curve, all the points on the line (except the endpoints) are preferred to
the endpoints. For this to be the case, it must be that the indifference
curve slopes upward ever more steeply as you move to the right along it.
You should convince yourself of this by making some drawings on scratch
18 PREFERENCES (Ch. 3)
paper. Draw an upward-sloping curve passing through the point (0,4)
and getting steeper as one moves to the right.
When you have completed this workout, we hope that you will be
able to do the following:
•Given the formula for an indifference curve, draw this curve, and find
•Determine whether a consumer prefers one bundle to another or is
•Draw indifference curves for the special cases of perfect substitutes
•Draw indifference curves for someone who likes goods up to a point
•Identify weakly preferred sets and determine whether these are con-
•Know what the marginal rate of substitution is and be able to deter-
•Determine whether a preference relation or any other relation be-
3.1 (0) Charlie likes both apples and bananas. He consumes nothing else.
The consumption bundle where Charlie consumes xAbushels of apples
peryearandxBbushels of bananas per year is written as (xA,x
B). Last
year, Charlie consumed 20 bushels of apples and 5 bushels of bananas. It
happens that the set of consumption bundles (xA,x
B) such that Charlie
is indifferent between (xA,x
B)and(20,5) is the set of all bundles such
that xB= 100/xA. The set of bundles (xA,x
B) such that Charlie is just
indifferent between (xA,x
B) and the bundle (10,15) is the set of bundles
such that xB= 150/xA.
(a) On the graph below, plot several points that lie on the indifference
curve that passes through the point (20,5), and sketch this curve, using
(b) Use pencil to shade in the set of commodity bundles that Charlie
weakly prefers to the bundle (10,15). Use blue ink to shade in the set
NAME 19
0102030
40
10
20
30
Apples
Bananas
40
Blue Curve
Pencil Shading
Red Curve
Blue Shading
For each of the following statements about Charlie’s preferences, write
“true” or “false.”
(h) A set is convex if for any two points in the set, the line segment
(i) Is the set of bundles that Charlie considers inferior to (20,5) a convex
(j) The slope of Charlie’s indifference curve through a point, (xA,x
B), is
20 PREFERENCES (Ch. 3)
(k) Remember that Charlie’s indifference curve through the point (10,10)
has the equation xB= 100/xA. Those of you who know calculus will
remember that the slope of a curve is just its derivative, which in this
case is −100/x2
A. (If you don’t know calculus, you will have to take our
word for this.) Find Charlie’s marginal rate of substitution at the point,
(m) What is his marginal rate of substitution at the point (20,5)?
(n) Do the indifference curves you have drawn for Charlie exhibit dimin-
3.2 (0) Ambrose consumes only nuts and berries. Fortunately, he likes
both goods. The consumption bundle where Ambrose consumes x1units
of nuts per week and x2units of berries per week is written as (x1,x
2).
The set of consumption bundles (x1,x
2) such that Ambrose is indifferent
between (x1,x
2)and(1,16) is the set of bundles such that x1≥0, x2≥0,
and x2=20−4√x1. The set of bundles (x1,x
2) such that (x1,x
2)∼
(36,0) is the set of bundles such that x1≥0, x2≥0andx2=24−4√x1.
(a) On the graph below, plot several points that lie on the indifference
curve that passes through the point (1,16), and sketch this curve, using
(b) Use pencil to shade in the set of commodity bundles that Ambrose
weakly prefers to the bundle (1,16). Use red ink to shade in the set of
(c) What is the slope of Ambrose’s indifference curve at the point (9,8)?
(Hint: Recall from calculus the way to calculate the slope of a curve. If
you don’t know calculus, you will have to draw your diagram carefully
NAME 21
(d) What is the slope of his indifference curve at the point (4,12)? −1.
0102030
40
10
20
30
Nuts
Berries
40
Pencil Shading
Red Curve
Red
Shading Blue Curve
(e) What is the slope of his indifference curve at the point (9,12)? −2/3
(f) Do the indifference curves you have drawn for Ambrose exhibit dimin-
3.3 (0) Shirley Sixpack is in the habit of drinking beer each evening
while watching “The Best of Bowlerama” on TV. She has a strong thumb
(a) On the graph below, draw some of Shirley’s indifference curves be-
22 PREFERENCES (Ch. 3)
0246
8
2
4
6
16-ounce
8-ounce
8
Blue Lines
Red Lines
(b) Lorraine Quiche likes to have a beer while she watches “Masterpiece
Theatre.” She only allows herself an 8-ounce glass of beer at any one
time. Since her cat doesn’t like beer and she hates stale beer, if there is
3.4 (0) Elmo finds himself at a Coke machine on a hot and dusty Sunday.
The Coke machine requires exact change—two quarters and a dime. No
other combination of coins will make anything come out of the machine.
No stores are open; no one is in sight. Elmo is so thirsty that the only
NAME 23
0246
8
2
4
6
Quarters
Dimes
8
Red
shading
Blue
shading
Black
lines
(a) If Elmo has 2 quarters and a dime in his pockets, he can buy 1 soft
2.
(b) Use red ink to shade in the area on the graph consisting of all com-
binations of quarters and dimes that Elmo thinks are just indifferent to
having 2 quarters and 1 dime. (Imagine that it is possible for Elmo to
(c) Does Elmo have convex preferences between dimes and quarters?
(f) If Elmo had arrived at the Coke machine on a Saturday, the drugstore
across the street would have been open. This drugstore has a soda foun-
tain that will sell you as much Coke as you want at a price of 4 cents an
ounce. The salesperson will take any combination of dimes and quarters
24 PREFERENCES (Ch. 3)
as if Elmo’s fractional quarters and fractional dimes are accepted at the
corresponding fraction of their value.) Describe these new indifference
3.5 (0) Randy Ratpack hates studying both economics and history. The
more time he spends studying either subject, the less happy he is. But
Randy has strictly convex preferences.
(a) Sketch an indifference curve for Randy where the two commodities
are hours per week spent studying economics and hours per week spent
studying history. Will the slope of an indifference curve be positive or
(b) Do Randy’s indifference curves get steeper or flatter as you move from
0246
8
2
4
6
Hours studying economics
Hours studying history
8
Preference
direction
3.6 (0) Flossy Toothsome likes to spend some time studying and some
time dating. In fact her indifference curves between hours per week spent
studying and hours per week spent dating are concentric circles around
her favorite combination, which is 20 hours of studying and 15 hours of
dating per week. The closer she is to her favorite combination, the happier
she is.
NAME 25
(a) Suppose that Flossy is currently studying 25 hours a week and dating
3 hours a week. Would she prefer to be studying 30 hours a week and
(b) On the axes below, draw a few of Flossy’s indifference curves and
0102030
40
10
20
30
Hours studying
Hours dating
40
(25,3)
(30,8)
(20,15)
Preference
direction
3.7 (0) Joan likes chocolate cake and ice cream, but after 10 slices of
cake, she gets tired of cake, and eating more cake makes her less happy.
Joan always prefers more ice cream to less. Joan’s parents require her to
eat everything put on her plate. In the axes below, use blue ink to draw a
set of indifference curves that depict her preferences between plates with
different amounts of cake and ice cream. Be sure to label the axes.
(a) Suppose that Joan’s preferences are as before, but that her parents
allow her to leave anything on her plate that she doesn’t want. On the
Blue curves
Red curves
Ice cream
Chocolate cake
10
Preference
direction
18 PREFERENCES (Ch. 3)
paper. Draw an upward-sloping curve passing through the point (0,4)
and getting steeper as one moves to the right.
When you have completed this workout, we hope that you will be
able to do the following:
•Given the formula for an indifference curve, draw this curve, and find
•Determine whether a consumer prefers one bundle to another or is
•Draw indifference curves for the special cases of perfect substitutes
•Draw indifference curves for someone who likes goods up to a point
•Identify weakly preferred sets and determine whether these are con-
•Know what the marginal rate of substitution is and be able to deter-
•Determine whether a preference relation or any other relation be-
3.1 (0) Charlie likes both apples and bananas. He consumes nothing else.
The consumption bundle where Charlie consumes xAbushels of apples
peryearandxBbushels of bananas per year is written as (xA,x
B). Last
year, Charlie consumed 20 bushels of apples and 5 bushels of bananas. It
happens that the set of consumption bundles (xA,x
B) such that Charlie
is indifferent between (xA,x
B)and(20,5) is the set of all bundles such
that xB= 100/xA. The set of bundles (xA,x
B) such that Charlie is just
indifferent between (xA,x
B) and the bundle (10,15) is the set of bundles
such that xB= 150/xA.
(a) On the graph below, plot several points that lie on the indifference
curve that passes through the point (20,5), and sketch this curve, using
(b) Use pencil to shade in the set of commodity bundles that Charlie
weakly prefers to the bundle (10,15). Use blue ink to shade in the set
NAME 19
0102030
40
10
20
30
Apples
Bananas
40
Blue Curve
Pencil Shading
Red Curve
Blue Shading
For each of the following statements about Charlie’s preferences, write
“true” or “false.”
(h) A set is convex if for any two points in the set, the line segment
(i) Is the set of bundles that Charlie considers inferior to (20,5) a convex
(j) The slope of Charlie’s indifference curve through a point, (xA,x
B), is
20 PREFERENCES (Ch. 3)
(k) Remember that Charlie’s indifference curve through the point (10,10)
has the equation xB= 100/xA. Those of you who know calculus will
remember that the slope of a curve is just its derivative, which in this
case is −100/x2
A. (If you don’t know calculus, you will have to take our
word for this.) Find Charlie’s marginal rate of substitution at the point,
(m) What is his marginal rate of substitution at the point (20,5)?
(n) Do the indifference curves you have drawn for Charlie exhibit dimin-
3.2 (0) Ambrose consumes only nuts and berries. Fortunately, he likes
both goods. The consumption bundle where Ambrose consumes x1units
of nuts per week and x2units of berries per week is written as (x1,x
2).
The set of consumption bundles (x1,x
2) such that Ambrose is indifferent
between (x1,x
2)and(1,16) is the set of bundles such that x1≥0, x2≥0,
and x2=20−4√x1. The set of bundles (x1,x
2) such that (x1,x
2)∼
(36,0) is the set of bundles such that x1≥0, x2≥0andx2=24−4√x1.
(a) On the graph below, plot several points that lie on the indifference
curve that passes through the point (1,16), and sketch this curve, using
(b) Use pencil to shade in the set of commodity bundles that Ambrose
weakly prefers to the bundle (1,16). Use red ink to shade in the set of
(c) What is the slope of Ambrose’s indifference curve at the point (9,8)?
(Hint: Recall from calculus the way to calculate the slope of a curve. If
you don’t know calculus, you will have to draw your diagram carefully
NAME 21
(d) What is the slope of his indifference curve at the point (4,12)? −1.
0102030
40
10
20
30
Nuts
Berries
40
Pencil Shading
Red Curve
Red
Shading Blue Curve
(e) What is the slope of his indifference curve at the point (9,12)? −2/3
(f) Do the indifference curves you have drawn for Ambrose exhibit dimin-
3.3 (0) Shirley Sixpack is in the habit of drinking beer each evening
while watching “The Best of Bowlerama” on TV. She has a strong thumb
(a) On the graph below, draw some of Shirley’s indifference curves be-
22 PREFERENCES (Ch. 3)
0246
8
2
4
6
16-ounce
8-ounce
8
Blue Lines
Red Lines
(b) Lorraine Quiche likes to have a beer while she watches “Masterpiece
Theatre.” She only allows herself an 8-ounce glass of beer at any one
time. Since her cat doesn’t like beer and she hates stale beer, if there is
3.4 (0) Elmo finds himself at a Coke machine on a hot and dusty Sunday.
The Coke machine requires exact change—two quarters and a dime. No
other combination of coins will make anything come out of the machine.
No stores are open; no one is in sight. Elmo is so thirsty that the only
NAME 23
0246
8
2
4
6
Quarters
Dimes
8
Red
shading
Blue
shading
Black
lines
(a) If Elmo has 2 quarters and a dime in his pockets, he can buy 1 soft
2.
(b) Use red ink to shade in the area on the graph consisting of all com-
binations of quarters and dimes that Elmo thinks are just indifferent to
having 2 quarters and 1 dime. (Imagine that it is possible for Elmo to
(c) Does Elmo have convex preferences between dimes and quarters?
(f) If Elmo had arrived at the Coke machine on a Saturday, the drugstore
across the street would have been open. This drugstore has a soda foun-
tain that will sell you as much Coke as you want at a price of 4 cents an
ounce. The salesperson will take any combination of dimes and quarters
24 PREFERENCES (Ch. 3)
as if Elmo’s fractional quarters and fractional dimes are accepted at the
corresponding fraction of their value.) Describe these new indifference
3.5 (0) Randy Ratpack hates studying both economics and history. The
more time he spends studying either subject, the less happy he is. But
Randy has strictly convex preferences.
(a) Sketch an indifference curve for Randy where the two commodities
are hours per week spent studying economics and hours per week spent
studying history. Will the slope of an indifference curve be positive or
(b) Do Randy’s indifference curves get steeper or flatter as you move from
0246
8
2
4
6
Hours studying economics
Hours studying history
8
Preference
direction
3.6 (0) Flossy Toothsome likes to spend some time studying and some
time dating. In fact her indifference curves between hours per week spent
studying and hours per week spent dating are concentric circles around
her favorite combination, which is 20 hours of studying and 15 hours of
dating per week. The closer she is to her favorite combination, the happier
she is.
NAME 25
(a) Suppose that Flossy is currently studying 25 hours a week and dating
3 hours a week. Would she prefer to be studying 30 hours a week and
(b) On the axes below, draw a few of Flossy’s indifference curves and
0102030
40
10
20
30
Hours studying
Hours dating
40
(25,3)
(30,8)
(20,15)
Preference
direction
3.7 (0) Joan likes chocolate cake and ice cream, but after 10 slices of
cake, she gets tired of cake, and eating more cake makes her less happy.
Joan always prefers more ice cream to less. Joan’s parents require her to
eat everything put on her plate. In the axes below, use blue ink to draw a
set of indifference curves that depict her preferences between plates with
different amounts of cake and ice cream. Be sure to label the axes.
(a) Suppose that Joan’s preferences are as before, but that her parents
allow her to leave anything on her plate that she doesn’t want. On the
Blue curves
Red curves
Ice cream
Chocolate cake
10
Preference
direction