Problem 2.1
a) It is reasonable to assume that $5 bills and $20 bills are perfect substitute (or very close
to perfect substitutes) goods. The MRS of $20 bills for $5 bills should be 4; that is, one
is willing to exchange that one $20 bill for 4 $5 bills. The following picture shows a set of
indifference curves for these two goods.
$5 bills
16 6
C
C
C
C
C
12
C
C
C
C
C
C
C
C
C
C
8
CC
4
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C C C C
C C C C
C C C C
C C C C
C C C C
C C C C
-
$20 bills
1 2 3 4
b) Right shoes and left shoes are perfect complements for a consumer who has two feet. The
left picture below show a set of indifference curves for a consumer with two feet. Notice
starting from a consumption bundle like (1, 1), and increasing the quantity of right (left)
shoes only, the consumer remains indifferent. You can check that an utility function like
the following U (qR , qL ) = min {qR , qL } yields indifference curves like those in the left graph
below.
The picture on the right shows a set of indifference curves for a consumer with one foot
only (the right foot). This consumer has no use for left shoes (her utility depends only
on the quantity of right shoes). Any two consumption bundles with the same quantity
of right shoes yield an identical level of utility and therefore are on the same indifference
curve. This results in horizontal indifference curves.
R
R
6
6
3
1
-
1
3
-
L
L
c) From the information given in the question we can conclude that the MRS of cocaine for
money is increasing is increasing in the amount of cocaine consumed. The more cocaine
the agent consumes, the more money she will need to be compensated for the loss of one
gram of cocaine. The picture below, depicts a set of indifference curves consistent with
the assumption given in the question. At x the agents consumes more cocaine than at
y. The MRS between cocaine and money is higher at x than at y consistently with the
assumption of the question.
Money
6
MRS
rx
MRS
ry
-
Cocaine
d) The assumption of the question implies that there exists some quantity of lobster tails,
say S, such that when the consumption of lobster tails exceed S lobster tails become a
”bad” (consuming more lobster tails, everithing else equal, makes the consumer worse off).
The picture below show indifference curves consistent with this assumption. In the region
on the left of the dashed line, the consumption of lobster tails is < S. In that region
lobster tails are a good so the indifference curves slope down. On the right of the dashed
line, the consumption of Lobster tails is > S, lobster tails are therefore a ”bad” and the
indifference curves slope up in that region. Staring at a point like x, if the consumption
of lobster inscreases, keeping the quantity of money constant, the consumer is worse off
(lower utility). One need to give the consumer more money to compensate her for for the
loss of utility due to the consumption of yet another lobster tail.
The second picture shows the case in when theconsumer can dispose at no cost of unwanted
lobsters. In this second case, the indifference curves are flat on the right of the dashed
line. Increasing the consumption of lobsters beyond S while keeping the quantity of money
constant does not increase the utility of our agent, but it does not make our consumer
worse off either (because of the free disposal assumption).
Money
Money
6
6
x
r
-
S
Lobster
-
S
Lobster
Problem 2.2
Let assume that the agent can buy TCA miles (M ) or a generic consumption good denoted
C. pm and pc denote the (full) prices of M and C respectively. I is the income of our agent.
The difficulty arises from the fact that the price paid for miles changes when the consumer buys
more than 30, 000 and more than 50, 000 miles respectively. Precisely the price of miles is pm
for the first 30, 000 miles (in the region on the left of the first dashed line), it is 0.8 ∗ pm for the
next 20, 000 miles (in the region between the two dashed lines) and it becomes 0.5 ∗ pm for any
mile above 50, 000 (the region on the right of the second dashed line).
We know that the slope of the budget constraint is the ratio of prices, when the price of
airmiles changes we should expect a change in the slope of the budget constraint. Precisely, the
m
slope of the budget constraint will be ppmc on the left of the first dashed line, 0.8∗p
between the
pc
0.5∗pm
two dashed lines and pc on the right of the second dashed line.
To draw the budget constraint, notice that the vertical intercept it is pIc , this is the amount
of C the agent can buy if she spends all her income buying the generic consumption good. The
budget constraint should have two kinks (two points at which the slope changes). The first
kink should be on the first dashed line (at that point the price of miles is reduced by 20% as
the airmile program kicks in). To find the exact location of the kink one needs to answr the
folowing question. How much consumption goos can he agent buy when she is buyng exactly
30, 000 miles? The cost of exactly 30, 000 miles is 30, 000 ∗ pm . After paying for the miles the
m
agent is left with I − 30, 000 ∗ pm which she can spend on C. Therefore she can buy I−30,000p
pc
units of C. (If I − 30, 000pm ≤ 0 the consumer cannot afford to buy more than 30, 000 miles.
In this case the budget constraint is linear with prices pm and pc . As the consumer is to poor to
afford more than 30, 000 miles she will not be able to take advantage of the air miles discount,
so the discount is ineffective.)
The second kink on the budget constraint will be on the second dashed line. At 50, 000
miles the price of miles changes from 0.8pm to 0.5pm as the second discount kicks in. The
cost of exactly 50, 000 miles is 30, 000pm + 20, 000 ∗ 0.8pm , the consumer will be left with I −
m)
(30, 000pm + 20, 000 ∗ 0.8pm ) so she can buy I−(30,000pm +20,000∗0.8p
of C (the same note as above
pc
applies).
The horizontal intercept: I − (30, 000pm + 20, 000 ∗ 0.8pm ) is the money the consumer is
left after buying 50, 000 miles. If she spends al of it buying more miles she can buy another
I−(30,000pm +20,000∗0.8pm )
miles. If consumer buys only miles she can buy a total of is 50, 000 +
0.5pm
I−(30,000pm +20,000∗0.8pm )
0.5pm
miles.
The picture below is drawn assuming I = 100, 000, pc = 1 and pm = 2.
C
I
pc
= 100, 000
6
A
A
A
A
A
A
I−30,000pm
pc
= 40, 000
I−(30,000pm +20,000∗0.8pm )
pc
A
A
A
J
J
J
JJ
I−(30,000pm +20,000∗0.8pm )
@
0.5pm
-+
@58, 000 = 50, 000
= 8, 000
30, 000
50, 000
M
Problem 2.3
a) We know that Rita will always buy a consumtion bundle on her budget line (she always
spens all her income) so x = (100, 50) lies on the budget line. At prices p1 = 2 and p2 = 4,
(100, 50) costs 400 which means that 400 is exactly Rita’s income. Once we know the
income we can draw the budget constraint as in the picture below (the thick budget line).
The circle represents the chosen consumption bundle x.
When Rita’s income become I 0 = 500 and the price or C1 becomes p03 = 3 we can draw a
new budget line (the thin line in the picture). Notice that now x costs 100∗3+50∗4 = 500
so it lies on the new budget line as well.
The assumption of smooth and convex indifference curves implies that the indifference
curve at x is tangent to the old budget line (has the same slope). The new budget line
is steeper, so the indifference curve through x will cross the new budget line. We can
therefore find another consumption bundle y affordable given the new income and prices
which lies above the indifference curve passing through x. We can conclude that Rita is
made better off by the change in price and income.
C2
I0
p2
= 125
I
p2
= 100
6
Z
Z
HHZ
HZ
y
HZ
q
HZ
HZ x
HH
r
ZH
ZH
ZHH
Z H
Z H
H
I0
p0
3
= 167
I
p1
= 200
C1
b) If the income grows by 97 dollars (I 00 = 497), the new budget line becomes the thin line
in the becture below. We cannot conclude that Rita’s is better off neither that Rita’s is
worse off. x is now above the new budget line. It could be that the indifference curve
through x is above the new budget line (as in the picture) and Rita is worse off. It could
also be though that the indifference curve through x cosses the new budget line in which
case Rita would be better off (try do draw this case).
C2
I 00
p2
= 124
I
p2
= 100
6
Z
HZ
HZ
HZ
HZ
HH x
ZH r
ZH
ZHH
Z H
Z HH
Z
H
H
Z
I 00
p0
3
= 166
I
p1
= 200
C1