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 commodities
at different dates as different commodities. This time, we invent
new commodities, which we call contingent commodities. If either of two
events A or B could happen, then we define one contingent commodity
as consumption if A happens and another contingent commodity as consumption
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 NeumannMorgenstern
utility. The assumption that utility takes this form is called
the expected utility hypothesis. If there are two events, 1 and 2 with
probabilities π1 and π2, and if the contingent consumptions are c1 and
c2, then the von Neumann-Morgenstern utility function has the special
functional form, U(c1, c2) = π1u(c1) + π2u(c2). The consumer’s behavior
is determined by maximizing this utility function subject to his budget
constraint.
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 Reds will
make it to the World Series is π = .2. If you don’t bet, you are certain to
have $1,000 to spend on consumption goods. Your behavior satisfies the
expected utility hypothesis and your von Neumann-Morgenstern utility
function is π1
√
c1 + π2
√
c2.
The contingent commodities are dollars if the Reds make the World
Series and dollars if the Reds don’t make the World Series. Let cW be
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 $x on 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 $x on the Reds and they made it to the Series, you
would have cW = 1, 000 + 10x to spend on consumption. If you bet $x
on the Reds and they didn’t make it to the Series, you would lose $x,
and you would have cNW = 1, 000 − x. By increasing the amount $x that
you bet, you can make cW larger and cNW smaller. (You could also bet
against the Reds at the same odds. If you bet $x against the Reds and
they fail to make it to the Series, you make a net gain of .1x and if they
make it to the Series, you lose $x. If you work through the rest of this
discussion for the case where you bet against the Reds, you will see that
the same equations apply, with x being a negative number.) We can use
the above two equations to solve for a budget equation. From the second
equation, we have x = 1, 000−cNW . Substitute this expression for x into
the first equation and rearrange terms to find cW + 10cNW = 11, 000, or
equivalently, .1cW + cNW = 1, 100. (The same budget equation can be
written in many equivalent ways by multiplying both sides by a positive
constant.)
Then you will choose your contingent consumption bundle (cW , cNW )
to maximize U(cW , cNW ) = .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
you set the marginal rate of substitution between cW and cNW equal to
the price ratio and simplify the resulting expression, you will find that
cNW = .16cW . This equation, together with the budget equation implies
that cW = $4, 230.77 and cNW = $676.92. You achieve this bundle by
betting $323.08 on the Reds. If the Reds make it to the Series, you will
have $1, 000 + 10 × 323.08 = $4, 230.80. If not, you will have $676.92.
(We rounded the solutions to the nearest penny.)
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 cA be 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, cNA) = .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 x shares of stock, and if the weapons system is approved,
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 x shares of stock, and if the weapons system is not approved,
then he will make a loss of $ per share. Thus the amount
he can consume, contingent on the system not being approved, is cNA =
.
(b) You can solve for Condor’s budget constraint on contingent commodity
bundles (cA, cNA) by eliminating x from these two equations. His budget
constraint can be written as cA+ cNA = 50, 000.
(c) Buzz Condor has no moral qualms about trading on inside information,
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 NeumannMorgenstern
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 =
(Reminder: Where a is any constant, the derivative of a ln x
with respect to x is a/x.)
(d) Condor finds that his optimal contingent commodity bundle is
(cA, cNA) = To acquire this contingent commodity
bundle, he must buy shares of stock in General Statics.
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 cF denote 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 , cNF ) = .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
will be (cF , cNF ) = .
(b) To buy insurance that pays him $x in 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 there is a
flood, he gets $x in insurance benefits. Suppose that Willy has contracted
for insurance that pays him $x in the event of a flood. Then after paying
his insurance premium, he will be able to consume cF =
If Willy has this amount of insurance and there is no flood, then he will
be able to consume cNF = .
(c) You can eliminate x from the two equations for cF and 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 which the “price” of cNF is 1 can be written as .9cNF + cF =
.
(d) Willy’s marginal rate of substitution between the two contingent commodities,
dollars if there is no flood and dollars if there is a flood, is
MRS(cF , cN F) = −
.1
√
cNF
.9
√
cF
. To find his optimal bundle of contingent
commodities, you must set this marginal rate of substitution equal to the
number Solving this equation, you find that Willy will choose
to consume the two contingent commodities in the ratio .
(e) Since you know the ratio in which he will consume cF and cNF , and
you know his budget equation, you can solve for his optimal consumption
bundle, which is (cF , cNF )= Willy will buy an insurance
policy that will pay him if there is a flood. The
amount of insurance premium that he will have to pay is .
12.3 (0) Clarence Bunsen is an expected utility maximizer. His preferences
among contingent commodity bundles are represented by the expected
utility function
u(c1, c2, π1, π2) = 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.”
(a) If Clarence accepts the bet, then in Event 1, he will have
dollars and in Event 2, he will have dollars.
(b) Since the probability of each event is 1/2, Clarence’s expected utility
for a gamble in which he gets c1 in Event 1 and c2 in Event 2 can be
described by the formula Therefore Clarence’s expected
utility if he accepts the bet with Hjalmer will be (Use
that calculator.)
(c) If Clarence decides not to bet, then in Event 1, he will have
dollars and in Event 2, he will have dollars. Therefore if he
doesn’t bet, his expected utility will be .
(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.
Does Clarence take the bet? .
12.4 (0) It is a slow day at Bunsen Motors, so since he has his calculator
warmed up, Clarence Bunsen (whose preferences toward risk were
described in the last problem) decides to study his expected utility function
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
bet would be , while his expected utility if he didn’t take the
bet would be Therefore he concludes that he would not take
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,
my expected utility would be If I didn’t take the bet, my
expected utility would be Therefore I should the
bet.”
(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 x in 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
The solution is x = .
(d) Your answer to the last part gives you two points on Clarence’s indifference
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
and money in Event 2 is On the graph below, label this point
B.
Money in Event 2 (×1, 000)
40
30
20
10
0 10 20 30 40
Money in Event 1 (×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 indifferent
between point B on the graph and a point in which he gets zero if
Event 2 happens and if Event 1 happens. Find this point on
the Figure above and label it C.
(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
up and where he wins dollars if tails turn up. (Hint: To
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 + x in 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 contingent
commodity bundles are represented by the expected utility function
u(c1, c2, π1, π2) = π1c2
1 + π2c2
2.
(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 have
dollars and if he lost the bet, he would have dollars.
Earl’s expected utility if he took the bet would be , and his
expected utility if he did not take the bet would be 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
bet? Would Earl be willing to take this bet?
.
(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 preferences
between income contingent on Event 1, c1, and income contingent
on Event 2, c2, can be represented by the equation Use
blue ink on the graph below to sketch Earl’s indifference curve passing
through the point (100, 100).
Money in Event 2
200
150
100
50
0 50 100 150 200
Money in Event 1
(d) On the same graph, let us draw Hjalmer’s son-in-law Earl’s indifference
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 contingent
on Event 2 are represented by the formula On
the graph, use red ink to show two of Earl’s indifference curves, including
the one that passes through (100, 100).
12.6 (1) Sidewalk Sam makes his living selling sunglasses at the boardwalk
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.
Cr
40
30
20
10
0 10 20 30 40
Cs
(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 negative
amounts of them.) What is the slope of Sam’s budget line at points
above and to the left of his initial endowment? .
(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 function
for consumption in the two states of nature
u(cs, cr, π) = c1−π
s cπ
r ,
where cs is the dollar value of his consumption if it shines, cr is the dollar
value of his consumption if it rains, and π is the probability that it will
rain. The probability that it will rain is π = .5.
(a) How many units of consumption is it optimal for Sam to consume
conditional on rain? .
(b) How many rain coupons is it optimal for Sam to buy? .
12.8 (0) Sidewalk Sam’s brother Morgan von Neumanstern is an expected
utility maximizer. His von Neumann-Morgenstern utility function
for wealth is u(c) = ln c. 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, cr)
be? .
(b) How does Morgan’s utility function compare to Sam’s? Is one a
monotonic transformation of the other?
.
(c) What will Morgan’s optimal pattern of consumption be? Answer:
Morgan will consume on the sunny days and on the
rainy days. How does this compare to Sam’s consumption?
.
12.9 (0) Billy John Pigskin of Mule Shoe, Texas, has a von NeumannMorgenstern
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 professional
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.
(a) What is Billy John’s expected utility?
.
(b) If Billy John pays $p for 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 − p no matter what happened
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.
.
(c) Solve this equation for p. .
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
settled, you will have a total of dollars if the Boars win and
dollars if they lose. On the graph below, use blue ink to draw a
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.
Money if the Boars lose
400
300
200
100
0 100 200 300 400
Money if the Boars win
(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,
how much money will you have left? If the Robber Barons
win, how much money will you have left after collecting your winnings
and paying off your losses? .
(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 D that 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
over lotteries that give you an amount x if Event 1 happens and y if
Event 1 does not happen is U(x, y, π) = π
√
x + (1 − π)
√
y, where π is the
probability that Event 1 happens and 1 − π is the probability that Event
1 does not happen.
(a) If π = .5, calculate the utility of a lottery that gives you $10,000 if
Event 1 happens and $100 if Event 1 does not happen.
.
(b) If you were sure to receive $4,900, what would your utility be?
(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 $x if Event 1 happens and
$y if Event 1 does not happen. .
(d) Calculate the certainty equivalent of receiving $10,000 if Event 1 happens
and $100 if Event 1 does not happen. .
12.12 (0) Dan Partridge is a risk averter who tries to maximize the
expected value of
√
c, where c is 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
buy fire insurance. .
(c) Suppose that he can buy insurance at a price of $1 per $100 of insurance.
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
what happens his after-tax wealth will be .
(d) Therefore if he buys full insurance, the certainty equivalent of his
wealth is , and his expected utility is .
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, where c is total wealth. What is the minimum price
at which she will sell the rights to her ship? .
Supply and demand problems are bread and butter for economists. In
the problems below, you will typically want to solve for equilibrium prices
and quantities by writing an equation that sets supply equal to demand.
Where the price received by suppliers is the same as the price paid by
demanders, one writes supply and demand as functions of the same price
variable, p, and solves for the price that equalizes supply and demand.
But if, as happens with taxes and subsidies, suppliers face different prices
from demanders, it is a good idea to denote these two prices by separate
variables, ps and pd. Then one can solve for equilibrium by solving a system
of two equations in the two unknowns ps and pd. The two equations
are the equation that sets supply equal to demand and the equation that
relates the price paid by demanders to the net price received by suppliers.
The demand function for commodity x is q = 1, 000 − 10pd, where pd
is the price paid by consumers. The supply function for x is q = 100 +
20ps, where ps is the price received by suppliers. For each unit sold, the
government collects a tax equal to half of the price paid by consumers. Let
us find the equilibrium prices and quantities. In equilibrium, supply must
equal demand, so that 1, 000 − 10pd = 100 + 20ps. Since the government
collects a tax equal to half of the price paid by consumers, it must be
that the sellers only get half of the price paid by consumers, so it must
be that ps = pd/2. Now we have two equations in the two unknowns, ps
and pd. Substitute the expression pd/2 for ps in the first equation, and
you have 1, 000 − 10pd = 100 + 10pd. Solve this equation to find pd = 45.
Then ps = 22.5 and q = 550.
16.1 (0) The demand for yak butter is given by 120 − 4pd and the
supply is 2ps − 30, where pd is the price paid by demanders and ps is
the price received by suppliers, measured in dollars per hundred pounds.
Quantities demanded and supplied are measured in hundred-pound units.
(a) On the axes below, draw the demand curve (with blue ink) and the
supply curve (with red ink) for yak butter.
Price
80
60
40
20
0 20 40 60 80 100 120
Yak butter
(b) Write down the equation that you would solve to find the equilibrium
price. .
(c) What is the equilibrium price of yak butter? What is the
equilibrium quantity? Locate the equilibrium price and quantity
on the graph, and label them p1 and q1.
(d) A terrible drought strikes the central Ohio steppes, traditional homeland
of the yaks. The supply schedule shifts to 2ps − 60. The demand
schedule remains as before. Draw the new supply schedule. Write down
the equation that you would solve to find the new equilibrium price of
yak butter. .
(e) The new equilibrium price is and the quantity is
Locate the new equilibrium price and quantity on the graph and label
them p2 and q2.
(f) The government decides to relieve stricken yak butter consumers and
producers by paying a subsidy of $5 per hundred pounds of yak butter to
producers. If pd is the price paid by demanders for yak butter, what is the
total amount received by producers for each unit they produce?
When the price paid by consumers is pd, how much yak butter
is produced? .
(g) Write down an equation that can be solved for the equilibrium price
paid by consumers, given the subsidy program.
What are the equilibrium price paid by consumers and the equilibrium
quantity of yak butter now? .
(h) Suppose the government had paid the subsidy to consumers rather
than producers. What would be the equilibrium net price paid by consumers?
The equilibrium quantity would be .
16.2 (0) Here are the supply and demand equations for throstles, where
p is the price in dollars:
D(p) = 40 − p
S(p) = 10 + p.
On the axes below, draw the demand and supply curves for throstles,
using blue ink.
Price
40
30
20
10
0 10 20 30 40
Throstles
(a) The equilibrium price of throstles is and the equilibrium
quantity is .
(b) Suppose that the government decides to restrict the industry to selling
only 20 throstles. At what price would 20 throstles be demanded?
How many throstles would suppliers supply at that price? At
what price would the suppliers supply only 20 units? .
(c) The government wants to make sure that only 20 throstles are bought,
but it doesn’t want the firms in the industry to receive more than the
minimum price that it would take to have them supply 20 throstles. One
way to do this is for the government to issue 20 ration coupons. Then
in order to buy a throstle, a consumer would need to present a ration
coupon along with the necessary amount of money to pay for the good.
If the ration coupons were freely bought and sold on the open market,
what would be the equilibrium price of these coupons? .
(d) On the graph above, shade in the area that represents the deadweight
loss from restricting the supply of throstles to 20. How much is this expressed
in dollars? (Hint: What is the formula for the area of a triangle?)
.
16.3 (0) The demand curve for ski lessons is given by D(pD) = 100−2pD
and the supply curve is given by S(pS) = 3pS.
(a) What is the equilibrium price? What is the equilibrium
quantity? .
(b) A tax of $10 per ski lesson is imposed on consumers. Write an equation
that relates the price paid by demanders to the price received by suppliers.
Write an equation that states that supply equals
demand. .
(c) Solve these two equations for the two unknowns pS and pD. With the
$10 tax, the equilibrium price pD paid by consumers would be
per lesson. The total number of lessons given would be .
(d) A senator from a mountainous state suggests that although ski lesson
consumers are rich and deserve to be taxed, ski instructors are poor and
deserve a subsidy. He proposes a $6 subsidy on production while maintaining
the $10 tax on consumption of ski lessons. Would this policy have
any different effects for suppliers or for demanders than a tax of $4 per
lesson? .
16.4 (0) The demand curve for salted codfish is D(P) = 200 − 5P and
the supply curve S(P) = 5P.
(a) On the graph below, use blue ink to draw the demand curve and the
supply curve. The equilibrium market price is and the equilibrium
quantity sold is .
Price
40
30
20
10
0 50 100 150 200
Quantity of codfish
(b) A quantity tax of $2 per unit sold is placed on salted codfish. Use
red ink to draw the new supply curve, where the price on the vertical
axis remains the price per unit paid by demanders. The new equilibrium
price paid by the demanders will be and the new price received
by the suppliers will be The equilibrium quantity sold will be
.
(c) The deadweight loss due to this tax will be On your
graph, shade in the area that represents the deadweight loss.
16.5 (0) The demand function for merino ewes is D(P) = 100/P, and
the supply function is S(P) = P.
(a) What is the equilibrium price? .
(b) What is the equilibrium quantity? .
(c) An ad valorem tax of 300% is imposed on merino ewes so that the price
paid by demanders is four times the price received by suppliers. What
is the equilibrium price paid by the demanders for merino ewes now?
What is the equilibrium price received by the suppliers for
merino ewes? What is the equilibrium quantity? .
16.6 (0) Schrecklich and LaMerde are two justifiably obscure nineteenthcentury
impressionist painters. The world’s total stock of paintings by
Schrecklich is 100, and the world’s stock of paintings by LaMerde is 150.
The two painters are regarded by connoisseurs as being very similar in
style. Therefore the demand for either painter’s work depends both on its
own price and the price of the other painter’s work. The demand function
for Schrecklichs is DS(P) = 200−4PS −2PL, and the demand function for
LaMerdes is DL(P) = 200 − 3PL − PS, where PS and PL are respectively
the price in dollars of a Schrecklich painting and a LaMerde painting.
(a) Write down two simultaneous equations that state the equilibrium
condition that the demand for each painter’s work equals supply.
.
(b) Solving these two equations, one finds that the equilibrium price of
Schrecklichs is and the equilibrium price of LaMerdes is .
(c) On the diagram below, draw a line that represents all combinations of
prices for Schrecklichs and LaMerdes such that the supply of Schrecklichs
equals the demand for Schrecklichs. Draw a second line that represents
those price combinations at which the demand for LaMerdes equals the
supply of LaMerdes. Label the unique price combination at which both
markets clear with the letter E.
PL
40
30
20
10
0 10 20 30 40
PS
(d) A fire in a bowling alley in Hamtramck, Michigan, destroyed one of
the world’s largest collections of works by Schrecklich. The fire destroyed
a total of 10 Schrecklichs. After the fire, the equilibrium price of Schrecklichs
was and the equilibrium price of LaMerdes was .
(e) On the diagram you drew above, use red ink to draw a line that shows
the locus of price combinations at which the demand for Schrecklichs
equals the supply of Schrecklichs after the fire. On your diagram, label
the new equilibrium combination of prices E′
.
16.7 (0) The price elasticity of demand for oatmeal is constant and
equal to −1. When the price of oatmeal is $10 per unit, the total amount
demanded is 6,000 units.
(a) Write an equation for the demand function.
Graph this demand function below with blue ink. (Hint: If the demand
curve has a constant price elasticity equal to ǫ, then D(p) = apǫ
for some
constant a. You have to use the data of the problem to solve for the
constants a and ǫ that apply in this particular case.)
Price
20
15
10
5
0 2 4 6 8 10 12
Quantity (thousands)
(b) If the supply is perfectly inelastic at 5,000 units, what is the equilibrium
price? Show the supply curve on your graph and label
the equilibrium with an E.
(c) Suppose that the demand curve shifts outward by 10%. Write down
the new equation for the demand function. Suppose
that the supply curve remains vertical but shifts to the right by 5%. Solve
for the new equilibrium price and quantity .
(d) By what percentage approximately did the equilibrium price rise?
Use red ink to draw the new demand curve
and the new supply curve on your graph.
(e) Suppose that in the above problem the demand curve shifts outward
by x% and the supply curve shifts right by y%. By approximately what
percentage will the equilibrium price rise? .
16.8 (0) An economic historian* reports that econometric studies indicate
for the pre–Civil War period, 1820–1860, the price elasticity of
demand for cotton from the American South was approximately −1. Due
to the rapid expansion of the British textile industry, the demand curve
for American cotton is estimated to have shifted outward by about 5%
per year during this entire period.
(a) If during this period, cotton production in the United States grew
by 3% per year, what (approximately) must be the rate of change of the
price of cotton during this period? .
(b) Assuming a constant price elasticity of −1, and assuming that when
the price is $20, the quantity is also 20, graph the demand curve for
cotton. What is the total revenue when the price is $20?
What is the total revenue when the price is $10? .
Price of cotton
40
30
20
10
0 10 20 30 40
Quantity of cotton
* Gavin Wright, The Political Economy of the Cotton South, W. W.
Norton, 1978.
(c) If the change in the quantity of cotton supplied by the United States is
to be interpreted as a movement along an upward-sloping long-run supply
curve, what would the elasticity of supply have to be? (Hint: From 1820
to 1860 quantity rose by about 3% per year and price rose by %
per year. [See your earlier answer.] If the quantity change is a movement
along the long-run supply curve, then the long-run price elasticity must
be what?) .
(d) The American Civil War, beginning in 1861, had a devastating effect
on cotton production in the South. Production fell by about 50% and
remained at that level throughout the war. What would you predict
would be the effect on the price of cotton?
.
(e) What would be the effect on total revenue of cotton farmers in
the South?
.
(f) The expansion of the British textile industry ended in the 1860s,
and for the remainder of the nineteenth century, the demand curve for
American cotton remained approximately unchanged. By about 1900, the
South approximately regained its prewar output level. What do you think
happened to cotton prices then? .
16.9 (0) The number of bottles of chardonnay demanded per year is
$1, 000, 000 − 60, 000P, where P is the price per bottle (in U.S. dollars).
The number of bottles supplied is 40, 000P.
(a) What is the equilibrium price? What is the equilibrium
quantity? .
(b) Suppose that the government introduces a new tax such that the
wine maker must pay a tax of $5 per bottle for every bottle that he
produces. What is the new equilibrium price paid by consumers?
What is the new price received by suppliers? What is
the new equilibrium quantity? .
16.10 (0) The inverse demand function for bananas is Pd = 18 − 3Qd
and the inverse supply function is Ps = 6+Qs, where prices are measured
in cents.
(a) If there are no taxes or subsidies, what is the equilibrium quantity?
What is the equilibrium market price? .
(b) If a subsidy of 2 cents per pound is paid to banana growers, then
in equilibrium it still must be that the quantity demanded equals the
quantity supplied, but now the price received by sellers is 2 cents higher
than the price paid by consumers. What is the new equilibrium quantity?
What is the new equilibrium price received by suppliers?
What is the new equilibrium price paid by demanders?
.
(c) Express the change in price as a percentage of the original price.
If the cross-elasticity of demand between bananas and
apples is +.5, what will happen to the quantity of apples demanded as a
consequence of the banana subsidy, if the price of apples stays constant?
(State your answer in terms of percentage change.) .
16.11 (1) King Kanuta rules a small tropical island, Nutting Atoll,
whose primary crop is coconuts. If the price of coconuts is P, then King
Kanuta’s subjects will demand D(P) = 1, 200 − 100P coconuts per week
for their own use. The number of coconuts that will be supplied per week
by the island’s coconut growers is S(p) = 100P.
(a) The equilibrium price of coconuts will be and the equilibrium
quantity supplied will be .
(b) One day, King Kanuta decided to tax his subjects in order to collect
coconuts for the Royal Larder. The king required that every subject
who consumed a coconut would have to pay a coconut to the king as a
tax. Thus, if a subject wanted 5 coconuts for himself, he would have
to purchase 10 coconuts and give 5 to the king. When the price that
is received by the sellers is pS, how much does it cost one of the king’s
subjects to get an extra coconut for himself? .
(c) When the price paid to suppliers is pS, how many coconuts will the
king’s subjects demand for their own consumption? (Hint: Express pD in
terms of pS and substitute into the demand function.)
.
(d) Since the king consumes a coconut for every coconut consumed by
the subjects, the total amount demanded by the king and his subjects is
twice the amount demanded by the subjects. Therefore, when the price
received by suppliers is pS, the total number of coconuts demanded per
week by Kanuta and his subjects is .
(e) Solve for the equilibrium value of pS , the equilibrium total
number of coconuts produced , and the equilibrium total number
of coconuts consumed by Kanuta’s subjects. .
(f) King Kanuta’s subjects resented paying the extra coconuts to the
king, and whispers of revolution spread through the palace. Worried by
the hostile atmosphere, the king changed the coconut tax. Now, the
shopkeepers who sold the coconuts would be responsible for paying the
tax. For every coconut sold to a consumer, the shopkeeper would have
to pay one coconut to the king. This plan resulted in
coconuts being sold to the consumers. The shopkeepers got
per coconut after paying their tax to the king, and the consumers paid a
price of per coconut.
16.12 (1) On August 29, 2005, Hurricane Katrina caused severe damage
to oil installations in the Gulf of Mexico. Although this damage could
eventually be repaired, it resulted in a substantial reduction in the short
run supply of gasoline in the United States. In many areas, retail gasoline
prices quickly rose by about 30% to an average of $3.06 per gallon.
Georgia governor Sonny Perdue suspended his state’s 7.5 cents-agallon
gas tax and 4% sales tax on gasoline purchases until Oct. 1. Governor
Perdue explained that, “I believe it is absolutely wrong for the state
to reap a tax windfall in this time of urgency and tragedy.” Lawmakers
in several other states were considering similar actions.
Let us apply supply and demand analysis to this problem. Before
the hurricane, the United States consumed about 180 million gallons of
gasoline per day, of which about 30 million gallons came from the Gulf
of Mexico. In the short run, the supply of gasoline is extremely inelastic
and is limited by refinery and transport capacity. Let us assume that the
daily short run supply of gasoline was perfectly inelastic at 180 million
gallons before the storm and perfectly inelastic at 150 million gallons after
the storm. Suppose that the demand function, measured in millions of
gallons per day, is given by Q = 240 − 30P where P is the dollar price,
including tax, that consumers pay for gasoline.
(a) What was the market equilibrium price for gasoline before the hurricane?
After the hurricane? .
(b) Suppose that both before and after the hurricane, a government tax of
10 cents is charged for every gallon of gasoline sold in the United States.
How much money would suppliers receive per gallon of gasoline before the
hurricane? After the hurricane? .
(c) Suppose that after the hurricane, the federal government removed the
gas tax. What would then be the equilibrium price paid by consumers?
How much money would suppliers receive per gallon of gasoline?
How much revenue would the government lose per day
by removing the tax? What is the net effect of removing
the tax on gasoline prices? Who are the gainers
and who are the losers from removing the tax?
.
(d) Suppose that after the hurricane, the ten-cent tax is removed in some
states but not in others. The states where the tax is removed constitute
just half of the demand in the United States. Thus the demand
schedule in each half of the country is Q = 120 − 15P where P is the
price paid by consumers in that part of the country. Let P∗
be the
equilibrium price for consumers in the part of the country where the tax
is removed. In equilibrium, suppliers must receive the same price per
gallon in all parts of the country. Therefore the equilibrium price for consumers
in states that keep the tax must be $P∗
+ $0.10. In equilibrium
it must be that the total amount of gasoline demanded in the two parts
of the country equals the total supply. Write an equation for total demand
as a function of P∗
. Set demand
equal to supply and solve for the price paid by consumers in the states
that remove the tax and for the price paid by consumers in
states that do not remove the tax. How much money
do suppliers receive per gallon of gasoline sold in every state?
How does the tax removal affect daily gasoline consumption in
each group of states?
.
(e) If half of the states remove the gasoline tax, as described above, some
groups will be better off and some worse off than they would be if the tax
were left in place. Describe the gains or losses for each of the following
groups.
Consumers in the states that remove the tax
.
Consumers in other states .
Gasoline suppliers .
Governments of the states that remove the tax
.
Governments of states that do not remove the tax
.