978-0393123982 Chapter 12 Solution Manual Part 1

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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-

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)

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.)

(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)

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

Money in Event 1 (x 1,000)

Money in Event 2 (x 1,000)

(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

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

Use blue ink on the graph below to sketch Earl’s indifference curve passing

through the point (100,100).