ECON 201, Summer 2016
July 6, 2016
Problem Set 1 Solutions
1. Suppose the John’s utility function for cans of colas and hamburgers
is given by U (C, H) = C 1/3 H 2/3 , where C and H are respectively, the
cans of cola and the number of hamburgers he consumes.
(a) Draw a couple of his indifference curves.
Consider U (C, H) = 1 then C 1/3 H 2/3 = 1 or CH 2 = 1.
√
If C = 1 then H = 1. If C = 12 then H = 2.
Now consider U (C, H) = 2 then C 1/3 H 2/3 = 2 or CH 2 = 8.
If C = 2 then H = 2. If C = 8 then H = 1.
(b) What is his marginal rate of substitution of colas for hamburgers
(M RSCH )? Does his M RSCH depends on how much of the goods
he has? What happens to M RSCH as he has more and more
colas?
1 −2/3 2/3
C
H
M UC
H
M RSCH =
= 32 1/3 −1/3 =
.
M UH
2C
C H
3
1
The M RSCH depends on how much C and H the consumer has,
H
in particular it depends on
.
2C
C
will deWhen he has more and more colas, the M RSCH =
2H
cline. Therefore, he has diminishing marginal rate of substitution.
(c) Suppose the price of a can of cola is pC = $1, the price of a
hamburger is pH = $3 and John’s income is m = $27. Draw his
budget line.
pC C + pH H = I
C + 3H = 27
(d) Find John’s optimal consumption bundle.
We know that at the optimal bundle the slope of John’s indifference curve (M RSCH ) and the slope of his bugdet line (pH /pC )
should be tangent to each other. In other words, the following
equality should be satisfied:
2
pH
pC
C
1
2C ∗
= ⇒ H∗ =
2H
3
3
M RSCH =
Next, we need to substitute this equality into the budget constraint:
C∗ + 3
2C ∗
= 27
3
3C ∗ = 27 ⇒ C ∗ = 9 and H ∗ = 6
So, John’s optimal bundle with the given prices, income and preferences is (9, 6)
(e) Suppose that the price of a can of cola is pC = $2 and the price of
a hamburger is pH = $6. Moreover, his income is $54. Draw his
bugdet line and find his optimal choice.
Let’s first write the bugdet constraint.
2C + 6H = 54 ⇒ C + 3H = 27,
which is the same as in part c. Since his preferences are not
changed, his optimal bundle should be the same, i.e. (9, 6).
2. Suppose that Gozde thinks of jam and honey as substitutes and she is
willing to trade 2 jars of jam (J) for 3 jars of honey (H). Moreover,
her friend Duygu is always consuming 1 cup of coffee (C) with 2 sugar
cubes (S).
(a) Find a utility function separately for Gozde and Duygu based on
their preferences.
For Gozde, honey and jam are substitutes and 1 jar of jam worths
3
3/2 jars of honey. Therefore, UG (H, J) = J + H. If we mul2
tiply UG (H, J) by 2, we get the following monotonic transformation VG (H, J) = 3J + 2H, representing the same preferences as
UG (H, J).
For Duygu, coffee and sugar are complements, since she wants
to consume them together. Moreover, the portions are 1 to 2.
3
Therefore UD (C, S) = min{C, S/2}, which equals to the number
of correctly sweetened cups of coffee. Again, we can multiply
it by 2 and obtain a monotonic transformation, which represents
the same preferences as UD (C, S) = min{C, S/2}, i.e. VD (C, S) =
min{2C, S}
(b) Draw a couple of indifference curves of Gozde and Duygu (Put H
on x-axis and J on y-axis for Gozde; Put C on x-axis and S on
y-axis for Duygu)
(c) Suppose that Gozde faces the following prices: pJ = $2 and pH =
$6. Moreover, she has an income of $120. What is the optimal
consumption of H and J? Show on the graph.
Let’s first write down the budget constraint:
pJ J + p H H = I
2J + 6H = 120
Morever, her marginal rate of substitution, i.e. the slope of the
indifference curve, equals to 2/3. Since the slope of the budget
line (pJ /pH ) equals to 3, her indifference curves are flatter than
the budget line. Let’s show it on the graph:
4
As we can see from the graph, her indifference curve crosses the
budget line at the bundle (60, 0). In other words, Honey is expensive and not valuable enough for Gozde as Jam, so that she does
not buy any jar of honey. Instead, she substitutes is with jam.
(d) Suppose that Duygu faces the following prices: pC = $10 and
pS = $5. Moreover, she has an income of $100. What is the
optimal consumption of C and S? Show on the graph.
Again, first we write the budget constraint:
pC C + pS S = I
10C + 5S = 100
At the optimal bundle, 2C ∗ = S ∗ . Then, we substitute this into
the budget constraint:
10C ∗ + 10C ∗ = 100
C ∗ = 5 ⇒ S ∗ = 10
5
(e) Assume that Duygu decides to quit sugar and therefore does not
consume tea with sugar anymore. She still likes consuming tea.
What can you tell about her utility function?
Since she is not consuming sugar and coffee together, they are not
complements for Duygu anymore. Moreover, since she decides to
quit sugar, it does not give her any utility, i.e. sugar will not be
a part of her utility function. Coffee still gives her utility, so her
utility function will depend only on coffee: VD (C).
3. Suppose that a consumer’s utility function for x1 and x2 takes the form
1/2 1/2
U (x1 , x2 ) = x1 x2 . Prices are denoted by p1 and p2 , and income by
m.
(a) Find the demand functions for x1 and x2 (as a function of m,p1
and p2 ). Are x1 and x2 normal goods? Why?
M RS for this utility function is given by
M U1
M U2
x2
=−
x1
M RS = −
6
From the optimality condition, M RS = −
p1
, we get the following:
p2
p1
x2
=−
x1
p2
∗
∗
x2 p 2 = x 1 p 1
−
Since we want demand functions of x1 and x2 , we need to write
these in terms of price and income only, not as a function of each
other. To find the demand function, we should remember the fact
that the optimal bundle has to lie on the budget constraint, that
is,
p1 x∗1 + p2 x∗2 = m
When we substitute x∗1 p1 = x∗2 p2 into the budget constraint we
get,
m
2p1
m
x∗2 (p1 , p2 , m) =
2p2
x∗1 (p1 , p2 , m) =
x1 and x2 are normal goods because
1
∂x∗1
=
>0
∂m
2p1
1
∂x∗2
=
>0
∂m
2p2
(b) Suppose that prices are given by p1 = 2,p2 = 1 and income by
m = 100. Find the optimal consumption bundle. What is the
utility level at the optimal bundle?
We can use the demand function that we found in part a:
1 100
= 25
2 2
1 100
x2 (2, 1, 100) =
= 50
2 1
x1 (2, 1, 100) =
7
Using the utility function, we can find the utility level:
√
U (25, 50) = (25)1/2 (50)1/2 = 25 2
(c) Suppose that government imposes a per-unit tax of $1 on x1 (that
is, for every unit of x1 they buy, the consumer has to pay $1
to the government in addition to the price). Find the new consumption bundle. What is the government tax revenue? The tax
increases the price of good 1 to p∗1 = 3, then the optimal demands
x∗1 (p∗1 , p2 , m) and x∗2 (p∗1 , p2 , m) are
1 100
50
=
= 16.6
2 3
3
1 100
50
x∗2 (p∗1 , p2 , m) =
=
= 50
2 1
1
x∗1 (p∗1 , p2 , m) =
The revenue of the government comes from the tax, i.e. (p∗1 −p1 )×
x∗1 (p∗1 , p2 , m) = $1 × x∗1 (3, 1, 100) = $16.6
(d) Suppose that the government uses a lump-sum tax instead of
quantity tax. What would the amount of the lump-sum tax need
to be, for the consumer to be just as well off with the lump-sum
tax as she is with the quantity tax? (Hint: A lump-sum tax does
not change prices, just changes income. For example, if t is the
imposed lump-sum tax, then income changes to m − t)
Let t denote the lump-sum tax. The tax does not change the price
but changes income to m−t. The utility levels should be the same
with either tax, so we have:
U (3, 1, 100) = U (2, 1, 100 − t)
1
1
100(3)−1/2 = (100 − t)(2)−1/2
2
2
r
2
t = 100(1 −
) = 18.35
3
The government revenue with the quantity tax is 16.6 whereas
with a lump-sum tax is 18.35. As we can see the latter policy is
better because it give more revenues to the governement and the
consumer is indifferent between these two policies since each of
them gives the same utility.
8
4. Lisa likes tea (denoted by t) but dislikes coffee (denoted by c). In
particular, in order to consume 1 cup of coffee, she has to be offered 3
cups of tea for compensation. Let pt denote the price of tea, pc denote
the price of coffee and m denote her income.
(a) Find her demand functions for tea and coffee.
m
and c(pt , pc , m) = 0
t(pt , pc , m) =
pt
Since she dislikes coffee, she would not demand any of it irrespective of prices and her income level.
(b) Write down three different utility functions that would represent
her preferences over tea and coffee.
Since she dislikes coffee, any amount of coffee should decrease her
utility. Therefore, we have UL (t, c) = t − 3c. Any monotonic
transformation would represent the same preferences: kUL (t, c) =
t − 3c and k ∈ {2, 3, 4}
5. Solve 4.2 and 4.12 in Workouts in Intermediate Microeconomics.
• (4.2) Let x1 denotes nuts and x2 denotes berries.
√
(a) With 9 nuts and 10 berries,√
his utility is 4 9 + 10 = 22. With
4 nuts he needs x2 = 22 − 4 4 = 14 berries to reach the same
utility level of 22.
(b) No. They are not on the same IC. The utility function assigns
different numbers to these two bundles.
(c) M RS = −3 at bundle (9, 10). M RS = −2/3 also at the
bundle (9, 20). What is the change
in utility when x1 changes
p
√
from 9 to say 10? This is 4( 10 − 9) = 4(3.16 − 3) = 0.64.
This is regardless of x2 . Then to compensate, x2 must be
decreased by 0.64. This is the M RS.
(d)
M RS =
∂U (x1 ,x2 )
∂x1
∂U (x1 ,x2 )
∂x2
∂U (x1 , x2 )
4
∂U (x1 , x2 )
= √ and
=1
∂x1
2 x1
∂x2
9
2
then M RS = √ . As we suspected, M RS only depends on
x1
x1 . It does not change as x2 changes.
• (4.12)
∂U (x1 , x2 )
∂U (x1 , x2 )
= 2x1 + 2x2 and M U2 =
=
∂x1
∂x2
2x1 + 2x2
2x1 + 2x2 then M RS =
=1
2x1 + 2x2
(b) M U1 = 1 and M U2 = 1 then M RS = 1/1 = 1
(c) Yes. Take u = x1 + x2 and apply f (u) = u2 .
(a) M U1 =
10