NAME_______________________________________________________________
FE461 Practice Problem
Second Homework Set
Due February 9th
1. (50 points) Suppose a monopolist can identify two groups of customers. The inverse
demand for each group is given by:
A : P  200  Q
B : P  120  2Q
a) (10 points) Find the market demand if the monopolist cannot price discriminate
(assume there is 1 of each type).
Sum horizontally: QA + QB = 200 – P + 60 – ½ P  QA+B = 260 – 3/2 P  P = 173.3 – 2/3 Q
P  200  Q if 120  P  200
2
P  173.3  Q if P  120
3
b) (5 points) If the monopolist faces a constant marginal cost equal to 40, what is
the profit maximizing quantity if the monopolist cannot price discriminate?
200
120
106.6
40
P = 173.3-2/3Q
MC
MR = 173.3-4/3Q
100
Quantity
4
MR  173.3  Q  40  MC
3
4
Q  133.3
3
Q  100
P  106.6
c) (5 points) What is the quantity purchased by each group?
QA=200-106.6=93.4
QB=60- ½ (106.6) = 6.7
d) (5 points) What is the firm’s profit?
Profit = (106.6 – 40)*100 = 6600
Now suppose the firm can price discriminate.
e) (10 points) What price will the firm charge each group?
P 200
P
PA= 120
120
PB= 80
40=MC
P=200-Q
80 MR=200-2Q
40=MC
P=120-2Q
20 MR=120-4Q Quantity
MRA=200-2Q = 40 = MC
160=2Q
Q=80
PA=120
MRB=120-4Q = 40 = MC
80=4Q
Q=20
PB=80
f) (5 points) What is the firm’s profit
Profit = (PA-MC)QA+(PB-MC)QB
Profit = (120 – 40)80 +(80 – 40)20 = 7200
g) (10 points) What happens to total surplus? (be sure to show specific numbers)
200
120
106.6
40
80
100
CS
= ½ (200 – 120)80 + (120 – 106.6)80 + ½ (120-106.6)(100 – 80) = 4424
CSPD = ½ (200 – 120)80 + ½ (120 – 80)20 =3600
TSNO PD =4424 + 6600 = 11,024
TSPD = 3600 +7200 =10,800
NO PD
2. (25 points) Frank Buckely sells his famous bad tasting but very effective cough medicine
in Toronto and Montreal. The inverse demand functions are given by:
PT  18  QT
PM  14  QM
If the firm can price discriminate, what price will the firm charge each group if the
marginal cost is given by: MC  1  2Q ?
PT  18  QT    MRT  18  2QT
PM  14  QM    MRM  14  2QM
Invert MR and aggregate MR.
1
MRT
2
1
QM  7  MR M
2
Q  16  MR
QT  9 
Reinvert MR:
Q  16  MR    MR  16  Q
Equate to MC
MR  16  Q  1  2Q  MC
15  3Q
Q5
MR  16  5  11
Now need to go to each individual MR to find the Q and the price:
MRT  18  2QT  11    QT  3.5
MRM  14  2QM  11    QM  1.5
Back to inverse demand functions for Prices:
PT  18  QT    PT  18  3.5  14.5
PM  14  QM    PM  14  1.5  12.5
3. A nightclub owner has both student and adult customers. Assume the owner faces a
constant marginal cost of $2 per drink. The demand for each group is given by:
Q A  10  2 P
QS  18  3P
a) (10 points) If the owner cannot discriminate, what are the owner’s profits?
If the club owner cannot price discriminate, she will consider the aggregate demand, which
is
Sum horizontally: QA + QS = 10 – 2P + 18 – 3P  QA+S = 28 – 5P
Inverting  P 
28 1
 Q
5 5
The marginal revenue
MR 
28 2
 Q
5 5
Equate marginal revenue with marginal cost, that is, 2
MR 
28 2
 Q  2  MC
5 5
Q9
Therefore,
P
28 1
28 9
19
 Q

  P 
5 5
5 5
5
Her profit without price discrimination:
19
9
81
  (  2)9  ( )9 
 16.2
5
5
5
b) (10 points) If the owner could separate the groups and practice third-degree price
discrimination what price per drink would be would be charged to each member?
PA=5- ½ Q
MRA=5-Q = 2 = MC
3=Q
PA=3.5
PS=6- 1/3 Q
MRS=6- 2/3 Q = 2 = MC
4(3/2)=Q
Q=6
PS=4
c) (15 points) Suppose instead, the owner can “card” patrons to separate students
from adults, and he can offer each group a cover charge and number of drink
tokens. What will be the cover charge and number of tokens per student? Per
adult?
In this scenario, she can practice two-part pricing. For each group, the number of token will
be equal to quantity demanded at price $2, which is the marginal cost of a drink. Number of
tokens for students QS  18  3P  18  3(2)  12 , and the number of tokens for the adults
QA  10  2P  10  2(2)  6 . Now, for each group, the cover charge should equal the
consumer surplus received at the given number of tokens.
1
12(6  2)  24  cov er
2
With Drink tokens and cover total 24  2(12)  48
1
That is, Cover charge for a adult CS  6(5  2)  9  cov er
2
With Drink tokens and cover total 9  2(6)  21
That is, Cover charge for a student CS 