26 PREFERENCES (Ch. 3)
3.8 (0) Professor Goodheart always gives two midterms in his commu-
nications class. He only uses the higher of the two scores that a student
gets on the midterms when he calculates the course grade.
(a) Nancy Lerner wants to maximize her grade in this course. Let x1be
her score on the first midterm and x2be her score on the second midterm.
Which combination of scores would Nancy prefer, x1=20andx2=70
(b) On the graph below, use red ink to draw an indifference curve showing
all of the combinations of scores that Nancy likes exactly as much as
x1=20andx2= 70. Also use red ink to draw an indifference curve
showing the combinations that Nancy likes exactly as much as x1=60
and x2= 60.
0204060
80
20
40
60
Grade on first midterm
Grade on second midterm
80
Preference
direction
Blue curves
Red
curves
(d) Nancy is also taking a course in economics from Professor Stern.
Professor Stern gives two midterms. Instead of discarding the lower grade,
Professor Stern discards the higher one. Let x1be her score on the first
midterm and x2be her score on the second midterm. Which combination
of scores would Nancy prefer, x1=20andx2=70orx1=60and
(e) On the graph above, use blue ink to draw an indifference curve showing
all of the combinations of scores on her econ exams that Nancy likes
exactly as well as x1=20andx2= 70. Also use blue ink to draw an
indifference curve showing the combinations that Nancy likes exactly as
well as x1=60andx2= 50. Does Nancy have convex preferences over
NAME 27
3.9 (0) Mary Granola loves to consume two goods, grapefruits and
avocados.
(a) On the graph below, the slope of an indifference curve through any
point where she has more grapefruits than avocados is −2. This means
that when she has more grapefruits than avocados, she is willing to give
(b) On the same graph, the slope of an indifference curve at points where
she has fewer grapefruits than avocados is −1/2. This means that when
(c) On this graph, draw an indifference curve for Mary through bundle
0102030
40
10
20
30
Avocados
Grapefruits
40
45
Slope -2
Slope -1/2
(d) Does Mary have convex preferences? Yes.
3.10 (2) Ralph Rigid likes to eat lunch at 12 noon. However, he also
likes to save money so he can buy other consumption goods by attending
the “early bird specials” and “late lunchers” promoted by his local diner.
Ralph has 15 dollars a day to spend on lunch and other stuff. Lunch at
(a) If Ralph eats lunch at noon, how much money does he have per day
28 PREFERENCES (Ch. 3)
(b) How much money per day would he have left for other stuff if he ate
(c) On the graph below, use blue ink to draw the broken line that shows
combinations of meal time and money for other stuff that Ralph can just
0
11 12 1 2
5
10
15
Time
Money
20
10
3.11 (0) Henry Hanover is currently consuming 20 cheeseburgers and 20
Cheeseb
ur
b
b
gers
r
r
Cherry Cok
e
k
k
1
0
2
0
30
4
0
0
4
0
30
2
0
1
0
NAME 29
(a) If someone offered to trade Henry one extra cheeseburger for every
(b) What if it were the other way around: for every cheeseburger Henry
(c) At what rate of exchange would Henry be willing to stay put at his
3.12 (1) Tommy Twit is happiest when he has 8 cookies and 4 glasses of
milk per day. Whenever he has more than his favorite amount of either
food, giving him still more makes him worse off. Whenever he has less
than his favorite amount of either food, giving him more makes him better
off. His mother makes him drink 7 glasses of milk and only allows him 2
cookies per day. One day when his mother was gone, Tommy’s sadistic
sister made him eat 13 cookies and only gave him 1 glass of milk, despite
(a) Use black ink to draw some indifference curves for Tommy that are
consistent with this story.
0
(8,4)
(13,1)
(2,7)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Cookies
1
2
3
4
5
6
7
8
9
10
11
12
Milk
30 PREFERENCES (Ch. 3)
(b) Tommy’s mother believes that the optimal amount for him to consume
is 7 glasses of milk and 2 cookies. She measures deviations by absolute
values. If Tommy consumes some other bundle, say, (c, m), she measures
his departure from the optimal bundle by D=|7−m|+|2−c|.The
3.13 (0) Coach Steroid likes his players to be big, fast, and obedient. If
player Ais better than player Bin two of these three characteristics, then
Coach Steroid prefers Ato B, but if Bis better than Ain two of these
three characteristics, then Steroid prefers Bto A. Otherwise, Steroid is
indifferent between them. Wilbur Westinghouse weighs 340 pounds, runs
very slowly, and is fairly obedient. Harold Hotpoint weighs 240 pounds,
runs very fast, and is very disobedient. Jerry Jacuzzi weighs 150 pounds,
runs at average speed, and is extremely obedient.
(a) Does Steroid prefer Westinghouse to Hotpoint or vice versa? He
(b) Does Steroid prefer Hotpoint to Jacuzzi or vice versa? He
(c) Does Steroid prefer Westinghouse to Jacuzzi or vice versa? He
(e) After several losing seasons, Coach Steroid decides to change his way of
judging players. According to his new preferences, Steroid prefers player
Ato player Bif player Ais better in all three of the characteristics that
Steroid values, and he prefers Bto Aif player Bis better at all three
things. He is indifferent between Aand Bif they weigh the same, are
equally fast, and are equally obedient. In all other cases, Coach Steroid
simply says “Aand Bare not comparable.”
NAME 31
3.14 (0) The Bear family is trying to decide what to have for din-
ner. Baby Bear says that his ranking of the possibilities is (honey, grubs,
Goldilocks). Mama Bear ranks the choices (grubs, Goldilocks, honey),
while Papa Bear’s ranking is (Goldilocks, honey, grubs). They decide to
take each pair of alternatives and let a majority vote determine the family
rankings.
(a) Papa suggests that they first consider honey vs. grubs, and then the
(b) Mama suggests instead that they consider honey vs. Goldilocks and
(c) What order should Baby Bear suggest if he wants to get his favorite
(d) Are the Bear family’s “collective preferences,” as determined by vot-
ing, transitive? No.
3.15 (0) Olson likes strong coffee, the stronger the better. But he can’t
distinguish small differences. Over the years, Mrs. Olson has discovered
that if she changes the amount of coffee by more than one teaspoon in
her six-cup pot, Olson can tell that she did it. But he cannot distinguish
differences smaller than one teaspoon per pot. Where Aand Bare two
different cups of coffee, let us write ABif Olson prefers cup Ato
cup B.LetuswriteABif Olson either prefers Ato B, or can’t tell
the difference between them. Let us write A∼Bif Olson can’t tell the
difference between cups Aand B. Suppose that Olson is offered cups A,
B,andCall brewed in the Olsons’ six-cup pot. Cup Awas brewed using
14 teaspoons of coffee in the pot. Cup Bwas brewed using 14.75 teaspoons
of coffee in the pot and cup Cwas brewed using 15.5 teaspoons of coffee
in the pot. For each of the following expressions determine whether it is
true of false.
32 PREFERENCES (Ch. 3)
(f) AB.True.
(g) BA.True.
(h) BC.True.
(m) BC.False.
(p) Is Olson’s “at-least-as-good-as” relation, , transitive? No.