Module 3-2 - Problem Set: Monopoly Behavior

Microeconomics II
Miguel de Quinto and Daniel Sánchez-Moscona
To be discussed in Seminars 5 and 6.

1. A market is supplied by a monopolist that shows a constant marginal cost of 5

and no fixed costs. In this market, there are two groups of consumers, A and B,
whose corresponding demand functions for the product in question are as follows:
xA (PA ) = 50 − PA
xB (PB ) = 30 − 2PB

(a) Show what prices would be offered to A and B if this monopolist was third-
degree price discriminating. What are the profits acquired by this monopoly
under this scheme?

Solution: To solve for the prices PA and PB , we need to calculate the

marginal revenue relative to each group of consumers to equate to the marginal
cost, 5. To get the marginal revenue, we first need the inverse demand func-
tions:
xA (PA ) = 50 − PA =⇒ PA (xA ) = 50 − xA =⇒ M RA (xA ) = 50 − 2xA
xB
xB (PB ) = 30 − 2PB =⇒ PB (xB ) = 15 − =⇒ M RB (xB ) = 15 − xB
2
The quantities sold to A will result from the following equation, which on
its turn will yield the following price:
M RA (xA ) = M C
50 − 2xA = 5
45
x∗A = = 22, 5
 2
45 45 55
=⇒ PA∗ x∗A = = 50 − = = 27, 5
2 2 2
Regarding B, we will have the following:
M RB (xB ) = M C
15 − xB = 5
x∗B = 10
10
=⇒ PB∗ (x∗B = 10) = 15 − = 10
2

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 With these prices and quantities, the profits of the monopoly would be:

π M ∗ = PA∗ x∗A + PB∗ x∗B − C(x∗A + x∗B ) =

 
55 45 45 2225
= · + 10 · 10 − 5 + 10 = = 556, 25
2 2 2 4

(b) Suppose that the government intervenes this market and determines that
this price discrimination by the monopoly is illegal. Find the (uniform and
unique) price P that the monopoly will offer in this market. What group of
consumers will decide to consume the product from now on? What are the
profits of the monopoly?

Solution: We have to aggregate a pooled inverse demand function of the

market. This, will be defined in sections. From, above, it can be deduced
that for prices above 15 (since PB (xB = 0) = 15 − 20 = 15), only type A
consumers will acquire the product. That is, for prices above 15, only the
demand function of A enters into the market. Therefore, the pooled demand
function X(P ) will be:
(
50 − P if P ≥ 15
X(P ) = xA (P ) + xB (P ) =
80 − 3P if P < 15

The inverse pooled demand function, P (X), would be:

(
50 − X if X ≤ xA (P = 15) = 50 − 15 = 35
P (X) = 80−X
3
if X > 35

As we know, the marginal revenue is just the inverse demand function with
twice the slope. Therefore,
(
50 − 2X if X ≤ 35
M R(X) = 80−2X
3
if X > 35

Now, as any monopolist would do in its position, it will equate marginal

revenues with the marginal cost, which is equal to 5. Regarding the first
section of the marginal cost, this equation is:

50 − 2X = 5
45
X= = 22, 5
2
For these quantities, the market price would be:
 
45 45 55
P X= = 50 − = = 27, 5
2 2 2

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 The profits acquired by the monopolist from this scheme would be:
   
M 45 55 45 55 45 2025
π X = ,P = = −5 = = 506, 25
2 2 2 2 2 4
However, we also need to check for what quantities, if any, the second section
of the marginal revenue meets the marginal costs. In that case, we would
have:
80 − 2X
=5
3
65
X= = 32, 5
2
These quantities are below the threshold above which this section of the
marginal revenue applies. Therefore, the monopoly would choose to sell
X ∗ = 45
2
only to the group A of consumers at a price of P ∗ = 552
. The profits
acquired by the monopolist would be π M ∗ = 2025
4
.

2. In a market served by a monopolist, there are two consumers, A and B, whose

utility functions are as follows:
x2
uA (mA , xA ) = mA + vA (x) = mA + 100x −
2
uB (mB , xB ) = mB + vB (x) = mB + 80x − x2
The monopolist shows a marginal cost of 10 and no fixed costs.

(a) Design the quantities (xi ) and price of the package (Ai ) that this monopolist
would design for each consumer, A and B, in the case that the monopolist
could perfectly observe who is what kind of consumer. What are the profits
acquired by the monopolist under this scenario?

Solution: Of course, here we will be applying the first degree price discrim-
ination for this monopolist. First, the monopolist will find the quantities
for each consumer by equating vi′ (xi ) = M C, where M C is constant at 10.
Therefore, the quantities xA and xB will be:
vA′ (xA ) = 100 − xA = 10 =⇒ xA = 90
vB′ (xB ) = 80 − 2xB = 10 =⇒ xB = 35
The monopolist will aim at extracting all the consumer surplus (i.e., the
value yielded by each type of consumer from consuming so many quantities)
from each consumer in the form of AA and AB . Therefore,
902
AA = vA (xA = 90) = 100 · 90 − = 4950
2
AB = vvB (xB = 35) = 80 · 35 − 352 = 1575

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 Therefore, the package sold to the A-type consumer would include 90 units
at a total price of 4950 and the one for the B-type would include 35 units at
a total price of 1575.

The profits by the monopolist would be:

π M = AA + AB − C(xA + xB ) = 4950 + 1575 − 10(90 + 35) = 5275

(b) For some reason, suddenly this monopolist is incapable of distinguishing

which consumer belongs to which type, A or B. If the monopolist kept
offering the same offerings, what package would each consumer, A and B,
choose? What would be the profits of the monopolist under this scenario?

Solution: We have to calculate the utility that each package yields to each
consumer, A or B. For A, we have that she would yield the following utility
from each package designed for A and B:
902
uA (mA − AA , xA ) = mA − 4950 + 100 · 90 − = mA
2
352
uA (mA − AB , xB ) = mA − 1575 + 100 · 35 − =
2
2625
= mA + = mA + 1312, 5
2
Therefore, consumer A is better-off by choosing the package designed for B.

On its turn, the utility that B would yield from each package would be:

uB (mB − AA , xA ) = mB − 4950 + 80 · 90 − 902 = mB − 5850

uB (mB − AB , xB ) = mB − 1575 + 80 · 35 − 352 = mB

Therefore, B is better-off by also choosing the package designed for herself.

That is, the monopolist will just sell twice the package designed for B and
her profits would be:

π M = 2AB − C(2 · 35) = 2 · 1575 − 10(2 · 35) = 2450

This result implies a noticeable drop in profits for this monopolist.

(c) To address the situation, the monopolist is aiming at reducing the quantity
and the total price offered to B (respectively, xB and AB ) in order to induce
the A-type consumer to choose the package designed for herself in 2a and to
extract all consumer surplus from B from the corresponding transaction like
before. What would be the profit of the monopolist after deploying this new

4
 package for B?

Solution: With the new package for B, we need that A collects the same
value as from the package designed for her, which yields 0. That is:

x2B x2
−AB + 100xB − = 0 ⇔ AB = 100xB − B
2 2
Regarding B, we still need that she chooses her package and to extract all
the rents from herself, which means that:

AB = 80xB − x2B

With the above, we have a system of two equations with two unknowns, AB
and xB :
x2
(
AB = 100xB − 2B
AB = 80xB − x2B

By equating both equations, we get that the new quantity should be xB =

−40. Of course, this is not coherent. The monopolist should instead choose
to drop completely the package designed for B and just offer the package
designed for A. In this case, only A would choose to acquire the package (as
we have seen in 2b), and the profits of the monopolist would be:

π M = AA − C(xA ) = 4950 − 10 · 90 = 4050

It is worth noting that these profits are higher than in the previous section.

3. A market supplied by a monopoly has two consumers, A and B, who show the
following utility functions:

x2
uA (mA , x) = mA + vA (x) = mA + 100x −
2
uB (mB , x) = mB + vB (x) = mB + 60x − x2

where mi stands for the money agent i holds and x for the quantity consumed of
the product supplied by the monopoly.

The monopolist shows the following cost function: C(x) = 5x.

(a) This monopoly intends to apply first degree price discrimination since it can
perfectly observe the utility function of each consumer as well as which con-
sumer holds each function. Design the menus intended for each consumer,
each one consisting of some quantity xi and a price for those quantities Ai ,
being i both A and B. Calculate the profits of the monopolist under this

5
scheme.

Solution: First, we need to obtain the marginal value that each consumer
obtains from the consumption of the product:
x2
vA (x) = 100x − =⇒ vA′ (x) = 100 − x
2
vB (x) = 60x − x2 =⇒ vB′ (x) = 60 − 2x
Also, the marginal cost of the monopoly is:
∂C(x)
M C(x) = =5
∂x
Now, for consumer A we can find the quantity to include in her package by
equating her marginal value to the marginal cost of the monopoly:
vA′ (x) = M C(x)
100 − x = 5
x∗A = 95
For this quantity, the monopoly can charge the value that consumer A ob-
tains from x∗A :
Z 95 95
x2

∗ 9975
AA = 100 − x dx = 100x − = = 4987, 5
0 2 0 2
For consumer B, we apply the same method. First, we calculate xB :
vB′ (x) = M C(x)
60 − 2x = 5
55
x∗B = = 27, 5
2
Now, we can calculate AB :
Z 55
∗
2  55 3575
60 − 2x dx = 60x − x2 02 =

AB = = 893, 75
0 4
Thus, the menu for A will be ∗ ∗ 9975
and the one for B will be


x A = 95; A A = 2
xB = 2 ; AB = 4 . Of course, the consumers will not extract any value
∗ 55 ∗ 3575

from the transaction, since the monopolist is acquiring all the rents from the
exchange.

The monopoly will obtain the following profits under this scheme:
π M ∗ =A∗A + A∗B − C(x∗A + x∗B ) =
 
9975 3575 55 21075
= + − 5 95 + = = 5268, 75
2 4 2 4

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(b) For some reason, suddenly the monopoly is not able to know which consumer
is A and B. Define the optimization problem so that the monopoly is maxi-
mizing its profit while consumers A and B choose their corresponding menu
(each one consisting on a quantity of product xi and a total price for the
bundle Ai ). Prove that each consumer will indeed acquire their correspond-
ing menu that you have designed.

Solution: We can define the optimization problem including the ICA (or
ICCA ) and the IRB (or P CB ):

max π M = AA + AB − C(xA + xB ) = AA + AB − 5(xA + xB )

xA ,AA ,xB ,AB

s.t. vA (xA ) − AA = vA (xB ) − AB

x2 x2
=⇒ 100xA − A − AA = 100xB − B − AB
2 2
vL (xL ) − AL = 0
=⇒ 60xB − x2B − AL = 0

Of course, we can rearrange the restrictions above so that they are expres-
sions for AA and AB , respectively. From the IRB we can get the following
expression for AB :

60xB − x2B − AB = 0 =⇒ AB = 60xB − x2B

From the ICA and the expression for AB above, we get the following expres-
sion for AA :

x2A x2
100xA −− AA = 100xB − B − AB
2 2
2 2
x x
=⇒ 100xA − A − AA = 100xB − B − 60xB + x2B
2 2
x2A x2B 200xA − x2A − 80xB − x2B
AA = 100xA − − 40xB − =
2 2 2
Now, we can plug these expressions for AA and AB in the objective function
above so that we can get rid of the restrictions. Thus:

200xA − x2A − 80xB − x2B

max π M = + 60xB − x2B − 5(xA + xB )
xA ,xB 2
∂π M 200 − 2xA
=⇒ = − 5 = 0 −→ x∗A = 95
∂xA 2
∂π M −80 − 2xB
=⇒ = + 60 − 2xB − 5 = 0 −→ x∗B = 5
∂xB 2

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With this values for x∗A and x∗B , we can obtain the prices for each menu, as
shown below:
200 · 95 − 952 − 80 · 5 − 52
A∗A (x∗A = 95; x∗B = 5) = = 4775
2
A∗B (x∗B = 5) = 60 · 5 − 52 = 275

Thus, the menus offered by the monopoly will be (x∗A = 95; A∗A = 4775) and
(x∗B = 5; A∗B = 275).

We can prove that each consumer A and B will acquire its menu by checking
the utility they obtain from not acquiring any menu and acquiring each one
of the menus. For A, we have the following:
• The utility of A from not acquiring any menu:

02
uA (mA , x = 0) = mA + 100 · 0 − = mA
2
• The utility of A from acquiring the menu designed for A:

952 425
uA (mA −A∗A , x∗A ) = mA −4775+100·95− = mA + = mA +212, 5
2 2
• The utility of A from acquiring the menu designed for B:

52 425
uB (mB − A∗B , x∗B ) = mA − 275 + 100 · 5 − = mA + = mA + 212, 5
2 2
Therefore, consumer A will be indifferent between acquiring any of both
menus, since the associated utility is the same (as the ICA or ICCA was
designed to achieve). This utility is bigger than the utility of not acquir-
ing any menu. Thus, consumer A will choose to acquire her own menu
(x∗A = 95; A∗A = 4775).

For B, we have the following utilities from each option available:

• The utility of B from not acquiring any menu:

uB (mB , x = 0) = mB + 60 · 0 − 02 = mB

• The utility of B from acquiring the menu designed for A:

uB (mB − A∗A , x∗A ) = mB − 4775 + 60 · 95 − 952 = mB − 8100

• The utility of B from acquiring the menu designed for B:

uB (mB − A∗B , x∗B ) = mB − 275 + 60 · 5 − 52 = mB

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 Therefore, we have that consumer B is worse-off by acquiring the menu meant
for A, since it provides a negative value to her. Additionally, this consumer
will be indifferent between not acquiring any menu and acquiring the menu
meant for herself. Thus, B will end up buying her menu, (x∗B = 5; A∗B = 275).

4. A monopolist faces two consumers, A and B, with quasi-linear preferences:

Ui (mi , xi ) = mi + vi (xi ) for i ∈ {A, B}

Where mi stands for the money agent i holds and xi for the quantity consumed
of the product supplied by the monopoly. Assume that,

vA′ (xA ) = 21 − xA and vB′ (xB ) = 18 − xB

x2
The monopolist’s cost function is C(x) = 2
.

(a) Assume that the monopolist can perfectly discriminate between consumers.
Write down the firm’s profit-maximization problem specifying the relevant
constraints. Find the profit-maximizing menus (xA , AA ), (xB , AB ).

Solution: The monopolist solves the following maximization problem:

max AA + AB − C(xA + xB )
(xA ,AA ),(xB ,AB )

s.t. AA ≤ vA (xA ) (P CA )
AB ≤ vB (xB ) (P CB )

Both (P CA ) and (P CB ) are binding. Hence, we can simplify the above

maximization problem as follows:

(xA + xB )2
max π = vA (xA ) + vB (xB ) −
xA ,xB 2
∂π
= vA′ (xA ) − (xA + xB ) = 0
∂xA
∂π
= vB′ (xB ) − (xA + xB ) = 0
∂xB
Which implies that:

21 − xA = xA + xB (1)
18 − xB = xA + xB (2)

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 Solving the system equations formed by (1) and (2) we get xA = 8, xB = 5,
and consequently,
8 8
x2A
Z 
AA = (21 − xA )dxA = 21xA − = 136
0 2 0
Z 5 5
x2B

AB = (18 − xB )dxB = 18xB − = 77.5
0 2 0

(b) Assume that the monopolist can no longer discriminate between consumers
and, hence, it needs to design the menus so that each agent selects the
menu designed for himself. Write down the firm’s profit-maximization prob-
lem specifying the relevant constraints. Find the profit-maximizing menus
(xA , AA ), (xB , AB ).

Solution: The monopolist solves the following maximization problem:

max AA + AB − C(xA + xB )
(xA ,AA ),(xB ,AB )

s.t.AA ≤ vA (xA ) (P CA )
AB ≤ vB (xB ) (P CB )
vA (xA ) − AA ≥ vA (xB ) − AB (ICA )
vB (xB ) − AB ≥ vB (xA ) − AA (ICB )

As shown in class, only (ICA ) and (P CB ) are binding:

AB = vB (xB )
AA = AB + vA (xA ) − vA (xB ) = vB (xB ) + vA (xA ) − vA (xB )

Thus we can simplify the former maximization problem as follows:

(xA + xB )2
max π = 2vB (xB ) + vA (xA ) − vA (xB ) −
xA ,xB 2
∂π
= vA′ (xA ) − (xA + xB ) = 0
∂xA
∂π
= 2vB′ (xB ) − vA′ (xB ) − (xA + xB ) = 0
∂xA
Which implies:

21 − xA = xA + xB (3)
2(18 − xB ) − (21 − xB ) = xA + xB (4)

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Solving the system of equations formed by (3) and (4) we get, x∗A = 9,
x∗B = 3, and consequently,
Z 3
AB = (18 − xB )dxB = 49.5
0
Z 9 Z 3
AA = AB + (21 − xA )dxA − (21 − xB )dxB = 139.5
0 0

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