58 CHOICE (Ch. 5)
04812
16
4
8
12
Y
16
X
(3,3)
Blue
curve
Black
curve
Red line
(b) On the same graph, use red ink to draw Linus’s budget line if the
price of xis 1 and the price of yis 2 and his income is 8. What bundle
(c) What bundle would Linus choose if the price of xis 1, the price of y
5.8 (2) Remember our friend Ralph Rigid from Chapter 3? His favorite
diner, Food for Thought, has adopted the following policy to reduce the
crowds at lunch time: if you show up for lunch thours before or after
12 noon, you get to deduct tdollars from your bill. (This holds for any
fraction of an hour as well.)
NAME 59
0
11 12 1 2
5
10
15
Time
Money
20
10
yyyyyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyyyyy
Red curves
Blue budget set
(a) Use blue ink to show Ralph’s budget set. On this graph, the horizontal
axis measures the time of day that he eats lunch, and the vertical axis
measures the amount of money that he will have to spend on things other
(b) Recall that Ralph’s preferred lunch time is 12 noon, but that he is
5.9 (0) Joe Grad has just arrived at the big U. He has a fellowship that
covers his tuition and the rent on an apartment. In order to get by, Joe
has become a grader in intermediate price theory, earning $100 a month.
Out of this $100 he must pay for his food and utilities in his apartment.
His utilities expenses consist of heating costs when he heats his apartment
and air-conditioning costs when he cools it. To raise the temperature of
60 CHOICE (Ch. 5)
02030405060
Temperature
20
40
60
80
100
Food
120
10 70 80 90 100
December September August
Black budget constraint
Blue budget constraint Red budget constraint
(a) When Joe first arrives in September, the temperature of his apartment
is 60 degrees. If he spends nothing on heating or cooling, the temperature
in his room will be 60 degrees and he will have $100 left to spend on food.
to spend on food. On the graph below, show Joe’s September budget
constraint (with black ink). (Hint: You have just found three points that
Joe can afford. Apparently, his budget set is not bounded by a single
straight line.)
(b) In December, the outside temperature is 30 degrees and in August
poor Joe is trying to understand macroeconomics while the temperature
(c) Draw a few smooth (unkinky) indifference curves for Joe in such a way
that the following are true. (i) His favorite temperature for his apartment
(d) In what months is the slope of Joe’s budget constraint equal to the
NAME 61
(e) In December Joe’s marginal rate of substitution between food and
(f) Since Joe neither heats nor cools his apartment in September, we
cannot determine his marginal rate of substitution exactly, but we do
5.10 (0) Central High School has $60,000 to spend on computers and
other stuff, so its budget equation is C+X=60,000, where Cis ex-
penditure on computers and Xis expenditures on other things. C.H.S.
currently plans to spend $20,000 on computers.
The State Education Commission wants to encourage “computer lit-
eracy” in the high schools under its jurisdiction. The following plans have
been proposed.
Plan B: This plan would give a $10,000 grant to any high school, so
Plan C: Plan C is a “matching grant.” For every dollar’s worth of
Plan D: This plan is like plan C, except that the maximum amount of
(a) Write an equation for Central High School’s budget if plan A is
get line for Central High School if plan A is adopted.
(b) If plan B is adopted, the boundary of Central High School’s budget set
has two separate downward-sloping line segments. One of these segments
describes the cases where C.H.S. spends at least $30,000 on computers.
This line segment runs from the point (C, X)=(70,000,0) to the point
(c) Another line segment corresponds to the cases where C.H.S. spends
less than $30,000 on computers. This line segment runs from (C, X)=
ink to draw these two line segments.
62 CHOICE (Ch. 5)
(d) If plan C is adopted and Central High School spends Cdollars on com-
puters, then it will have X=60,000 −.5Cdollars left to spend on other
Use blue ink to draw this budget line.
(e) If plan Dis adopted, the school district’s budget consists of two
line segments that intersect at the point where expenditure on comput-
(f) The slope of the flatter line segment is −.5.The slope of the
steeper segment is −1.Use pencil to draw this budget line.
02030405060
Thousands of dollars worth of computers
10
20
30
40
50
Thousands of dollars worth of other things
60
10
Red budget line
(plan B)
Black budget line (plan A)
Pencil budget line
(plan D)
Blue
budget
line
(plan C)
5.11 (0) Suppose that Central High School has preferences that can
be represented by the utility function U(C, X)=CX2. Let us try to
determine how the various plans described in the last problem will affect
the amount that C.H.S. spends on computers.
NAME 63
(a) If the state adopts none of the new plans, find the expenditure on
computers that maximizes the district’s utility subject to its budget con-
(b) If plan A is adopted, find the expenditure on computers that maxi-
(c) On your graph, sketch the indifference curve that passes through the
point (30,000, 40,000) if plan B is adopted. At this point, which is steeper,
(d) If plan B is adopted, find the expenditure on computers that maxi-
mizes the district’s utility subject to its budget constraint. (Hint: Look
(e) If plan C is adopted, find the expenditure on computers that maxi-
(f) If plan D is adopted, find the expenditure on computers that maxi-
5.12 (0) The telephone company allows one to choose between two
different pricing plans. For a fee of $12 per month you can make as
many local phone calls as you want, at no additional charge per call.
Alternatively, you can pay $8 per month and be charged 5 cents for each
local phone call that you make. Suppose that you have a total of $20 per
month to spend.
(a) On the graph below, use black ink to sketch a budget line for someone
who chooses the first plan. Use red ink to draw a budget line for someone
who chooses the second plan. Where do the two budget lines cross?
64 CHOICE (Ch. 5)
0 40 60 80 100
Local phone calls
4
8
12
16
Other goods
20 120
Black line
Red line
Pencil curve
Blue curve
(b) On the graph above, use pencil to draw indifference curves for some-
one who prefers the second plan to the first. Use blue ink to draw an
indifference curve for someone who prefers the first plan to the second.
5.13 (1) This is a puzzle—just for fun. Lewis Carroll (1832-1898),
author of Alice in Wonderland and Through the Looking Glass, was a
mathematician, logician, and political scientist. Carroll loved careful rea-
soning about puzzling things. Here Carroll’s Alice presents a nice bit
of economic analysis. At first glance, it may seem that Alice is talking
nonsense, but, indeed, her reasoning is impeccable.
“I should like to buy an egg, please.” she said timidly. “How do you
sell them?”
“Fivepence farthing for one—twopence for two,” the Sheep replied.
“Then two are cheaper than one?” Alice said, taking out her purse.
“Only you must eat them both if you buy two,” said the Sheep.
“Then I’ll have one please,” said Alice, as she put the money down
on the counter. For she thought to herself, “They mightn’t be at all nice,
you know.”
(a) Let us try to draw a budget set and indifference curves that are
consistent with this story. Suppose that Alice has a total of 8 pence to
spend and that she can buy either 0, 1, or 2 eggs from the Sheep, but no
(b) The point where she buys 2 eggs is (2,6).Plot this point and
label it C. If Alice chooses to buy 1 egg, she must like the bundle Bbetter
NAME 65
0123
4
2
4
6
Eggs
Other goods
8
b
a
c
5.14 (1) You will remember Harry Mazzola, who consumes only corn
chips and french fries. Harry’s utility function is u(x1,x
2)=min{x1+
2x2,2x1+x2}where x1is his consumption of corn chips and x2is his
consumption of french fries.
(a) On the graph below, use black ink to draw in the indifference curve
along which Harry’s utility is 6. Use red ink to draw the budget line for
Harry if the price of corn chips is p1= 3, the price of french fries is p2=2,
and income is m= 10. How many units of corn chips and how many units
of french fries should he consume to maximize his utility subject to this
(b) On the graph below use blue ink to draw Harry’s budget line if the
price of corn chips is 1, the price of french fries is 3, and Harry’s income
is 6. How many units of corn chips and how many units of french fries
66 CHOICE (Ch. 5)
0246
8
2
4
6
Corn chips
French fries
8
Pencil line
Red
line
Blue
lines
Black line
Blue
lines
(c) At what prices will Harry consume only corn chips and no french fries?
(d) At what price-income combinations does Harry choose to consume