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40 UTILITY (Ch. 4)
4.5 (0) As you may recall, Nancy Lerner is taking Professor Stern’s
economics course. She will take two examinations in the course, and her
score for the course is the minimum of the scores that she gets on the two
exams. Nancy wants to get the highest possible score for the course.
(a) Write a utility function that represents Nancy’s preferences over al-
ternative combinations of test scores x1and x2on tests 1 and 2 re-
4.6 (0) Remember Shirley Sixpack and Lorraine Quiche from the last
chapter? Shirley thinks a 16-ounce can of beer is just as good as two
8-ounce cans. Lorraine only drinks 8 ounces at a time and hates stale
beer, so she thinks a 16-ounce can is no better or worse than an 8-ounce
can.
(a) Write a utility function that represents Shirley’s preferences between
commodity bundles comprised of 8-ounce cans and 16-ounce cans of beer.
Let Xstand for the number of 8-ounce cans and Ystand for the number
(b) Now write a utility function that represents Lorraine’s preferences.
(c) Would the function utility U(X, Y ) = 100X+200Yrepresent Shirley’s
(d) Give an example of two commodity bundles such that Shirley likes
the first bundle better than the second bundle, while Lorraine likes the
4.7 (0) Harry Mazzola has the utility function u(x1,x
2)=min{x1+
2x2,2x1+x2},wherex1is his consumption of corn chips and x2is his
consumption of french fries.
(a) On the graph below, use a pencil to draw the locus of points along
which x1+2x2=2x1+x2. Use blue ink to show the locus of points for
NAME 41
(b) On the graph you have drawn, shade in the region where both of the
following inequalities are satisfied: x1+2x2≥12 and 2x1+x2≥12.
(c) Use black ink to sketch in the indifference curve along which Harry’s
utility is 12. Use red ink to sketch in the indifference curve along which
0246
8
2
4
6
Corn chips
French fries
8
Pencil line
Red
line
Blue
lines
Black line
Blue
lines
(d) At the point where Harry is consuming 5 units of corn chips and 2
units of french fries, how many units of corn chips would he be willing to
trade for one unit of french fries? 2.
(a) Use pencil to draw the locus of points at which x=y.Whatpoint
on this gives Vanna a utility of 10? (10,10).Useblueinktodraw
the line along which 2y−x= 10. When min{2x−y, 2y−x}=2y−x,
42 UTILITY (Ch. 4)
there are (more men than women, more women than men)? More
women. Draw a squiggly red line over the part of the blue line for which
men than women, more women than men)? More women. Now
draw a blue line along which 2x−y= 10. Draw a squiggly red line over
(b) Suppose that there are 9 men and 10 women at Vanna’s party. Would
Vanna think it was a better party or a worse party if 5 more men came
(c) If Vanna has 16 women at her party and more men than women, and
if she thinks the party is exactly as good as having 10 men and 10 women,
at her party and more women than men, and if she thinks the party is
exactly as good as having 10 men and 10 women, how many men does
(d) Vanna’s indifference curves are shaped like what letter of the alpha-
bet? V.
0 5 10 15 20
5
10
15
x
y
20
Pencil
line
Blue
lines
Squiggly
red
lines
4.9 (0) Suppose that the utility functions u(x, y)andv(x, y) are related
NAME 43
calculus users: A differentiable function f(u) is an increasing function of
uif its derivative is positive.)
(d) f(u)=log
10 u.Yes.
4.10 (0) Martha Modest has preferences represented by the utility func-
tion U(a, b)=ab/100, where ais the number of ounces of animal crackers
that she consumes and bis the number of ounces of beans that she con-
sumes.
(a) On the graph below, sketch the locus of points that Martha finds
indifferent to having 8 ounces of animal crackers and 2 ounces of beans.
Also sketch the locus of points that she finds indifferent to having 6 ounces
of animal crackers and 4 ounces of beans.
0246
8
2
4
6
Animal crackers
Beans
8
(8,2)
(6,4)
44 UTILITY (Ch. 4)
(b) Bertha Brassy has preferences represented by the utility function
V(a, b)=1,000a2b2,whereais the number of ounces of animal crack-
ers that she consumes and bis the number of ounces of beans that she
consumes. On the graph below, sketch the locus of points that Bertha
0246
8
2
4
6
Animal crackers
Beans
8
(8,2)
(6,4)
(e) How could you tell this was going to happen without having to draw
4.11 (0) Willy Wheeler’s preferences over bundles that contain non-
negative amounts of x1and x2are represented by the utility function
U(x1,x
2)=x2
1+x2
2.
(a) Draw a few of his indifference curves. What kind of geometric fig-
NAME 45
0246
8
2
4
6
x1
x2
8
Calculus 4.12 (0) Joe Bob has a utility function given by u(x1,x
2)=x2
1+2x1x2+
x2
2.
(a) Compute Joe Bob’s marginal rate of substitution: MRS(x1,x
2)=
−1.
(b) Joe Bob’s straight cousin, Al, has a utility function v(x1,x
2)=x2+x1.
Compute Al’s marginal rate of substitution. MRS(x1,x
2)= −1.
4.13 (0) The idea of assigning numerical values to determine a preference
ordering over a set of objects is not limited in application to commodity
bundles. The Bill James Baseball Abstract argues that a baseball player’s
batting average is not an adequate measure of his offensive productivity.
Batting averages treat singles just the same as extra base hits. Further-
more they do not give credit for “walks,” although a walk is almost as
good as a single. James argues that a double in two at-bats is better than
a single, but not as good as two singles. To reflect these considerations,
James proposes the following index, which he calls “runs created.” Let A
be the number of hits plus the number of walks that a batter gets in a sea-
son. Let Bbe the number of total bases that the batter gets in the season.
(Thus, if a batter has Ssingles, Wwalks, Ddoubles, Ttriples, and H
46 UTILITY (Ch. 4)
home runs, then A=S+D+T+H+Wand B=S+W+2D+3T+4H.)
Let Nbe the number of times the batter bats. Then his index of runs
created in the season is defined to be AB/N and will be called his RC.
(a) In 1987, George Bell batted 649 times. He had 39 walks, 105 singles,
32 doubles, 4 triples, and 47 home runs. In 1987, Wade Boggs batted 656
times. He had 105 walks, 130 singles, 40 doubles, 6 triples, and 24 home
runs. In 1987, Alan Trammell batted 657 times. He had 60 walks, 140
singles, 34 doubles, 3 triples, and 28 home runs. In 1987, Tony Gwynn
batted 671 times. He had 82 walks, 162 singles, 36 doubles, 13 triples, and
7 home runs. We can calculate A, the number of hits plus walks, Bthe
number of total bases, and RC, the runs created index for each of these
players. For Bell, A= 227, B= 408, RC = 143. For Boggs, A= 305,
B= 429, RC = 199. For Trammell, A= 265, B= 389, RC = 157. For
(b) If somebody has a preference ordering among these players, based only
on the runs-created index, which player(s) would she prefer to Trammell?
(c) The differences in the number of times at bat for these players are
small, and we will ignore them for simplicity of calculation. On the graph
below, plot the combinations of Aand Bachieved by each of the players.
Draw four “indifference curves,” one through each of the four points you
have plotted. These indifference curves should represent combinations of
Aand Bthat lead to the same number of runs-created.
0120 180 240 300 360
Number of hits plus walks
80
160
240
320
400
Number of total bases
480
60
Bell
Trammell
Gwynn
Boggs
NAME 47
4.14 (0) This problem concerns the runs-created index discussed in the
preceding problem. Consider a batter who bats 100 times and always
either makes an out, hits for a single, or hits a home run.
(a) Let xbe the number of singles and ybe the number of home runs
in 100 at-bats. Suppose that the utility function U(x, y)bywhichwe
evaluate alternative combinations of singles and home runs is the runs-
(b) Let’s try to find out about the shape of an indifference curve between
singles and home runs. Hitting 10 home runs and no singles would give
(c) Where xis the number of singles you solved for in the previous part,
mark the point (x/2,5) on your graph. Is U(x/2,5) greater than or less
0 5 10 15 20
5
10
15
Singles
Home runs
20
(0,10)
(20,0)
(10,5)
Preference
direction
