Chapter 5 NAME
Choice
Introduction. You have studied budgets, and you have studied prefer-
ences. 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 bud-
get. 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)akinkinanindifference
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,x
2)ifthey
have the same slope at that point. Now the slope of an indifference curve
at (x1,x
2)istheratio−MU1(x1,x
2)/M U2(x1,x
2). (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
Example: A consumer has the utility function U(x1,x
2)=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,x
2)=2x1x2and MU2(x1,x
2)=x2
1.There–
fore his marginal rate of substitution is −MU1(x1,x
2)/M U2(x1,x
2)=
−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, x1and 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
∗Some people have trouble remembering whether the marginal rate
of substitution is −MU1/M U2or −MU2/M U1. 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.
50 CHOICE (Ch. 5)
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.
Example: A consumer has the utility function U(x1,x
2)=min{x1,3x2}.
The price of x1is 2, the price of x2is 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 x1and
x2. The second equation is her budget equation, which is 2x1+x2= 140.
Solve these two equations to find that x1=60andx2= 20. So we know
that the consumer chooses the bundle (x1,x
2)=(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
•Find the best affordable bundle, given prices and income for a con-
•Recognize standard examples where the best bundle a consumer can
•Apply the methods you have learned to choices made with some kinds
5.1 (0) We begin again with Charlie of the apples and bananas. Recall
that Charlie’s utility function is U(xA,x
B)=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.
NAME 51
0102030
40
10
20
30
Apples
Bananas
40
a
e
Blue
budget
line
Red
curves
Black curve
Pencil line
(b) Can Charlie afford any bundles that give him a utility of 150? Yes.
(c) Can Charlie afford any bundles that give him a utility of 300? No.
(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,x
B), Charlie’s
marginal rate of substitution is a function of xAand xB.Infact,ifyou
(f) Write an equation that implies that the budget line is tangent to an
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.
52 CHOICE (Ch. 5)
(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
(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
5.2 (0) Clara’s utility function is U(X, Y )=(X+2)(Y+1),where X
is her consumption of good Xand Yis her consumption of good Y.
(a) Write an equation for Clara’s indifference curve that goes through the
point (X, Y )=(2,8). Y=36
Clara’s indifference curve for U= 36.
04812
16
4
8
12
Y
16
11
11
U=36
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
NAME 53
(c) At the commodity bundle, (X, Y ), Clara’s marginal rate of substitu-
(d) If we set the absolute value of the MRS equal to the price ratio, we
(f) Solving these two equations for the two unknowns, Xand Y, we find
5.3 (0) Ambrose, the nut and berry consumer, has a utility function
U(x1,x
2)=4
√x1+x2,wherex1is his consumption of nuts and x2is his
consumption of berries.
(a) The commodity bundle (25,0) gives Ambrose a utility of 20. Other
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
(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
54 CHOICE (Ch. 5)
010152025
Nuts
5
10
15
20
Berries
5 30
Red
curve
Blue line
Red curve
Pencil line
Blue
line
(f) Now let us explore a case where there is a “boundary solution.” Sup-
pose 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
(h) Which is steeper at this point, the budget line or the indifference
(i) Can Ambrose afford any bundles that he likes better than the point
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.
NAME 55
(a) On the graph below, draw a “budget line” showing the various com-
binations 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 “indif-
ference 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.
0 40 60 80 100
Score on test 1
20
40
60
80
Score on test 2
20 120
a
Budget line
“L” shaped
indifference
curves
(b) Write an equation for the line passing through the kinks of Nancy’s
(d) Solve these two equations in two unknowns to determine the intersec-
(e) Given that she spends a total of 1,200 minutes studying, Nancy will
tion.
5.5 (1) In her communications course, Nancy also takes two examina-
tions. 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 m1minutes studying
56 CHOICE (Ch. 5)
for the first examination, her score on this exam will be x1=m1/5. If
she spends m2minutes 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 combi-
nations 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
0204060
80
20
40
60
Score on test 1
Score on test 2
80
Preference
direction
Max.
overall
score
5.6 (0) Elmer’s utility function is U(x, y)=min{x, y2}.
(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).
NAME 57
(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
(g) On the same graph, use black ink to draw Elmer’s budget line when
the price of xis 1, the price of yis 2, and his income is 8. What bundle
0 8 12 16 20
x
4
8
12
16
y
424
Black
line (16,5)
Blue
curves
Blue curve
Black curve
Chosen
bundle
(h) Suppose that the price of xis 10 and the price of yis 15 and Elmer
you might think there is too little information to answer this question.
But think about how much yhe 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.