CHAPTER 3: Preferences
TRUE/FALSE
1. If preferences are transitive, more is always preferred to less.
2. A person with reflexive preferences is someone who does not shop carefully.
3. If someone has the utility function U = 1,000 + 2min x, y, then x and y are perfect complements for
that person.
4. A consumer with convex preferences who is indifferent between the bundles (5, 2) and (11, 6) will like
the bundle (8, 4) at least as well as either of the first two bundles.
5. A consumer with convex preferences who is indifferent between the bundles (5, 1) and (11, 3) will like
the bundle (8, 2) at least as well as either of the first two bundles.
6. If there are two goods, if a consumer prefers more of each good to less, and if she has a diminishing
marginal rate of substitution, then her preferences are convex.
7. If preferences are convex, then for any commodity bundle x, the set of commodity bundles that are
worse than x is a convex set.
8. Bill Katz prefers more of good 1 to less and he prefers less of good 2 to more. Bill has convex
preferences. If we draw his indifference curves with good 1 on the horizontal axis and good 2 on the
vertical axis, then his indifference curves have positive slope but get steeper as they rise.
9. The marginal rate of substitution measures the distance between one indifference curve and the next
one.
10. Ambrose has an indifference curve with equation x2 = 20 − 4x1/21. When Ambrose is consuming the
bundle (4, 16), his marginal rate of substitution is 25/4.
11. Nancy’s psychology teacher will give her a course grade that is the maximum of her scores on three
midterm examinations. Nancy has convex preferences over the possible combinations of midterm
scores.
12. If Melody has more classical records than rock and roll records, she is willing to exchange exactly 1
classical record for 2 rock and roll records, but if she has more rock and roll records than classical
records, then she is willing to exchange exactly 1 rock and roll record for 2 classical records. Melody
has convex preferences.
13. Josephine buys 3 quarts of milk and 2 pounds of butter when milk sells for $2 a quart and butter sells
for $1 a pound. Wilma buys 2 quarts of milk and 3 pounds of butter at the same prices. Josephine’s
marginal rate of substitution between milk and butter is greater than Wilma’s.
14. A consumer who is unable to detect small differences in the amount of water in her beer could have a
transitive strict preference relation but is unlikely to have a transitive indifference relation.
MULTIPLE CHOICE
1. Colette consumes goods x and y. Her indifference curves are described by the formula y = k/(x + 7).
Higher values of k correspond to better indifference curves.
a.
Colette likes good y and hates good x.
b.
Colette prefers bundle (12, 16) to bundle (16, 12).
c.
Colette prefers bundle (8, 5) to bundle (5, 8).
d.
Colette likes good x and hates good y.
e.
More than one of the above statements are true.
2. Angela consumes goods x and y. Her indifference curves are described by the formula y = k/(x + 3).
Higher values of k correspond to better indifference curves.
a.
Angela prefers bundle (8, 9) to bundle (9, 8).
b.
Angela likes good y and hates good x.
c.
Angela prefers bundle (11, 9) to bundle (9, 11).
d.
Angela likes good x and hates good y.
e.
More than one of the above statements are true.
3. Nick’s indifference curves are circles, all of which are centered at (12, 12). Of any two indifference
circles, he would rather be on the inner one than the outer one.
a.
Nick’s preferences are not complete.
b.
Nick prefers (16, 17) to (10, 10).
c.
Nick prefers (10, 17) to (10, 10).
d.
Nick prefers (8, 8) to (17, 21).
e.
More than one of the above statements are true.
4. Steven’s indifference curves are circles, all of which are centered at (15, 13). Of any two indifference
circles, he would rather be on the inner one than the outer one.
a.
Steven prefers (19, 22) to (13, 7).
b.
Steven prefers (13, 22) to (13, 7).
c.
Steven prefers (12, 10) to (22, 18).
d.
Steven’s preferences are not complete.
e.
More than one of the above statements are true.
5. Tim consumes only apples and bananas. He prefers more apples to fewer, but he gets tired of bananas.
If he consumes fewer than 29 bananas per week, he thinks that 1 banana is a perfect substitute for 1
apple. But you would have to pay him 1 apple for each banana beyond 29 that he consumes. The
indifference curve that passes through the consumption bundle with 30 apples and 39 bananas also
passes through the bundle with A apples and 21 bananas, where A equals
a.
25.
b.
28.
c.
34.
d.
36.
e.
None of the above.
6. Leo consumes only apples and bananas. He prefers more apples to fewer, but he gets tired of bananas.
If he consumes fewer than 24 bananas per week, he thinks that 1 banana is a perfect substitute for 1
apple. But you would have to pay him 1 apple for each banana beyond 24 that he consumes. The
indifference curve that passes through the consumption bundle with 31 apples and 36 bananas also
passes through the bundle with A apples and 18 bananas, where A equals
a.
29.
b.
23.
c.
31.
d.
25.
e.
None of the above.
7. If two goods are both desirable and preferences are convex, then
a.
there must be a kink in the indifference curves.
b.
indifference curves must be straight lines.
c.
if two bundles are indifferent, then an average of the two bundles is worse than either one.
d.
the marginal rate of substitution is constant along indifference curves.
e.
None of the above.
8. If there are only two goods, if more of good 1 is always preferred to less, and if less of good 2 is
always preferred to more, then indifference curves
a.
slope downward.
b.
slope upward.
c.
may cross.
d.
could take the form of ellipses.
e.
None of the above.
9. If two goods are perfect complements,
a.
there is a bliss point and the indifference curves surround this point.
b.
consumers will only buy the cheaper of the two goods.
c.
indifference curves have a positive slope.
d.
None of the above.
10. The relation “is preferred to” between commodity bundles is just one example of a binary relation.
Another example is the relation “is a full brother of” defined over the set of all human beings. Let xRy
mean person x is a full brother of person y.
a.
The relation R is reflexive, transitive, and complete.
b.
The relation R is transitive and complete but not reflexive.
c.
The relation R is transitive but not complete or reflexive.
d.
The relation R is complete but not transitive or reflexive.
e.
The relation R is not reflexive, transitive, or complete.
11. Preferences are said to be monotonic if
a.
all goods must be consumed in fixed proportions.
b.
all goods are perfect substitutes.
c.
more is always preferred to less.
d.
there is a diminishing marginal rate of substitution.
e.
None of the above.
12. Toby Talkalot subscribes to a local phone service that charges a fixed fee of $10 per month and allows
him to place as many local phone calls as he likes without further charge. Let good 1 be an aggregate
of commodities other than local phone use and let good 2 be local phone use. (Measure good 1 on the
horizontal axis and good 2 on the vertical axis.) On Monday, Toby didn’t use the telephone at all. The
slope m of his indifference curve at the consumption bundle he chose on Monday was
a.
positive.
b.
less than or equal to 0.
c.
0.
d.
greater than or equal to 0.
e.
negative.
13. Professor Goodheart’s colleague Dr. Kremepuff gives 3 midterm exams. He drops the lowest score
and gives each student her average score on the other two exams. Polly Sigh is taking his course and
has a 60 on her first exam. Let x2 be her score on the second exam and x3 be her score on the third
exam. If we draw her indifference curves for scores on the second and third exams with x2 represented
by the horizontal axis and x3 represented by the vertical axis, then her indifference curve through the
point (x2, x3) = (50,70) is
a.
L-shaped with a kink where x2 = x3.
b.
three line segments, one vertical, one horizontal, and one running from (70, 60) to
(60, 70).
c.
a straight line, running from (0, 120) to (120, 0).
d.
three line segments, one vertical, one horizontal, and one running from (70, 50) to
(50, 70).
e.
a V-shaped curve with its point at (50, 70).
14. Charlie’s indifference curves have the equation xB = constant/xA, where larger constants denote better
indifference curves. Charlie strictly prefers the bundle (6, 16) to
a.
the bundle (16, 6).
b.
the bundle (7, 15).
c.
the bundle (10, 11).
d.
more than one of these bundles.
e.
none of these bundles.
15. Charlie’s indifference curves have the equation xB = constant/xA where larger constants denote better
indifference curves. Charlie strictly prefers the bundle (10, 17) to
a.
the bundle (11, 16).
b.
the bundle (17, 10).
c.
the bundle (12, 15).
d.
more than one of these bundles.
e.
none of these bundles.
16. Ambrose has indifference curves with the equation x2 = constant − 4x1/21, where larger constants
correspond to higher indifference curves. If good 1 is drawn on the horizontal axis and good 2 on the
vertical axis, what is the slope of Ambrose’s indifference curve when his consumption bundle is
(16,17)?
a.
−16/17
b.
−17/16
c.
−0.50
d.
−21
e.
−4
17. Ambrose has indifference curves with the equation x2 = constant − 4x1/21, where larger constants
correspond to higher indifference curves. If good 1 is drawn on the horizontal axis and good 2 on the
vertical axis, what is the slope of Ambrose’s indifference curve when his consumption bundle is
(9, 5)?
a.
−0.67
b.
−8
c.
−9/5
d.
−5/9
e.
−3
18. Nancy Lerner is taking a course from Professor Goodheart who will count only her best midterm grade
and from Professor Stern who will count only her worst midterm grade. In one of her classes, Nancy
has scores of 30 on her first midterm and 50 on her second midterm. When the first midterm score is
measured on the horizontal axis and her second midterm score on the vertical, her indifference curve
has a slope of zero at the point (30, 50). Therefore this class could
a.
be Professor Goodheart’s but could not be Professor Stern’s.
b.
be Professor Stern’s but could not be Professor Goodheart’s.
c.
not be either Professor Goodheart’s or Professor Stern’s.
d.
be either Professor Goodheart’s or Professor Stern’s.
e.
None of the above.
19. Nancy Lerner is taking a course from Professor Goodheart who will count only her best midterm grade
and from Professor Stern who will count only her worst midterm grade. In one of her classes, Nancy
has scores of 20 on her first midterm and 70 on her second midterm. When the first midterm score is
measured on the horizontal axis and her second midterm score on the vertical, her indifference curve
has a slope of zero at the point (20, 70). Therefore this class could
a.
be Professor Goodheart’s but could not be Professor Stern’s.
b.
not be either Professor Goodheart’s or Professor Stern’s.
c.
be either Professor Goodheart’s or Professor Stern’s.
d.
be Professor Stern’s but could not be Professor Goodheart’s.
e.
None of the above.
20. If we graph Mary Granola’s indifference curves with avocados on the horizontal axis and grapefruits
on the vertical axis, then whenever she has more grapefruits than avocados, the slope of her
indifference curve is 22. Whenever she has more avocados than grapefruits, the slope is 21/2. Mary
would be indifferent between a bundle with 22 avocados and 37 grapefruits and another bundle that
has 37 avocados and
a.
27 grapefruits.
b.
32 grapefruits.
c.
17 grapefruits.
d.
22 grapefruits.
e.
24.5 grapefruits.
21. If we graph Mary Granola’s indifference curves with avocados on the horizontal axis and grapefruits
on the vertical axis, then whenever she has more grapefruits than avocados, the slope of her
indifference curve is 22. Whenever she has more avocados than grapefruits, the slope is 21/2. Mary
would be indifferent between a bundle with 14 avocados and 20 grapefruits and another bundle that
has 26 avocados and
a.
11 grapefruits.
b.
18 grapefruits.
c.
6 grapefruits.
d.
16 grapefruits.
e.
13.5 grapefruits.
22. Recall that Tommy Twit’s mother measures the departure of any bundle from her favorite bundle for
Tommy by the sum of the absolute values of the differences. Her favorite bundle for Tommy is (2, 7),
that is, 2 cookies and 7 glasses of milk. Tommy’s mother’s indifference curve that passes through the
point (c, m) = (4, 5) also passes through
a.
the point (6, 3).
b.
the points (2, 3), (6, 7), and (4, 9).
c.
the point (2, 7).
d.
the points (4, 7), (2, 5), and (2, 9).
e.
None of the above.
23. Recall that Tommy Twit’s mother measures the departure of any bundle from her favorite bundle for
Tommy by the sum of the absolute values of the differences. Her favorite bundle for Tommy is (2, 7),
that is, 2 cookies and 7 glasses of milk. Tommy’s mother’s indifference curve that passes through the
point (c, m) = (5, 4) also passes through
a.
the points (5, 7), (2, 4), and (2, 10).
b.
the points (2, 1), (8, 7), and (5, 10).
c.
the point (8, 1).
d.
the point (2, 7).
e.
None of the above.
24. Scholastica is taking a class from Professor Chaos. Professor Chaos gives two tests in this course and
determines a student’s grade as follows. He determines the smaller of the following two numbers: half
of the score on the first test (which is a relatively easy test) and the total score on the second test. He
gives each student a numerical score equal to the smaller number and then ranks the students.
Scholastica would like to be ranked as high as possible in Professor Chaos’s rankings. If we represent
her score on the first exam on the horizontal axis and her score on the second exam on the vertical
axis, then her indifference curves
a.
are L-shaped with kinks where the two exam scores are equal.
b.
have sections with a slope 22 and sections with a slope 1/2.
c.
are positively sloped.
d.
are L-shaped with kinks where the exam 1 score is twice the exam 2 score.
e.
are straight lines with a slope of 21/2.
25. In Professor Meanscore’s class, the first midterm exam and the second midterm exam are weighted
equally toward the final grade. With the first midterm’s score on the horizontal axis, and the second
midterm’s score on the vertical axis, indifference curves between the two exam scores are
a.
L-shaped with lines extending upward and to the right.
b.
L-shaped with lines extending downward and to the left.
c.
parabola shaped.
d.
straight lines with slope 21.
e.
straight lines with slope 2.
26. Professor Stern’s colleague, Dr. Schmertz, gives one midterm exam and a final exam. He weights the
final twice as heavily as the midterm to determine the course grade. No grades can be dropped. If the
midterm score is represented on the horizontal axis and the final score on the vertical axis, and if a
student in Dr. Schmertz’s class cares only about her course grade, her indifference curve is
a.
a line with slope 22.
b.
a line with slope 21.
c.
a line with slope 20.5.
d.
L-shaped with the kink at (x, 2x).
e.
L-shaped with the kink at (2x, x).
27. I prefer 6 apples and 1 orange to 5 apples and 2 oranges. My preferences
a.
are transitive.
b.
are complete.
c.
are convex.
d.
obey the Law of Demand.
e.
None of the above.
PROBLEM
1. Draw graphs with quantities of pepperoni pizza on the horizontal axis and quantities of anchovy pizza
on the vertical axis to illustrate the following situations. In each case draw two different indifference
curves and make a little arrow pointing in the direction of greater preference.
a. Marvin loves pepperoni pizza and hates anchovy pizza.
b. Mavis hates anchovy pizza and is completely indifferent about pepperoni pizza.
2. Coach Steroid likes his players to be big, fast, and obedient. If player A is better than player B in two
of these three characteristics, Steroid will prefer A to B. Three players try out for quarterback. Wilbur
Westinghouse weighs 320 pounds, runs very slowly, and is quite obedient. Harold Hotpoint weighs
240 pounds, runs extremely fast, and is extremely disobedient. Jerry Jacuzzi weighs 150 pounds, runs
at average speed, and is extremely obedient. Does Coach Steroid have transitive preferences? Explain
your answer.
3. Belinda loves chocolate and always thinks that more is better than less. Belinda thinks that a few piano
lessons would be worse than none at all, but if she had enough piano lessons to get good at playing the
piano, she would prefer more lessons to less. Draw a graph with piano lessons on the horizontal axis
and chocolate on the vertical axis. On your graph sketch two indifference curves for Belinda that
would be consistent with this story. Label the better of the two indifference curves AA and the worse
one BB.
4. Mac Rowe doesn’t sweat the petty stuff. In fact, he just cannot detect small differences. He consumes
two goods, x and y. He prefers the bundle (x, y) to the bundle (x’, y’) if and only if (xy − x’y’ 1).
Otherwise he is indifferent between the two bundles.
a. Show that the relation of indifference is not transitive for Mac. (Hint: Give an example.)
b. Show that the preferred relation is transitive for Mac.
5. Blanche Carter has devised a system for rating the males in her economics class. She cares about their
intelligence and their looks. She has ranked each male on a scale of 1 to 5 for intelligence and 1 to 3
for looks. She defines a preference relation, R, as follows: xRy if boy x scores at least as high as boy y
in either looks or intelligence. Give an example to show that Blanche’s method of determining
preferences might not lead to transitive preferences.
6. Explain how it would be possible to cheat someone who had intransitive preferences. Be explicit about
what you would offer him if you were trying to exploit his intransitivity and what he would do in
response.
7. If good X is measured on the horizontal axis and good Y on the vertical, what can you say about the
preferences of someone whose indifference curves are
a. parallel to the Y axis?
b. positively sloped with more desirable indifference curves as one moves to the right?
c. negatively sloped with more desirable indifference curves as one moves to the left?
8. Suppose that there are two commodities and a consumer prefers more to less of each good. If the
consumer has transitive preferences, can her in difference curves cross? Sketch a brief proof of your
answer, and illustrate with a diagram.