You have studied budgets, and you have studied preferences. Now is the
time to put these two ideas together and do something with them. In this
chapter you study the commodity bundle chosen by a utility-maximizing
consumer from a given budget.
Given prices and income, you know how to graph a consumer’s budget.
If you also know the consumer’s preferences, you can graph some of
his indiﬀerence curves. The consumer will choose the “best” indiﬀerence
curve that he can reach given his budget. But when you try to do this, you
have to ask yourself, “How do I ﬁnd the most desirable indiﬀerence curve
that the consumer can reach?” The answer to this question is “look in the
likely places.” Where are the likely places? As your textbook tells you,
there are three kinds of likely places. These are: (i) a tangency between
an indiﬀerence curve and the budget line; (ii) a kink in an indiﬀerence
curve; (iii) a “corner” where the consumer specializes in consuming just
one good.
Here is how you ﬁnd a point of tangency if we are told the consumer’s
utility function, the prices of both goods, and the consumer’s income. The
budget line and an indiﬀerence curve are tangent at a point (x1, x2) if they
have the same slope at that point. Now the slope of an indiﬀerence curve
at (x1, x2) is the ratio −MU1(x1, x2)/MU2(x1, x2). (This slope is also
known as the marginal rate of substitution.) The slope of the budget line
is −p1/p2. Therefore an indiﬀerence curve is tangent to the budget line
at the point (x1, x2) when MU1(x1, x2)/MU2(x1, x2) = p1/p2. This gives
us one equation in the two unknowns, x1 and x2. If we hope to solve
for the x’s, we need another equation. That other equation is the budget
equation p1x1 + p2x2 = m. With these two equations you can solve for
(x1, x2).∗
A consumer has the utility function U(x1, x2) = x2
1x2. The price of good
1 is p1 = 1, the price of good 2 is p2 = 3, and his income is 180. Then,
MU1(x1, x2) = 2x1x2 and MU2(x1, x2) = x2
1. Therefore his marginal rate
of substitution is −MU1(x1, x2)/MU2(x1, x2) = −2x1x2/x2
1 = −2x2/x1.
This implies that his indiﬀerence curve will be tangent to his budget line
when −2x2/x1 = −p1/p2 = −1/3. Simplifying this expression, we have
6x2 = x1. This is one of the two equations we need to solve for the two
unknowns, x1 and x2. The other equation is the budget equation. In this
case the budget equation is x1 + 3x2 = 180. Solving these two equations
in two unknowns, we ﬁnd x1 = 120 and x2 = 20. Therefore we know that
∗
Some people have trouble remembering whether the marginal rate
of substitution is −MU1/MU2 or −MU2/MU1. It isn’t really crucial to
remember which way this goes as long as you remember that a tangency
happens when the marginal utilities of any two goods are in the same
proportion as their prices.
the consumer chooses the bundle (x1, x2) = (120, 20).
For equilibrium at kinks or at corners, we don’t need the slope of
the indiﬀerence curves to equal the slope of the budget line. So we don’t
have the tangency equation to work with. But we still have the budget
equation. The second equation that you can use is an equation that tells
you that you are at one of the kinky points or at a corner. You will see
exactly how this works when you work a few exercises.
A consumer has the utility function U(x1, x2) = min{x1, 3x2}. The price
of x1 is 2, the price of x2 is 1, and her income is 140. Her indiﬀerence
curves are L-shaped. The corners of the L’s all lie along the line x1 = 3x2.
She will choose a combination at one of the corners, so this gives us one
of the two equations we need for ﬁnding the unknowns x1 and x2. The
second equation is her budget equation, which is 2x1 + x2 = 140. Solve
these two equations to ﬁnd that x1 = 60 and x2 = 20. So we know that
the consumer chooses the bundle (x1, x2) = (60, 20).
When you have ﬁnished these exercises, we hope that you will be
able to do the following:
• Calculate the best bundle a consumer can aﬀord at given prices and
income in the case of simple utility functions where the best aﬀordable
bundle happens at a point of tangency.
• Find the best aﬀordable bundle, given prices and income for a consumer
with kinked indiﬀerence curves.
• Recognize standard examples where the best bundle a consumer can
aﬀord happens at a corner of the budget set.
• Draw a diagram illustrating each of the above types of equilibrium.
• Apply the methods you have learned to choices made with some kinds
of nonlinear budgets that arise in real-world situations.
5.1 (0) We begin again with Charlie of the apples and bananas. Recall
that Charlie’s utility function is U(xA, xB) = xAxB. Suppose that the
price of apples is 1, the price of bananas is 2, and Charlie’s income is 40.
(a) On the graph below, use blue ink to draw Charlie’s budget line. (Use
a ruler and try to make this line accurate.) Plot a few points on the
indiﬀerence curve that gives Charlie a utility of 150 and sketch this curve
with red ink. Now plot a few points on the indiﬀerence curve that gives
Charlie a utility of 300 and sketch this curve with black ink or pencil.
Bananas
40
30
20
10
0 10 20 30 40
Apples
(b) Can Charlie aﬀord any bundles that give him a utility of 150?
.
(c) Can Charlie aﬀord any bundles that give him a utility of 300?
(d) On your graph, mark a point that Charlie can aﬀord and that gives
him a higher utility than 150. Label that point A.
(e) Neither of the indiﬀerence curves that you drew is tangent to Charlie’s
budget line. Let’s try to ﬁnd one that is. At any point, (xA, xB), Charlie’s
marginal rate of substitution is a function of xA and xB. In fact, if you
calculate the ratio of marginal utilities for Charlie’s utility function, you
will ﬁnd that Charlie’s marginal rate of substitution is MRS(xA, xB) =
−xB/xA. This is the slope of his indiﬀerence curve at (xA, xB). The
slope of Charlie’s budget line is (give a numerical answer).
(f) Write an equation that implies that the budget line is tangent to
an indiﬀerence curve at (xA, xB). There are many
solutions to this equation. Each of these solutions corresponds to a point
on a diﬀerent indiﬀerence curve. Use pencil to draw a line that passes
through all of these points.
(g) The best bundle that Charlie can aﬀord must lie somewhere on the
line you just penciled in. It must also lie on his budget line. If the point
is outside of his budget line, he can’t aﬀord it. If the point lies inside
of his budget line, he can aﬀord to do better by buying more of both
goods. On your graph, label this best aﬀordable bundle with an E. This
happens where xA= and xB= Verify your answer by
solving the two simultaneous equations given by his budget equation and
the tangency condition.
(h) What is Charlie’s utility if he consumes the bundle (20, 10)?
(i) On the graph above, use red ink to draw his indiﬀerence curve through
(20,10). Does this indiﬀerence curve cross Charlie’s budget line, just touch
it, or never touch it? .
5.2 (0) Clara’s utility function is U(X, Y ) = (X + 2)(Y + 1), where X
is her consumption of good X and Y is her consumption of good Y .
(a) Write an equation for Clara’s indiﬀerence curve that goes through
the point (X, Y ) = (2, 8). Y = On the axes below, sketch
Clara’s indiﬀerence curve for U = 36.
Y
16
12
8
4
0 4 8 12 16
X
(b) Suppose that the price of each good is 1 and that Clara has an income
of 11. Draw in her budget line. Can Clara achieve a utility of 36 with
this budget? .
(c) At the commodity bundle, (X, Y ), Clara’s marginal rate of substitution
is .
(d) If we set the absolute value of the MRS equal to the price ratio, we
have the equation .
(e) The budget equation is .
(f) Solving these two equations for the two unknowns, X and Y , we ﬁnd
X = and Y = .
5.3 (0) Ambrose, the nut and berry consumer, has a utility function
U(x1, x2) = 4
√
x1 + x2, where x1 is his consumption of nuts and x2 is his
consumption of berries.
(a) The commodity bundle (25, 0) gives Ambrose a utility of 20. Other
points that give him the same utility are (16, 4), (9, ), (4,
), (1, ), and (0, ). Plot these points on the
axes below and draw a red indiﬀerence curve through them.
(b) Suppose that the price of a unit of nuts is 1, the price of a unit of
berries is 2, and Ambrose’s income is 24. Draw Ambrose’s budget line
with blue ink. How many units of nuts does he choose to buy?
(c) How many units of berries? .
(d) Find some points on the indiﬀerence curve that gives him a utility of
25 and sketch this indiﬀerence curve (in red).
(e) Now suppose that the prices are as before, but Ambrose’s income is
34. Draw his new budget line (with pencil). How many units of nuts will
he choose? How many units of berries? .
Berries
20
15
10
5
0 5 10 15 20 25 30
Nuts
(f) Now let us explore a case where there is a “boundary solution.” Suppose
that the price of nuts is still 1 and the price of berries is 2, but
Ambrose’s income is only 9. Draw his budget line (in blue). Sketch the
indiﬀerence curve that passes through the point (9, 0). What is the slope
of his indiﬀerence curve at the point (9, 0)? .
(g) What is the slope of his budget line at this point? .
(h) Which is steeper at this point, the budget line or the indiﬀerence
curve? .
(i) Can Ambrose aﬀord any bundles that he likes better than the point
(9, 0)? .
5.4 (1) Nancy Lerner is trying to decide how to allocate her time in
studying for her economics course. There are two examinations in this
course. Her overall score for the course will be the minimum of her scores
on the two examinations. She has decided to devote a total of 1,200
minutes to studying for these two exams, and she wants to get as high an
overall score as possible. She knows that on the ﬁrst examination if she
doesn’t study at all, she will get a score of zero on it. For every 10 minutes
that she spends studying for the ﬁrst examination, she will increase her
score by one point. If she doesn’t study at all for the second examination
she will get a zero on it. For every 20 minutes she spends studying for
the second examination, she will increase her score by one point.
(a) On the graph below, draw a “budget line” showing the various combinations
of scores on the two exams that she can achieve with a total of
1,200 minutes of studying. On the same graph, draw two or three “indifference
curves” for Nancy. On your graph, draw a straight line that goes
through the kinks in Nancy’s indiﬀerence curves. Label the point where
this line hits Nancy’s budget with the letter A. Draw Nancy’s indiﬀerence
curve through this point.
Score on Test 2
80
60
40
20
0 20 40 60 80 100 120
Score on Test 1
(b) Write an equation for the line passing through the kinks of Nancy’s
indiﬀerence curves. .
(c) Write an equation for Nancy’s budget line. .
(d) Solve these two equations in two unknowns to determine the intersection
of these lines. This happens at the point (x1, x2) = .
(e) Given that she spends a total of 1,200 minutes studying, Nancy will
maximize her overall score by spending minutes studying for the
ﬁrst examination and minutes studying for the second examina-
tion.
5.5 (1) In her communications course, Nancy also takes two examinations.
Her overall grade for the course will be the maximum of her scores
on the two examinations. Nancy decides to spend a total of 400 minutes
studying for these two examinations. If she spends m1 minutes studying
for the ﬁrst examination, her score on this exam will be x1 = m1/5. If
she spends m2 minutes studying for the second examination, her score on
this exam will be x2 = m2/10.
(a) On the graph below, draw a “budget line” showing the various combinations
of scores on the two exams that she can achieve with a total of 400
minutes of studying. On the same graph, draw two or three “indiﬀerence
curves” for Nancy. On your graph, ﬁnd the point on Nancy’s budget line
that gives her the best overall score in the course.
(b) Given that she spends a total of 400 minutes studying, Nancy will
maximize her overall score by achieving a score of on the ﬁrst
examination and on the second examination.
(c) Her overall score for the course will then be .
Score on Test 2
80
60
40
20
0 20 40 60 80
Score on Test 1
5.6 (0) Elmer’s utility function is U(x, y) = min{x, y2
}.
(a) If Elmer consumes 4 units of x and 3 units of y, his utility is .
(b) If Elmer consumes 4 units of x and 2 units of y, his utility is .
(c) If Elmer consumes 5 units of x and 2 units of y, his utility is .
(d) On the graph below, use blue ink to draw the indiﬀerence curve for
Elmer that contains the bundles that he likes exactly as well as the bundle
(4, 2).
(e) On the same graph, use blue ink to draw the indiﬀerence curve for
Elmer that contains bundles that he likes exactly as well as the bundle
(1, 1) and the indiﬀerence curve that passes through the point (16, 5).
(f) On your graph, use black ink to show the locus of points at which
Elmer’s indiﬀerence curves have kinks. What is the equation for this
curve? .
(g) On the same graph, use black ink to draw Elmer’s budget line when
the price of x is 1, the price of y is 2, and his income is 8. What bundle
does Elmer choose in this situation? .
y
16
12
8
4
0 4 8 12 16 20 24
x
(h) Suppose that the price of x is 10 and the price of y is 15 and Elmer
buys 100 units of x. What is Elmer’s income? (Hint: At ﬁrst
you might think there is too little information to answer this question.
But think about how much y he must be demanding if he chooses 100
units of x.)
5.7 (0) Linus has the utility function U(x, y) = x + 3y.
(a) On the graph below, use blue ink to draw the indiﬀerence curve passing
through the point (x, y) = (3, 3). Use black ink to sketch the indiﬀerence
curve connecting bundles that give Linus a utility of 6.
y
16
12
8
4
0 4 8 12 16
x
(b) On the same graph, use red ink to draw Linus’s budget line if the
price of x is 1 and the price of y is 2 and his income is 8. What bundle
does Linus choose in this situation? .
(c) What bundle would Linus choose if the price of x is 1, the price of y
is 4, and his income is 8? .
5.8 (2) Remember our friend Ralph Rigid from Chapter 3? His favorite
diner, Food for Thought, has adopted the following policy to reduce the
crowds at lunch time: if you show up for lunch t hours before or after
12 noon, you get to deduct t dollars from your bill. (This holds for any
fraction of an hour as well.)
Money
20
15
10
5
10 11 12 1 2
Time
(a) Use blue ink to show Ralph’s budget set. On this graph, the horizontal
axis measures the time of day that he eats lunch, and the vertical axis
measures the amount of money that he will have to spend on things other
than lunch. Assume that he has $20 total to spend and that lunch at
noon costs $10. (Hint: How much money would he have left if he ate at
noon? at 1 P.M.? at 11 A.M.?)
(b) Recall that Ralph’s preferred lunch time is 12 noon, but that he is
willing to eat at another time if the food is suﬃciently cheap. Draw
some red indiﬀerence curves for Ralph that would be consistent with his
choosing to eat at 11 A.M.
5.9 (0) Joe Grad has just arrived at the big U. He has a fellowship that
covers his tuition and the rent on an apartment. In order to get by, Joe
has become a grader in intermediate price theory, earning $100 a month.
Out of this $100 he must pay for his food and utilities in his apartment.
His utilities expenses consist of heating costs when he heats his apartment
and air-conditioning costs when he cools it. To raise the temperature of
his apartment by one degree, it costs $2 per month (or $20 per month
to raise it ten degrees). To use air-conditioning to cool his apartment by
a degree, it costs $3 per month. Whatever is left over after paying the
utilities, he uses to buy food at $1 per unit.
Food
120
100
80
60
40
20
0 10 20 30 40 50 60 70 80 90 100
Temperature
(a) When Joe ﬁrst arrives in September, the temperature of his apartment
is 60 degrees. If he spends nothing on heating or cooling, the temperature
in his room will be 60 degrees and he will have $100 left to spend on food.
If he heated the room to 70 degrees, he would have left to spend
on food. If he cooled the room to 50 degrees, he would have left
to spend on food. On the graph below, show Joe’s September budget
constraint (with black ink). (Hint: You have just found three points that
Joe can aﬀord. Apparently, his budget set is not bounded by a single
straight line.)
(b) In December, the outside temperature is 30 degrees and in August
poor Joe is trying to understand macroeconomics while the temperature
outside is 85 degrees. On the same graph you used above, draw Joe’s
budget constraints for the months of December (in blue ink) and August
(in red ink).
(c) Draw a few smooth (unkinky) indiﬀerence curves for Joe in such a way
that the following are true. (i) His favorite temperature for his apartment
would be 65 degrees if it cost him nothing to heat it or cool it. (ii) Joe
chooses to use the furnace in December, air-conditioning in August, and
neither in September. (iii) Joe is better oﬀ in December than in August.
(d) In what months is the slope of Joe’s budget constraint equal to the
slope of his indiﬀerence curve? .
(e) In December Joe’s marginal rate of substitution between food and
degrees Fahrenheit is In August, his MRS is .
(f) Since Joe neither heats nor cools his apartment in September, we
cannot determine his marginal rate of substitution exactly, but we do
know that it must be no smaller than and no larger than
(Hint: Look carefully at your graph.)
5.10 (0) Central High School has $60,000 to spend on computers and
other stuﬀ, so its budget equation is C + X = 60, 000, where C is expenditure
on computers and X is expenditures on other things. C.H.S.
currently plans to spend $20,000 on computers.
The State Education Commission wants to encourage “computer literacy”
in the high schools under its jurisdiction. The following plans have
been proposed.
Plan A: This plan would give a grant of $10,000 to each high school in
the state that the school could spend as it wished.
Plan B: This plan would give a $10,000 grant to any high school, so
long as the school spent at least $10,000 more than it currently spends on
computers. Any high school can choose not to participate, in which case it
does not receive the grant, but it doesn’t have to increase its expenditure
on computers.
Plan C: Plan C is a “matching grant.” For every dollar’s worth of
computers that a high school orders, the state will give the school 50
cents.
Plan D: This plan is like plan C, except that the maximum amount of
matching funds that any high school could get from the state would be
limited to $10,000.
(a) Write an equation for Central High School’s budget if plan A is
adopted. Use black ink to draw the budget line for
Central High School if plan A is adopted.
(b) If plan B is adopted, the boundary of Central High School’s budget set
has two separate downward-sloping line segments. One of these segments
describes the cases where C.H.S. spends at least $30,000 on computers.
This line segment runs from the point (C, X) = (70, 000, 0) to the point
(C, X) = .
(c) Another line segment corresponds to the cases where C.H.S. spends
less than $30,000 on computers. This line segment runs from (C, X) =
to the point (C, X) = (0, 60, 000). Use red ink to draw
these two line segments.
(d) If plan C is adopted and Central High School spends C dollars on
computers, then it will have X = 60, 000 − .5C dollars left to spend on
other things. Therefore its budget line has the equation
Use blue ink to draw this budget line.
(e) If plan D is adopted, the school district’s budget consists of two line
segments that intersect at the point where expenditure on computers is
and expenditure on other instructional materials is
.
(f) The slope of the ﬂatter line segment is The slope of the
steeper segment is Use pencil to draw this budget line.
Thousands of dollars worth of other things
60
50
40
30
20
10
0 10 20 30 40 50 60
Thousands of dollars worth of computers
5.11 (0) Suppose that Central High School has preferences that can
be represented by the utility function U(C, X) = CX2
. Let us try to
determine how the various plans described in the last problem will aﬀect
the amount that C.H.S. spends on computers.
(a) If the state adopts none of the new plans, ﬁnd the expenditure on
computers that maximizes the district’s utility subject to its budget constraint.
.
(b) If plan A is adopted, ﬁnd the expenditure on computers that maximizes
the district’s utility subject to its budget constraint. .
(c) On your graph, sketch the indiﬀerence curve that passes through the
point (30,000, 40,000) if plan B is adopted. At this point, which is steeper,
the indiﬀerence curve or the budget line? .
(d) If plan B is adopted, ﬁnd the expenditure on computers that maximizes
the district’s utility subject to its budget constraint. (Hint: Look
at your graph.) .
(e) If plan C is adopted, ﬁnd the expenditure on computers that maximizes
the district’s utility subject to its budget constraint. .
(f) If plan D is adopted, ﬁnd the expenditure on computers that maximizes
the district’s utility subject to its budget constraint. .
5.12 (0) The telephone company allows one to choose between two
diﬀerent pricing plans. For a fee of $12 per month you can make as
many local phone calls as you want, at no additional charge per call.
Alternatively, you can pay $8 per month and be charged 5 cents for each
local phone call that you make. Suppose that you have a total of $20 per
month to spend.
(a) On the graph below, use black ink to sketch a budget line for someone
who chooses the ﬁrst plan. Use red ink to draw a budget line for someone
who chooses the second plan. Where do the two budget lines cross?
.
Other goods
16
12
8
4
0 20 40 60 80 100 120
Local phone calls
(b) On the graph above, use pencil to draw indiﬀerence curves for someone
who prefers the second plan to the ﬁrst. Use blue ink to draw an
indiﬀerence curve for someone who prefers the ﬁrst plan to the second.
5.13 (1) This is a puzzle—just for fun. Lewis Carroll (1832-1898),
author of Alice in Wonderland and Through the Looking Glass, was a
mathematician, logician, and political scientist. Carroll loved careful reasoning
about puzzling things. Here Carroll’s Alice presents a nice bit
of economic analysis. At ﬁrst glance, it may seem that Alice is talking
nonsense, but, indeed, her reasoning is impeccable.
“I should like to buy an egg, please.” she said timidly. “How do you
sell them?”
“Fivepence farthing for one—twopence for two,” the Sheep replied.
“Then two are cheaper than one?” Alice said, taking out her purse.
“Only you must eat them both if you buy two,” said the Sheep.
“Then I’ll have one please,” said Alice, as she put the money down
on the counter. For she thought to herself, “They mightn’t be at all nice,
you know.”
(a) Let us try to draw a budget set and indiﬀerence curves that are
consistent with this story. Suppose that Alice has a total of 8 pence to
spend and that she can buy either 0, 1, or 2 eggs from the Sheep, but no
fractional eggs. Then her budget set consists of just three points. The
point where she buys no eggs is (0, 8). Plot this point and label it A. On
your graph, the point where she buys 1 egg is (1, 23
4 ). (A farthing is 1/4
of a penny.) Plot this point and label it B.
(b) The point where she buys 2 eggs is Plot this point and
label it C. If Alice chooses to buy 1 egg, she must like the bundle B better
than either the bundle A or the bundle C. Draw indiﬀerence curves for
Alice that are consistent with this behavior.
Other goods
8
6
4
2
0 1 2 3 4
Eggs
In the last section, you were given a consumer’s preferences and then you
solved for his or her demand behavior. In this chapter we turn this process
around: you are given information about a consumer’s demand behavior
and you must deduce something about the consumer’s preferences. The
main tool is the weak axiom of revealed preference. This axiom says the
following. If a consumer chooses commodity bundle A when she can aﬀord
bundle B, then she will never choose bundle B from any budget in which
she can also aﬀord A. The idea behind this axiom is that if you choose A
when you could have had B, you must like A better than B. But if you
like A better than B, then you will never choose B when you can have A.
If somebody chooses A when she can aﬀord B, we say that for her, A is
directly revealed preferred to B. The weak axiom says that if A is directly
revealed preferred to B, then B is not directly revealed preferred to A.
Let us look at an example of how you check whether one bundle is revealed
preferred to another. Suppose that a consumer buys the bundle
(xA
1 , xA
2 ) = (2, 3) at prices (pA
1 , pA
2 ) = (1, 4). The cost of bundle (xA
1 , xA
2 )
at these prices is (2 × 1) + (3 × 4) = 14. Bundle (2, 3) is directly revealed
preferred to all the other bundles that she can aﬀord at prices (1, 4), when
she has an income of 14. For example, the bundle (5, 2) costs only 13 at
prices (1, 4), so we can say that for this consumer (2, 3) is directly revealed
preferred to (1, 4).
You will also have some problems about price and quantity indexes.
A price index is a comparison of average price levels between two diﬀerent
times or two diﬀerent places. If there is more than one commodity, it is not
necessarily the case that all prices changed in the same proportion. Let us
suppose that we want to compare the price level in the “current year” with
the price level in some “base year.” One way to make this comparison
is to compare the costs in the two years of some “reference” commodity
bundle. Two reasonable choices for the reference bundle come to mind.
One possibility is to use the current year’s consumption bundle for the
reference bundle. The other possibility is to use the bundle consumed
in the base year. Typically these will be diﬀerent bundles. If the baseyear
bundle is the reference bundle, the resulting price index is called the
Laspeyres price index. If the current year’s consumption bundle is the
reference bundle, then the index is called the Paasche price index.
Suppose that there are just two goods. In 1980, the prices were (1, 3) and
a consumer consumed the bundle (4, 2). In 1990, the prices were (2, 4) and
the consumer consumed the bundle (3, 3). The cost of the 1980 bundle at
1980 prices is (1 × 4) + (3 × 2) = 10. The cost of this same bundle at 1990
prices is (2 × 4) + (4 × 2) = 16. If 1980 is treated as the base year and
1990 as the current year, the Laspeyres price ratio is 16/10. To calculate
the Paasche price ratio, you ﬁnd the ratio of the cost of the 1990 bundle
at 1990 prices to the cost of the same bundle at 1980 prices. The 1990
bundle costs (2 × 3) + (4 × 3) = 18 at 1990 prices. The same bundle cost
(1 × 3) + (3 × 3) = 12 at 1980 prices. Therefore the Paasche price index
is 18/12. Notice that both price indexes indicate that prices rose, but
because the price changes are weighted diﬀerently, the two approaches
give diﬀerent price ratios.
Making an index of the “quantity” of stuﬀ consumed in the two
periods presents a similar problem. How do you weight changes in the
amount of good 1 relative to changes in the amount of good 2? This time
we could compare the cost of the two periods’ bundles evaluated at some
reference prices. Again there are at least two reasonable possibilities, the
Laspeyres quantity index and the Paasche quantity index. The Laspeyres
quantity index uses the base-year prices as the reference prices, and the
Paasche quantity index uses current prices as reference prices.
In the example above, the Laspeyres quantity index is the ratio of the
cost of the 1990 bundle at 1980 prices to the cost of the 1980 bundle at
1980 prices. The cost of the 1990 bundle at 1980 prices is 12 and the cost
of the 1980 bundle at 1980 prices is 10, so the Laspeyres quantity index
is 12/10. The cost of the 1990 bundle at 1990 prices is 18 and the cost
of the 1980 bundle at 1990 prices is 16. Therefore the Paasche quantity
index is 18/16.
When you have completed this section, we hope that you will be able
to do the following:
• Decide from given data about prices and consumption whether one
commodity bundle is preferred to another.
• Given price and consumption data, calculate Paasche and Laspeyres
price and quantity indexes.
• Use the weak axiom of revealed preferences to make logical deductions
about behavior.
• Use the idea of revealed preference to make comparisons of well-being
across time and across countries.
7.1 (0) When prices are (4, 6), Goldie chooses the bundle (6, 6), and
when prices are (6, 3), she chooses the bundle (10, 0).
(a) On the graph below, show Goldie’s ﬁrst budget line in red ink and
her second budget line in blue ink. Mark her choice from the ﬁrst budget
with the label A, and her choice from the second budget with the label
B.
(b) Is Goldie’s behavior consistent with the weak axiom of revealed preference?
.
Good 2
20
15
10
5
0 5 10 15 20
Good 1
7.2 (0) Freddy Frolic consumes only asparagus and tomatoes, which are
highly seasonal crops in Freddy’s part of the world. He sells umbrellas for
a living, which provides a ﬂuctuating income depending on the weather.
But Freddy doesn’t mind; he never thinks of tomorrow, so each week he
spends as much as he earns. One week, when the prices of asparagus and
tomatoes were each $1 a pound, Freddy consumed 15 pounds of each. Use
blue ink to show the budget line in the diagram below. Label Freddy’s
consumption bundle with the letter A.
(a) What is Freddy’s income? .
(b) The next week the price of tomatoes rose to $2 a pound, but the price
of asparagus remained at $1 a pound. By chance, Freddy’s income had
changed so that his old consumption bundle of (15,15) was just aﬀordable
at the new prices. Use red ink to draw this new budget line on the graph
below. Does your new budget line go through the point A?
What is the slope of this line? .
(c) How much asparagus can he aﬀord now if he spent all of his income
on asparagus? .
(d) What is Freddy’s income now? .
(e) Use pencil to shade the bundles of goods on Freddy’s new red budget
line that he deﬁnitely will not purchase with this budget. Is it possible
that he would increase his consumption of tomatoes when his budget
changes from the blue line to the red one? .
Tomatoes
40
30
20
10
0 10 20 30 40
Asparagus
7.3 (0) Pierre consumes bread and wine. For Pierre, the price of bread
is 4 francs per loaf, and the price of wine is 4 francs per glass. Pierre has
an income of 40 francs per day. Pierre consumes 6 glasses of wine and 4
loaves of bread per day.
Bob also consumes bread and wine. For Bob, the price of bread is
1/2 dollar per loaf and the price of wine is 2 dollars per glass. Bob has
an income of $15 per day.
(a) If Bob and Pierre have the same tastes, can you tell whether Bob is
better oﬀ than Pierre or vice versa? Explain.
.
(b) Suppose prices and incomes for Pierre and Bob are as above and that
Pierre’s consumption is as before. Suppose that Bob spends all of his income.
Give an example of a consumption bundle of wine and bread such
that, if Bob bought this bundle, we would know that Bob’s tastes are not
the same as Pierre’s tastes.
.
7.4 (0) Here is a table of prices and the demands of a consumer named
Ronald whose behavior was observed in 5 diﬀerent price-income situa-
tions.
Situation p1 p2 x1 x2
A 1 1 5 35
B 1 2 35 10
C 1 1 10 15
D 3 1 5 15
E 1 2 10 10
(a) Sketch each of his budget lines and label the point chosen in each case
by the letters A, B, C, D, and E.
(b) Is Ronald’s behavior consistent with the Weak Axiom of Revealed
Preference? .
(c) Shade lightly in red ink all of the points that you are certain are worse
for Ronald than the bundle C.
(d) Suppose that you are told that Ronald has convex and monotonic
preferences and that he obeys the strong axiom of revealed preference.
Shade lightly in blue ink all of the points that you are certain are at least
as good as the bundle C.
x2
40
30
20
10
0 10 20 30 40
x1
7.5 (0) Horst and Nigel live in diﬀerent countries. Possibly they have
diﬀerent preferences, and certainly they face diﬀerent prices. They each
consume only two goods, x and y. Horst has to pay 14 marks per unit of
x and 5 marks per unit of y. Horst spends his entire income of 167 marks
on 8 units of x and 11 units of y. Good x costs Nigel 9 quid per unit and
good y costs him 7 quid per unit. Nigel buys 10 units of x and 9 units of
y.
(a) Which prices and income would Horst prefer, Nigel’s income and
prices or his own, or is there too little information to tell? Explain
your answer.
.
(b) Would Nigel prefer to have Horst’s income and prices or his own, or
is there too little information to tell?
.
7.6 (0) Here is a table that illustrates some observed prices and choices
for three diﬀerent goods at three diﬀerent prices in three diﬀerent situa-
tions.
Situation p1 p2 p3 x1 x2 x3
A 1 2 8 2 1 3
B 4 1 8 3 4 2
C 3 1 2 2 6 2
(a) We will ﬁll in the table below as follows. Where i and j stand for any
of the letters A, B, and C in Row i and Column j of the matrix, write
the value of the Situation-j bundle at the Situation-i prices. For example,
in Row A and Column A, we put the value of the bundle purchased in
Situation A at Situation A prices. From the table above, we see that in
Situation A, the consumer bought bundle (2, 1, 3) at prices (1, 2, 8). The
cost of this bundle A at prices A is therefore (1×2)+(2×1)+(8×3) = 28,
so we put 28 in Row A, Column A. In Situation B the consumer bought
bundle (3, 4, 2). The value of the Situation-B bundle, evaluated at the
situation-A prices is (1 × 3) + (2 × 4) + (8 × 2) = 27, so put 27 in Row
A, Column B. We have ﬁlled in some of the boxes, but we leave a few for
you to do.
Prices/Quantities A B C
A 28 27
B 32 30
C 13 17
(b) Fill in the entry in Row i and Column j of the table below with a D if
the Situation-i bundle is directly revealed preferred to the Situation-j bundle.
For example, in Situation A the consumer’s expenditure is $28. We
see that at Situation-A prices, he could also aﬀord the Situation-B bundle,
which cost 27. Therefore the Situation-A bundle is directly revealed
preferred to the Situation-B bundle, so we put a D in Row A, Column
B. Now let us consider Row B, Column A. The cost of the Situation-B
bundle at Situation-B prices is 32. The cost of the Situation-A bundle
at Situation-B prices is 33. So, in Situation B, the consumer could not
aﬀord the Situation-A bundle. Therefore Situation B is not directly revealed
preferred to Situation A. So we leave the entry in Row B, Column
A blank. Generally, there is a D in Row i Column j if the number in the
ij entry of the table in part (a) is less than or equal to the entry in Row
i, Column i. There will be a violation of WARP if for some i and j, there
is a D in Row i Column j and also a D in Row j, Column i. Do these
observations violate WARP? .
Situation A B C
A —
B — D
C —
(c) Now ﬁll in Row i, Column j with an I if observation i is indirectly
revealed preferred to j. Do these observations violate the Strong Axiom
of Revealed Preference? .
7.7 (0) It is January, and Joe Grad, whom we met in Chapter 5, is
shivering in his apartment when the phone rings. It is Mandy Manana,
one of the students whose price theory problems he graded last term.
Mandy asks if Joe would be interested in spending the month of February
in her apartment. Mandy, who has switched majors from economics to
political science, plans to go to Aspen for the month and so her apartment
will be empty (alas). All Mandy asks is that Joe pay the monthly service
charge of $40 charged by her landlord and the heating bill for the month
of February. Since her apartment is much better insulated than Joe’s,
it only costs $1 per month to raise the temperature by 1 degree. Joe
thanks her and says he will let her know tomorrow. Joe puts his earmuﬀs
back on and muses. If he accepts Mandy’s oﬀer, he will still have to pay
rent on his current apartment but he won’t have to heat it. If he moved,
heating would be cheaper, but he would have the $40 service charge. The
outdoor temperature averages 20 degrees Fahrenheit in February, and it
costs him $2 per month to raise his apartment temperature by 1 degree.
Joe is still grading homework and has $100 a month left to spend on food
and utilities after he has paid the rent on his apartment. The price of
food is still $1 per unit.
(a) Draw Joe’s budget line for February if he moves to Mandy’s apartment
and on the same graph, draw his budget line if he doesn’t move.
(b) After drawing these lines himself, Joe decides that he would be better
oﬀ not moving. From this, we can tell, using the principle of revealed
preference that Joe must plan to keep his apartment at a temperature of
less than .
(c) Joe calls Mandy and tells her his decision. Mandy oﬀers to pay half
the service charge. Draw Joe’s budget line if he accepts Mandy’s new
oﬀer. Joe now accepts Mandy’s oﬀer. From the fact that Joe accepted
this oﬀer we can tell that he plans to keep the temperature in Mandy’s
apartment above .
Food
120
100
80
60
40
20
0 10 20 30 40 50 60 70 80
Temperature
7.8 (0) Lord Peter Pommy is a distinguished criminologist, schooled
in the latest techniques of forensic revealed preference. Lord Peter is investigating
the disappearance of Sir Cedric Pinchbottom who abandoned
his aging mother on a street corner in Liverpool and has not been seen
since. Lord Peter has learned that Sir Cedric left England and is living
under an assumed name somewhere in the Empire. There are three suspects,
R. Preston McAfee of Brass Monkey, Ontario, Canada, Richard
Manning of North Shag, New Zealand, and Richard Stevenson of Gooey
Shoes, Falkland Islands. Lord Peter has obtained Sir Cedric’s diary, which
recorded his consumption habits in minute detail. By careful observation,
he has also discovered the consumption behavior of McAfee, Manning, and
Stevenson. All three of these gentlemen, like Sir Cedric, spend their entire
incomes on beer and sausage. Their dossiers reveal the following:
• Sir Cedric Pinchbottom — In the year before his departure, Sir
Cedric consumed 10 kilograms of sausage and 20 liters of beer per
week. At that time, beer cost 1 English pound per liter and sausage
cost 1 English pound per kilogram.
• R. Preston McAfee — McAfee is known to consume 5 liters of beer
and 20 kilograms of sausage. In Brass Monkey, Ontario beer costs 1
Canadian dollar per liter and sausage costs 2 Canadian dollars per
kilogram.
• Richard Manning — Manning consumes 5 kilograms of sausage
and 10 liters of beer per week. In North Shag, a liter of beer costs
2 New Zealand dollars and sausage costs 2 New Zealand dollars per
kilogram.
• Richard Stevenson — Stevenson consumes 5 kilograms of sausage
and 30 liters of beer per week. In Gooey Shoes, a liter of beer costs 10
Falkland Island pounds and sausage costs 20 Falkland Island pounds
per kilogram.
(a) Draw the budget line for each of the three fugitives, using a diﬀerent
color of ink for each one. Label the consumption bundle that each chooses.
On this graph, superimpose Sir Cedric’s budget line and the bundle he
chose.
Sausage
40
30
20
10
0 10 20 30 40
Beer
(b) After pondering the dossiers for a few moments, Lord Peter announced.
“Unless Sir Cedric has changed his tastes, I can eliminate one
of the suspects. Revealed preference tells me that one of the suspects is
innocent.” Which one? .
(c) After thinking a bit longer, Lord Peter announced. “If Sir Cedric
left voluntarily, then he would have to be better oﬀ than he was before.
Therefore if Sir Cedric left voluntarily and if he has not changed his tastes,
he must be living in .
7.9 (1) The McCawber family is having a tough time making ends meet.
They spend $100 a week on food and $50 on other things. A new welfare
program has been introduced that gives them a choice between receiving
a grant of $50 per week that they can spend any way they want, and
buying any number of $2 food coupons for $1 apiece. (They naturally
are not allowed to resell these coupons.) Food is a normal good for the
McCawbers. As a family friend, you have been asked to help them decide
on which option to choose. Drawing on your growing fund of economic
knowledge, you proceed as follows.
(a) On the graph below, draw their old budget line in red ink and label
their current choice C. Now use black ink to draw the budget line that
they would have with the grant. If they chose the coupon option, how
much food could they buy if they spent all their money on food coupons?
How much could they spend on other things if they bought
no food? Use blue ink to draw their budget line if they
choose the coupon option.
Other things
180
150
120
90
60
30
0 30 60 90 120 150 180 210 240
Food
(b) Using the fact that food is a normal good for the McCawbers, and
knowing what they purchased before, darken the portion of the black
budget line where their consumption bundle could possibly be if they
chose the lump-sum grant option. Label the ends of this line segment A
and B.
(c) After studying the graph you have drawn, you report to the McCawbers.
“I have enough information to be able to tell you which choice to
make. You should choose the because
.
(d) Mr. McCawber thanks you for your help and then asks, “Would you
have been able to tell me what to do if you hadn’t known whether food
was a normal good for us?” On the axes below, draw the same budget
lines you drew on the diagram above, but draw indiﬀerence curves for
which food is not a normal good and for which the McCawbers would be
better oﬀ with the program you advised them not to take.
Other things
180
150
120
90
60
30
0 30 60 90 120 150 180 210 240
Food
7.10 (0) In 1933, the Swedish economist Gunnar Myrdal (who later won
a Nobel prize in economics) and a group of his associates at Stockholm
University collected a fantastically detailed historical series of prices and
price indexes in Sweden from 1830 until 1930. This was published in a
book called The Cost of Living in Sweden. In this book you can ﬁnd
100 years of prices for goods such as oat groats, hard rye bread, salted
codﬁsh, beef, reindeer meat, birchwood, tallow candles, eggs, sugar, and
coﬀee. There are also estimates of the quantities of each good consumed
by an average working-class family in 1850 and again in 1890.
The table below gives prices in 1830, 1850, 1890, and 1913, for ﬂour,
meat, milk, and potatoes. In this time period, these four staple foods
accounted for about 2/3 of the Swedish food budget.
Prices of Staple Foods in Sweden
Prices are in Swedish kronor per kilogram, except for milk, which is in
Swedish kronor per liter.
1830 1850 1890 1913
Grain Flour .14 .14 .16 .19
Meat .28 .34 .66 .85
Milk .07 .08 .10 .13
Potatoes .032 .044 .051 .064
Based on the tables published in Myrdal’s book, typical consumption
bundles for a working-class Swedish family in 1850 and 1890 are
listed below. (The reader should be warned that we have made some
approximations and simpliﬁcations to draw these simple tables from the
much more detailed information in the original study.)
Quantities Consumed by a Typical Swedish Family
Quantities are measured in kilograms per year, except for milk, which is
measured in liters per year.
1850 1890
Grain Flour 165 220
Meat 22 42
Milk 120 180
Potatoes 200 200
(a) Complete the table below, which reports the annual cost of the 1850
and 1890 bundles of staple foods at various years’ prices.
Cost of 1850 and 1890 Bundles at Various Years’ Prices
Cost 1850 bundle 1890 bundle
Cost at 1830 Prices 44.1 61.6
Cost at 1850 Prices
Cost at 1890 Prices
Cost at 1913 Prices 78.5 113.7
(b) Is the 1890 bundle revealed preferred to the 1850 bundle? .
(c) The Laspeyres quantity index for 1890 with base year 1850 is the ratio
of the value of the 1890 bundle at 1850 prices to the value of the 1850
bundle at 1850 prices. Calculate the Laspeyres quantity index of staple
food consumption for 1890 with base year 1850. .
(d) The Paasche quantity index for 1890 with base year 1850 is the ratio
of the value of the 1890 bundle at 1890 prices to the value of the 1850
bundle at 1890 prices. Calculate the Paasche quantity index for 1890 with
base year 1850. .
(e) The Laspeyres price index for 1890 with base year 1850 is calculated
using 1850 quantities for weights. Calculate the Laspeyres price index for
1890 with base year 1850 for this group of four staple foods. .
(f) If a Swede were rich enough in 1850 to aﬀord the 1890 bundle of staple
foods in 1850, he would have to spend times as much on these
foods as does the typical Swedish worker of 1850.
(g) If a Swede in 1890 decided to purchase the same bundle of food staples
that was consumed by typical 1850 workers, he would spend the fraction
of the amount that the typical Swedish worker of 1890 spends
on these goods.
7.11 (0) This question draws from the tables in the previous question.
Let us try to get an idea of what it would cost an American family at
today’s prices to purchase the bundle consumed by an average Swedish
family in 1850. In the United States today, the price of ﬂour is about $.40
per kilogram, the price of meat is about $3.75 per kilogram, the price of
milk is about $.50 per liter, and the price of potatoes is about $1 per
kilogram. We can also compute a Laspeyres price index across time and
across countries and use it to estimate the value of a current US dollar
relative to the value of an 1850 Swedish kronor.
(a) How much would it cost an American at today’s prices to buy the bundle
of staple food commodities purchased by an average Swedish workingclass
family in 1850? .
(b) Myrdal estimates that in 1850, about 2/3 of the average family’s
budget was spent on food. In turn, the four staples discussed in the last
question constitute about 2/3 of the average family’s food budget. If the
prices of other goods relative to the price of the food staples are similar
in the United States today to what they were in Sweden in 1850, about
how much would it cost an American at current prices to consume the
same overall consumption bundle consumed by a Swedish working-class
family in 1850? .
(c) Using the Swedish consumption bundle of staple foods in 1850 as
weights, calculate a Laspeyres price index to compare prices in current
American dollars relative to prices in 1850 Swedish kronor.
If we use this to estimate the value of current dollars relative to 1850
Swedish kronor, we would say that a U.S. dollar today is worth about
1850 Swedish kronor.
7.12 (0) Suppose that between 1960 and 1985, the price of all goods
exactly doubled while every consumer’s income tripled.
(a) Would the Laspeyres price index for 1985, with base year 1960 be
less than 2, greater than 2, or exactly equal to 2? What
about the Paasche price index? .
(b) If bananas are a normal good, will total banana consumption increase?
If everybody has homothetic preferences, can you
determine by what percentage total banana consumption must have increased?
Explain.
.
7.13 (1) Norm and Sheila consume only meat pies and beer. Meat pies
used to cost $2 each and beer was $1 per can. Their gross income used
to be $60 per week, but they had to pay an income tax of $10. Use red
ink to sketch their old budget line for meat pies and beer.
Beer
60
50
40
30
20
10
0 10 20 30 40 50 60
Meat pies
(a) They used to buy 30 cans of beer per week and spent the rest of their
income on meat pies. How many meat pies did they buy? .
(b) The government decided to eliminate the income tax and to put a
sales tax of $1 per can on beer, raising its price to $2 per can. Assuming
that Norm and Sheila’s pre-tax income and the price of meat pies did not
change, draw their new budget line in blue ink.
(c) The sales tax on beer induced Norm and Sheila to reduce their beer
consumption to 20 cans per week. What happened to their consumption
of meat pies? How much revenue did this tax
raise from Norm and Sheila? .
(d) This part of the problem will require some careful thinking. Suppose
that instead of just taxing beer, the government decided to tax both beer
and meat pies at the same percentage rate, and suppose that the price
of beer and the price of meat pies each went up by the full amount of
the tax. The new tax rate for both goods was set high enough to raise
exactly the same amount of money from Norm and Sheila as the tax on
beer used to raise. This new tax collects $ for every bottle of beer
sold and $ for every meat pie sold. (Hint: If both goods are
taxed at the same rate, the eﬀect is the same as an income tax.) How
large an income tax would it take to raise the same revenue as the $1 tax
on beer? Now you can ﬁgure out how big a tax on each good
is equivalent to an income tax of the amount you just found.
(e) Use black ink to draw the budget line for Norm and Sheila that corresponds
to the tax in the last section. Are Norm and Sheila better oﬀ
having just beer taxed or having both beer and meat pies taxed if both
sets of taxes raise the same revenue? (Hint: Try to use the
principle of revealed preference.)