)2+.4(60-58)2+.2(100-58)2=736

Investment B gives 49 for sure, therefore its mean is 49 and its variance is 0.

  1. Which investment has the highest expected monetary payoff?

Investment A.

  1. 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?

We calculate Albert’s expected utility of both investments:

U(A)= .1*5(40.25)+.2*5(50.25)+.4*5(60.25)+.2*5(100.25)= 12.645

U(B)=5(49.25)=13.229

Albert would choose investment B, because his expected utility is higher for investment B.

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

Albert’s expected utility from investment A needs to be the same as his utility from this sure amount of money. We already know that his expected utility from investment A is .1*5(40.25) +.2*5(50.25) +.4*5(60.25) +.2*5(100.25) = 12.645, we therefore need to find which amount for sure gives him the same utility. That is, the amount for sure, x, would give him U(x) = 5(x.25), and we now want to find when U(x) =12.645.

Solve 5(x.25)= 12.645, dividing both sides by 5, (x.25)= 12.645/5=2.529 and noting that .25 is equal to ¼, you need to take both sides to the exponent of 4 to find x: x = 2.5294, and x = 40.91.

That is, if you offer Albert approximately $40.91 for sure, he would be as well off as with investment A. This result makes sense as we have seen that if he is offered $49 for sure he would be better off with $49 for sure than with investment A, and therefore we would expect the amount for sure that makes him indifferent between this amount and investment A to be lower than $49.

  1. 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?

We calculate Britney’s expected utility of both investments:

U(A)= .1(40.5)+.2(50.5)+.4(60.5)+.2(100.5)=7.1451

U(B)=49.5=7.

Britney would choose investment A, because her expected utility is higher for investment A.

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

Britney’s expected utility from investment A needs to be the same as her utility from this sure amount of money. We already know that her expected utility from investment A is .1(40.5)+.2(50.5)+.4(60.5)+.2(100.5)=7.1451, we therefore need to find which amount for sure gives her the same utility. That is, the amount for sure, x, would give her U(x) = (x.5), and we now want to find when U(x) =7.1451.

Solve (x.5)= 7.1451, and noting that .5 is equal to 1/2, you need to take both sides to the exponent of 2 to find x: x = 7.14512, and x = 51.05.

That is, if you offer Britney approximately $51.05 for sure, she would be as well off as with investment A. This result makes sense as we have seen that if she is offered $49 for sure she would be better off with investment A, and therefore we would expect the amount for sure that makes her indifferent between this amount and investment A to be higher than $49.

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:

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

The person is risk averse. This can be seen in the graph below because the person’s utility of money is strictly concave, i.e. her marginal utility of money is decreasing.

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

It is equal to .25u(100) +.75u(200) = .25*55 + .75*90 = 81.25

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

In order to find the certainty equivalent, first draw a straight line through the points (100,55) and (200,90). Then go up at 175 (since the gamble’s expected monetary payoff is 175) until you hit this line. This is how you find the expected utility of the gamble. Holding expected utility constant, see where you hit the utility function over sure amounts of money. At this point go down to find out what the sure amount of money is that gives you the same expected utility as the gamble. This is the certainty equivalent.

  1. 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?

It’s 83.

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?

Clarice will have her $10,000 and gets an additional $1,000 if tails comes up: 10,000+1,000=11,000.

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

.5*(9,000) ½ + .5*(11,000) ½ = 99.875.

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

If she does not accept the bet, she’ll always have her $10,000. Her expected utility in this case is (10,000) ½ = 100.

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

If she does not accept the bet, she’ll always have her $10,000. Her expected utility in this case is (10,000) ½ = 100.

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.

Clarice needs to get at least the expected utility of 100 from this bet, because this is the utility she obtains from keeping her $10,000. If head comes up, she is left with zero; if tails comes up, she can keep her 10,000 and gets an additional amount x.

100 = .5*(0) ½ + .5 (10,000 + x) ½

100 = .5 (10,000 + x) ½

200 = (10,000 + x) ½

40,000 = 10,000 + x

x=30,000

Clarice would have to win at least 30,000 in the event of tails coming up in order to make the bet worthwhile.

5. Suppose that a consumer has the utility of wealth function U(w) = 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.

EV = (3/5)(100) + (2/5)(200) = 140.

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?

V(EV) = 1402 = 19,600.

c.  Now calculate the expected utility of the gamble.

EU = (3/5)(1002) + (2/5)(2002) = 6,000 + 16,000 = 22,000.

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.

EU > V(EV) ⇒the consumer is risk loving. We could also have reached this conclusion by evaluating the second derivative of V(w). V’’(w) = 2 > 0 ⇒the marginal utility of wealth is increasing in wealth ⇒the consumer is risk loving.

e.  Calculate the certainty equivalent of the gamble.

We know that the consumer gets EU of 2,000 from the gamble. The amount of money with certainty that would yield this utility is the certainty equivalent (CE) of the gamble. So we know that V(CE) = CE2 = 22,000 ⇒CE = $148.32. Note that the CE is greater than the EV, which is the opposite inequality to the case where the consumer is risk averse. Which makes sense, since we know that a risk loving consumer would prefer a risky gamble to a certain outcome that has the same EV as the risky gamble.

f.  Calculate the risk premium of the gamble.

Risk premium = EV – CE = 140 – 148.32 = - 8.32 < 0 (risk loving).

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?

Jane has a decreasing marginal utility of money. She is therefore risk-averse.

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

The expected monetary value of Jane’s gamble is .8*100,000+.2*30,000=86,000.

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

Insurance is guaranteeing Jane the same amount of money whether her house burns down or not. Therefore the maximum amount she is willing to pay for full insurance is the difference between 100,000 and the amount of money for sure that yields the same expected utility as the gamble. The expected utility of the gamble is equal to .8*(100,000½)+.2*(30,000½)=252.98+34.64=287.62. The amount for sure is therefore found by setting Jane’s expected utility from the gamble equal to her utility if she receives an amount for sure, x: x ½=287.62 and therefore x=287.622=82,725. Thus, Jane is willing to pay up to 17,275 in insurance premium.

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?

Homer is indifferent between selling insurance and not selling insurance to Jane if the premium is fair, that is if the premium is equal to the expected damages. This is equal to .2*70,000=14,000. An insurance premium between 14,000 and 17,275 would make both Jane and Homer better off.

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 Vw=ln⁡(w).

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

The consumer’s expected utility function is EU = (0.1 X ln wB) + (0.9 X ln wG). At her contingent endowment, her expected utility is thus equal to 12.502. The certainty equivalent is the dollar amount that – if the consumer received it with certainty – would also yield her 12.502 in utility. So we are looking for a level of w = CE such that ln(CE) = 12.502. That level of wealth is $268,787.54.

The risk premium is the EV of the gamble minus the certainty equivalent. The EV of this gamble is $280,000, so the risk premium = EV – CE = $280,000 - $268,787.54 = $11,212,46.

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.

Note that the BL has slope (negative) 1/4, but is only defined for wB > $100k. So the equation for the BL is wG = 325,000 - (1/4)wB, for wB ≥ 100. (How did I calculate the vertical intercept? Slope = rise over run, remember....)

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

The consumer’s expected utility function is EU = (0.1 X ln wB) + (0.9 X ln wG) so her