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168 UNCERTAINTY (Ch. 12)
0 50 100 150 200
50
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
2.
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
40
10
20
30
Cs
Cr
40
e
a
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.
(c) On the same graph, use blue ink to draw the budget line representing
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
r,
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
r)
(b) How does Morgan’s utility function compare to Sam’s? Is one a
(c) What will Morgan’s optimal pattern of consumption be? Answer:
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
e
c
d
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.
(c) After careful thought you decide to bet $50 on the Boars. Label the
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.)
(c) Given this utility function and π=.5, write a general formula for the
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
(c) Suppose that he can buy insurance at a price of $1 per $100 of in-
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
