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Chapter 12 NAME
Uncertainty
Introduction. In Chapter 11, you learned some tricks that allow you to
use techniques you already know for studying intertemporal choice. Here
you will learn some similar tricks, so that you can use the same methods
to study risk taking, insurance, and gambling.
One of these new tricks is similar to the trick of treating commodi-
ties at different dates as different commodities. This time, we invent
new commodities, which we call contingent commodities.Ifeitheroftwo
events Aor Bcould happen, then we define one contingent commodity
as consumption if A happens and another contingent commodity as con-
sumption if B happens. The second trick is to find a budget constraint
that correctly specifies the set of contingent commodity bundles that a
consumer can afford.
This chapter presents one other new idea, and that is the notion
of von Neumann-Morgenstern utility. A consumer’s willingness to take
various gambles and his willingness to buy insurance will be determined
by how he feels about various combinations of contingent commodities.
Often it is reasonable to assume that these preferences can be expressed
by a utility function that takes the special form known as von Neumann-
Morgenstern utility. The assumption that utility takes this form is called
the expected utility hypothesis. If there are two events, 1 and 2 with
probabilities π1and π2, and if the contingent consumptions are c1and
c2, then the von Neumann-Morgenstern utility function has the special
functional form, U(c1,c
2)=π1u(c1)+π2u(c2). The consumer’s behavior
is determined by maximizing this utility function subject to his budget
constraint.
Example: You are thinking of betting on whether the Cincinnati Reds
will make it to the World Series this year. A local gambler will bet with
you at odds of 10 to 1 against the Reds. You think the probability that
The contingent commodities are dollars if the Reds make the World
Series and dollars if the Reds don’t make the World Series.LetcWbe
your consumption contingent on the Reds making the World Series and
cNW be your consumption contingent on their not making the Series.
Betting on the Reds at odds of 10 to 1 means that if you bet $xon the
Reds, then if the Reds make it to the Series, you make a net gain of $10x,
but if they don’t, you have a net loss of $x. Since you had $1,000 before
betting, if you bet $xon the Reds and they made it to the Series, you
would have cW=1,000 + 10xto spend on consumption. If you bet $x
on the Reds and they didn’t make it to the Series, you would lose $x,
162 UNCERTAINTY (Ch. 12)
Then you will choose your contingent consumption bundle (cW,c
NW)
to maximize U(cW,c
NW)=.2√cW+.8√cNW subject to the budget
constraint, .1cW+cNW =1,100. Using techniques that are now familiar,
you can solve this consumer problem. From the budget constraint, you
see that consumption contingent on the Reds making the World Series
costs 1/10 as much as consumption contingent on their not making it. If
12.1 (0) In the next few weeks, Congress is going to decide whether
or not to develop an expensive new weapons system. If the system is
approved, it will be very profitable for the defense contractor, General
Statics. Indeed, if the new system is approved, the value of stock in
General Statics will rise from $10 per share to $15 a share, and if the
project is not approved, the value of the stock will fall to $5 a share. In
his capacity as a messenger for Congressman Kickback, Buzz Condor has
discovered that the weapons system is much more likely to be approved
than is generally thought. On the basis of what he knows, Condor has
decided that the probability that the system will be approved is 3/4 and
the probability that it will not be approved is 1/4. Let cAbe Condor’s
consumption if the system is approved and cNA be his consumption if
the system is not approved. Condor’s von Neumann-Morgenstern utility
function is U(cA,c
NA)=.75 ln cA+.25 ln cNA. Condor’s total wealth is
$50,000, all of which is invested in perfectly safe assets. Condor is about
to buy stock in General Statics.
(a) If Condor buys xshares of stock, and if the weapons system is ap-
proved, he will make a profit of $5 per share. Thus the amount he can
consume, contingent on the system being approved, is cA= $50,000+5x.
If Condor buys xshares of stock, and if the weapons system is not ap-
proved, then he will make a loss of $ 5per share. Thus the
NAME 163
amount he can consume, contingent on the system not being approved, is
(b) You can solve for Condor’s budget constraint on contingent commod-
ity bundles (cA,c
NA) by eliminating xfrom these two equations. His bud-
(c) Buzz Condor has no moral qualms about trading on inside informa-
tion, nor does he have any concern that he will be caught and punished.
To decide how much stock to buy, he simply maximizes his von Neumann-
Morgenstern utility function subject to his budget. If he sets his marginal
rate of substitution between the two contingent commodities equal to
their relative prices and simplifies the equation, he finds that cA/cNA =
3. (Reminder: Where ais any constant, the derivative of aln x
with respect to xis a/x.)
(d) Condor finds that his optimal contingent commodity bundle is
12.2 (0) Willy owns a small chocolate factory, located close to a river
that occasionally floods in the spring, with disastrous consequences. Next
summer, Willy plans to sell the factory and retire. The only income he
will have is the proceeds of the sale of his factory. If there is no flood,
the factory will be worth $500,000. If there is a flood, then what is left
of the factory will be worth only $50,000. Willy can buy flood insurance
at a cost of $.10 for each $1 worth of coverage. Willy thinks that the
probability that there will be a flood this spring is 1/10. Let cFdenote the
contingent commodity dollars if there is a flood and cNF denote dollars
if there is no flood. Willy’s von Neumann-Morgenstern utility function is
U(cF,c
NF)=.1√cF+.9√cNF .
(a) If he buys no insurance, then in each contingency, Willy’s consumption
will equal the value of his factory, so Willy’s contingent commodity bundle
(b) To buy insurance that pays him $xin case of a flood, Willy must
pay an insurance premium of .1x. (The insurance premium must be
paid whether or not there is a flood.) If Willy insures for $x, then if
thereisaflood,hegets$xin insurance benefits. Suppose that Willy has
contracted for insurance that pays him $xin the event of a flood. Then
after paying his insurance premium, he will be able to consume cF=
164 UNCERTAINTY (Ch. 12)
(c) You can eliminate xfrom the two equations for cFand cNF that
you found above. This gives you a budget equation for Willy. Of course
there are many equivalent ways of writing the same budget equation,
since multiplying both sides of a budget equation by a positive constant
yields an equivalent budget equation. The form of the budget equation in
(d) Willy’s marginal rate of substitution between the two contingent com-
modities, dollars if there is no flood and dollars if there is a flood,is
MRS(cF,c
NF)=−.1√cNF
.9√cF. To find his optimal bundle of contingent
commodities, you must set this marginal rate of substitution equal to
(e) Since you know the ratio in which he will consume cFand cNF ,and
you know his budget equation, you can solve for his optimal consumption
flood. The amount of insurance premium that he will have to pay is
$45,000.
12.3 (0) Clarence Bunsen is an expected utility maximizer. His pref-
erences among contingent commodity bundles are represented by the ex-
pected utility function
u(c1,c
2,π
1,π
2)=π1√c1+π2√c2.
Clarence’s friend, Hjalmer Ingqvist, has offered to bet him $1,000 on the
outcome of the toss of a coin. That is, if the coin comes up heads, Clarence
must pay Hjalmer $1,000 and if the coin comes up tails, Hjalmer must
pay Clarence $1,000. The coin is a fair coin, so that the probability of
heads and the probability of tails are both 1/2. If he doesn’t accept the
bet, Clarence will have $10,000 with certainty. In the privacy of his car
dealership office over at Bunsen Motors, Clarence is making his decision.
(Clarence uses the pocket calculator that his son, Elmer, gave him last
Christmas. You will find that it will be helpful for you to use a calculator
too.) Let Event 1 be “coin comes up heads” and let Event 2 be “coin
comes up tails.”
NAME 165
(Use that calculator.)
(c) If Clarence decides not to bet, then in Event 1, he will have
(d) Having calculated his expected utility if he bets and if he does not bet,
Clarence determines which is higher and makes his decision accordingly.
12.4 (0) It is a slow day at Bunsen Motors, so since he has his calcu-
lator warmed up, Clarence Bunsen (whose preferences toward risk were
described in the last problem) decides to study his expected utility func-
tion more closely.
(a) Clarence first thinks about really big gambles. What if he bet his
entire $10,000 on the toss of a coin, where he loses if heads and wins if
tails? Then if the coin came up heads, he would have 0 dollars and if it
came up tails, he would have $20,000. His expected utility if he took the
such a bet.
(b) Clarence then thinks, “Well, of course, I wouldn’t want to take a
chance on losing all of my money on just an ordinary bet. But, what
if somebody offered me a really good deal. Suppose I had a chance to
bet where if a fair coin came up heads, I lost my $10,000, but if it came
up tails, I would win $50,000. Would I take the bet? If I took the bet,
166 UNCERTAINTY (Ch. 12)
(c) Clarence later asks himself, “If I make a bet where I lose 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?” After some trial and error, Clarence found the answer. You,
too, might want to find the answer by trial and error, but it is easier to
find the answer by solving an equation. On the left side of your equation,
you would write down Clarence’s utility if he doesn’t bet. On the right
side of the equation, you write down an expression for Clarence’s utility
if he makes a bet such that he is left with zero consumption in Event 1
and xin Event 2. Solve this equation for x. The answer to Clarence’s
question is where x=10,000. The equation that you should write is
(d) Your answer to the last part gives you two points on Clarence’s in-
difference curve between the contingent commodities, money in Event 1
and money in Event 2. (Poor Clarence has never heard of indifference
curves or contingent commodities, so you will have to work this part for
him, while he heads over to the Chatterbox Cafe for morning coffee.) One
of these points is where money in both events is $10,000. On the graph
below, label this point A. The other is where money in Event 1 is zero
point B.
0102030
40
10
20
30
Money in Event 1 (x 1,000)
Money in Event 2 (x 1,000)
40
a
b
c
d
(e) You can quickly find a third point on this indifference curve. The
coin is a fair coin, and Clarence cares whether heads or tails turn up only
because that determines his prize. Therefore Clarence will be indifferent
between two gambles that are the same except that the assignment of
prizes to outcomes are reversed. In this example, Clarence will be indif-
ferent between point Bon the graph and a point in which he gets zero if
on the Figure above and label it C.
NAME 167
(f) Another gamble that is on the same indifference curve for Clarence
as not gambling at all is the gamble where he loses $5,000 if heads turn
solve this problem, put the utility of not betting on the left side of an
equation and on the right side of the equation, put the utility of having
$10,000 −$5,000 in Event 1 and $10,000 + xin Event 2. Then solve the
resulting equation for x.) On the axes above, plot this point and label it
D. Now sketch in the entire indifference curve through the points that
you have labeled.
12.5 (0) Hjalmer Ingqvist’s son-in-law, Earl, has not worked out very
well. It turns out that Earl likes to gamble. His preferences over contin-
gent commodity bundles are represented by the expected utility function
(a) Just the other day, some of the boys were down at Skoog’s tavern
when Earl stopped in. They got to talking about just how bad a bet they
could get him to take. At the time, Earl had $100. Kenny Olson shuffled
a deck of cards and offered to bet Earl $20 that Earl would not cut a spade
from the deck. Assuming that Earl believed that Kenny wouldn’t cheat,
the probability that Earl would win the bet was 1/4 and the probability
that Earl would lose the bet was 3/4. If he won the bet, Earl would
Therefore he refused the bet.
(b) Just when they started to think Earl might have changed his ways,
Kenny offered to make the same bet with Earl except that they would
bet $100 instead of $20. What is Earl’s expected utility if he takes that
(c) Let Event 1 be the event that a card drawn from a fair deck of cards is
a spade. Let Event 2 be the event that the card is not a spade. Earl’s pref-
erences between income contingent on Event 1, c1, and income contingent
on Event 2, c2, can be represented by the equation u=1
4c2
1+3
4c2
2.
Use blue ink on the graph below to sketch Earl’s indifference curve passing
through the point (100,100).
