978-0393123982 Chapter 12 Solution Manual Part 2

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168 UNCERTAINTY (Ch. 12)

0 50 100 150 200

100

150

Money in Event 1

Money in Event 2

200

Blue curve

Red curves

(d) On the same graph, let us draw Hjalmer’s son-in-law Earl’s indif-

ference curves between contingent commodities where the probabilities

are different. Suppose that a card is drawn from a fair deck of cards.

Let Event 1 be the event that the card is black. Let event 2 be the event

that the card drawn is red. Suppose each event has probability 1/2. Then

Earl’s preferences between income contingent on Event 1 and income con-

tingent on Event 2 are represented by the formula u=1

2c2

1+1

2c2

On the graph, use red ink to show two of Earl’s indifference curves, in-

cluding the one that passes through (100,100).

12.6 (1) Sidewalk Sam makes his living selling sunglasses at the board-

walk in Atlantic City. If the sun shines Sam makes $30, and if it rains

Sam only makes $10. For simplicity, we will suppose that there are only

two kinds of days, sunny ones and rainy ones.

(a) One of the casinos in Atlantic City has a new gimmick. It is accepting

bets on whether it will be sunny or rainy the next day. The casino sells

dated “rain coupons” for $1 each. If it rains the next day, the casino will

give you $2 for every rain coupon you bought on the previous day. If it

doesn’t rain, your rain coupon is worthless. In the graph below, mark

Sam’s “endowment” of contingent consumption if he makes no bets with

the casino, and label it E.

NAME 169

0102030

Blue

line

Red

line

(b) On the same graph, mark the combination of consumption contingent

on rain and consumption contingent on sun that he could achieve by

buying 10 rain coupons from the casino. Label it A.

all of the other patterns of consumption that Sam can achieve by buying

rain coupons. (Assume that he can buy fractional coupons, but not neg-

ative amounts of them.) What is the slope of Sam’s budget line at points

(d) Suppose that the casino also sells sunshine coupons. These tickets

also cost $1. With these tickets, the casino gives you $2 if it doesn’t rain

and nothing if it does. On the graph above, use red ink to sketch in the

budget line of contingent consumption bundles that Sam can achieve by

buying sunshine tickets.

(e) If the price of a dollar’s worth of consumption when it rains is set equal

to 1, what is the price of a dollar’s worth of consumption if it shines?

12.7 (0) Sidewalk Sam, from the previous problem, has the utility func-

tion for consumption in the two states of nature

u(cs,c

r,π)=c1−π

scπ

where csis the dollar value of his consumption if it shines, cris the dollar

value of his consumption if it rains, and πis the probability that it will

rain. The probability that it will rain is π=.5.

170 UNCERTAINTY (Ch. 12)

(a) How many units of consumption is it optimal for Sam to consume

12.8 (0) Sidewalk Sam’s brother Morgan von Neumanstern is an ex-

pected utility maximizer. His von Neumann-Morgenstern utility function

for wealth is u(c)=lnc. Sam’s brother also sells sunglasses on another

beach in Atlantic City and makes exactly the same income as Sam does.

He can make exactly the same deal with the casino as Sam can.

(a) If Morgan believes that there is a 50% chance of rain and a 50% chance

of sun every day, what would his expected utility of consuming (cs,c

(b) How does Morgan’s utility function compare to Sam’s? Is one a

12.9 (0) Billy John Pigskin of Mule Shoe, Texas, has a von Neumann-

Morgenstern utility function of the form u(c)=√c. Billy John also weighs

about 300 pounds and can outrun jackrabbits and pizza delivery trucks.

Billy John is beginning his senior year of college football. If he is not

seriously injured, he will receive a $1,000,000 contract for playing pro-

fessional football. If an injury ends his football career, he will receive a

$10,000 contract as a refuse removal facilitator in his home town. There

is a 10% chance that Billy John will be injured badly enough to end his

career.

NAME 171

(b) If Billy John pays $pfor an insurance policy that would give him

$1,000,000 if he suffered a career-ending injury while in college, then he

would be sure to have an income of $1,000,000 −pno matter what hap-

pened to him. Write an equation that can be solved to find the largest

price that Billy John would be willing to pay for such an insurance policy.

12.10 (1) You have $200 and are thinking about betting on the Big

Game next Saturday. Your team, the Golden Boars, are scheduled to

play their traditional rivals the Robber Barons. It appears that the going

odds are 2 to 1 against the Golden Boars. That is to say if you want

to bet $10 on the Boars, you can find someone who will agree to pay

you $20 if the Boars win in return for your promise to pay him $10 if

the Robber Barons win. Similarly if you want to bet $10 on the Robber

Barons, you can find someone who will pay you $10 if the Robber Barons

win, in return for your promise to pay him $20 if the Robber Barons lose.

Suppose that you are able to make as large a bet as you like, either on

the Boars or on the Robber Barons so long as your gambling losses do

not exceed $200. (To avoid tedium, let us ignore the possibility of ties.)

(a) If you do not bet at all, you will have $200 whether or not the Boars

win. If you bet $50 on the Boars, then after all gambling obligations are

line that represents all of the combinations of “money if the Boars win”

and “money if the Robber Barons win” that you could have by betting

from your initial $200 at these odds.

172 UNCERTAINTY (Ch. 12)

0 100 200 300 400

100

200

300

Money if the Boars win

Money if the Boars lose

400

Red line

Blue line

(b) Label the point on this graph where you would be if you did not bet

at all with an E.

point you have chosen on the graph with a C. Suppose that after you have

made this bet, it is announced that the star Robber Baron quarterback

suffered a sprained thumb during a tough economics midterm examination

and will miss the game. The market odds shift from 2 to 1 against the

Boars to “even money” or 1 to 1. That is, you can now bet on either

team and the amount you would win if you bet on the winning team is

the same as the amount that you would lose if you bet on the losing team.

You cannot cancel your original bet, but you can make new bets at the

new odds. Suppose that you keep your first bet, but you now also bet

$50 on the Robber Barons at the new odds. If the Boars win, then after

you collect your winnings from one bet and your losses from the other,

(d) Use red ink to draw a line on the diagram you made above, showing

the combinations of “money if the Boars win” and “money if the Robber

Barons win” that you could arrange for yourself by adding possible bets

at the new odds to the bet you made before the news of the quarterback’s

misfortune. On this graph, label the point Dthat you reached by making

the two bets discussed above.

12.11 (2) The certainty equivalent of a lottery is the amount of money

you would have to be given with certainty to be just as well-off with that

lottery. Suppose that your von Neumann-Morgenstern utility function

NAME 173

over lotteries that give you an amount xif Event 1 happens and yif

Event 1 does not happen is U(x, y, π)=π√x+(1−π)√y,whereπis the

probability that Event 1 happens and 1 −πis the probability that Event

1 does not happen.

(a) If π=.5, calculate the utility of a lottery that gives you $10,000

(Hint: If you receive $4,900 with certainty, then you receive $4,900 in

both events.)

certainty equivalent of a lottery that gives you $xif Event 1 happens and

(d) Calculate the certainty equivalent of receiving $10,000 if Event 1 hap-

12.12 (0) Dan Partridge is a risk averter who tries to maximize the

expected value of √c,wherecis his wealth. Dan has $50,000 in safe

assets and he also owns a house that is located in an area where there

are lots of forest fires. If his house burns down, the remains of his house

and the lot it is built on would be worth only $40,000, giving him a total

wealth of $90,000. If his home doesn’t burn, it will be worth $200,000

and his total wealth will be $250,000. The probability that his home will

burn down is .01.

(a) Calculate his expected utility if he doesn’t buy fire insurance.

(b) Calculate the certainty equivalent of the lottery he faces if he doesn’t

surance. For example if he buys $100,000 worth of insurance, he will pay

$1,000 to the company no matter what happens, but if his house burns,

he will also receive $100,000 from the company. If Dan buys $160,000

worth of insurance, he will be fully insured in the sense that no matter

174 UNCERTAINTY (Ch. 12)

(d) Therefore if he buys full insurance, the certainty equivalent of his

12.13 (1) Portia has been waiting a long time for her ship to come in

and has concluded that there is a 25% chance that it will arrive today.

If it does come in today, she will receive $1,600. If it does not come

in today, it will never come and her wealth will be zero. Portia has a

von Neumann-Morgenstern utility such that she wants to maximize the

expected value of √c,wherecis total wealth. What is the minimum price