Industrial Organization
ECNS 406
Fall 2015
Homework #: 4
Due by the beginning of class on: Tuesday September 29, 2015
Name:
Instructions:
There are 4 questions worth a total of 100 points. Answer each question clearly and
concisely. You must show your work to receive credit. You are allowed to work with others,
but all work must be your own.
Clearly print your name above and in the space provided on the next page. You must
turn in both sides of this cover sheet along with your responses. You do not need to turn
in the questions, only your responses with the cover sheet. All pages must be stapled to be
graded.
ECNS 406
Homework #: 4
Due: 9/29/2015
Bertrand Model With Asymmetric Costs
Here are some different variations on the Bertrand Model.
1. The inverse demand function is P = 120 − 2Q, costs for Firm 1 are C(q1 ) = 20q1 , and
costs for Firm 2 are C(q2 ) = c2 q2 . The two firms compete in prices.
(a) Plot the best response functions assuming c2 > 20
Solution: First determine the range of prices for Firm 2. Firm 2’s minimum
price is c2 . If Firm 2 sets a price below c2 , Firm 2’s profits are negative. Firm 2’s
maximum price is determined by finding the price Firm 2 would set if it behaved
2
= 120 − 4q2 − c2 .
like a monopolist. In this case, π2 = 120q2 − 2q22 − c2 q2 so dπ
dq2
Setting the first order condition to zero and solving for q2 , we get q2 = 30 − c42
and the maximum price Firm 2 would set is P2max = 120 − 2(30 − c42 ) = 60 + c22 .
Therefore the range of Firm 2’s price is c2 ≤ P2 ≤ 60 + c22 .
The range of Firm 1’s price is determined the same way except that c2 = 20 for
Firm 1, so 20 ≤ P1 ≤ 70.
The best response functions are as follows. If 20 < P2 < 70, Firm 1 wants to
set P1 right below P2 . This makes Firm 1’s best response:


for P2 ≤ 20
20
P1 = P2 − for 20 < P2 < 70 .


70
for P2 ≥ 70
If c2 < P1 < 60 +
response is:
c2
,
2
Firm 2 wants to set P2 right below P1 . Firm 2’s best


c 2
P 2 = P1 − 

60 + c22
for P1 ≤ c2
for c2 < P1 < 60 +
for P1 ≥ 60 + c22
Page 1 of 6
c2
2
(25)
ECNS 406
Homework #: 4
Due: 9/29/2015
P2
Firm 1’s
Best Response 45◦
60 +
c2
2
Firm 2’s
Best Response
c2
20 c2 − 70
P1
(b) Describe the equilibrium of the model as c2 increases for c2 > 0.
Solution: For c2 < 20, the equilibrium prices are P2 = 20− and P1 = 20. The
market equilibrium quantities are q2 = 50 + 2 and q1 = 0. Firm 2 undercuts
Firm 1’s marginal cost by and captures the entire market.
For 70 ≥ c2 > 20, the situation is reversed. The equilibrium prices are P2 = c2
and P1 = c2 − . The market equilibrium quantities are q1 = 60 − c22 + 2 and
q2 = 0. Firm 1 undercuts Firm 2’s marginal cost by and captures the entire
market.
For c2 > 70, P2 = c2 and P1 = 70. The market equilibrium quantities are
q1 = 25 and q2 = 0. Firm 1 captures the entire market and behaves like a
monopolist.
Bertrand Model With Capacity Constraints
2. The inverse demand function is P = 120 − 2Q. There are two firms which compete in
prices and have constant marginal costs of 0. Each firm has a production capacity of 20.
Page 2 of 6
(25)
ECNS 406
Homework #: 4
Due: 9/29/2015
(a) Assuming Firm 2 produces q2 = 20, what is Firm 1’s residual demand curve?
Solution: Firm 1’s residual demand is P = 120 − 2q1 − 2(20) = 80 − 2q1 .
(b) Assuming Firm 2 produces q2 = 20, what is Firm 1’s marginal revenue from the
residual demand curve?
Solution: Firm 1’s revenue from residual demand is R = 80q1 − 2q12 . Firm 1’s
marginal revenue from residual demand is M R = 80 − 4q1 .
(c) What is the equilibrium price and quantity?
Solution: If Firm 1 maximized profits (from the residual demand above), set
the marginal revenue equal to the marginal cost. The marginal cost is zero so
80 − 4q1 = 0 and q1 = 20. This is feasible since the capacity of each firm is 20.
Alternatively if Firm 1 produces 20, Firm 2 produces 20 as well which implies
this is a Nash equilibrium. When each firm produces 20, Q = 40 and the price
is P = 120 − 2(40) = 40.
(d) How much profit does each firm make?
Solution: Firm costs are zero so the profit equals revenue. The equilibrium
price is 40 and each firm produces 20, so each firm makes a profit of 800.
(e) How does the equilibrium of this model compare to the equilibrium if firms compete
in quantities rather than prices?
Solution: If firms compete in quantities, the equilibrium is characterized by the
Cournot duopoly model. The profit for Firm 1 is π1 = 120q1 − 2q12 − 2q2 q1 . Firm
1’s reaction function is q1 = 30 − q22 . Imposing symmetry, we get equilibrium
quantities of q ∗ = 20. The equilibrium of the model is the same if firms compete
in quantities rather than prices.
(f) Does the introduction of capacity constraints relax price competition?
Solution: Yes, rather than the perfectly competitive outcome, the equilibrium
is the same as the Cournot duopoly.
3. The inverse demand function is P = 120 − 2Q. There are two firms which compete in
prices and have constant marginal costs of 20. Each firm has a production capacity of
20.
(a) Assuming Firm 2 produces q2 = 20, what is Firm 1’s residual demand curve?
Page 3 of 6
(25)
ECNS 406
Homework #: 4
Due: 9/29/2015
Solution: Firm 1’s residual demand is P = 120 − 2q1 − 2(20) = 80 − 2q1 .
(b) Assuming Firm 2 produces q2 = 20, what is Firm 1’s marginal revenue from the
residual demand curve?
Solution: Firm 1’s revenue from residual demand is R = 80q1 − 2q12 . Firm 1’s
marginal revenue from residual demand is M R = 80 − 4q1 .
(c) What is the equilibrium price and quantity?
Solution: The equilibrium does not exist. If Firm 1 sets M R = M C, we get
q1 = 15. If the market quantity is Q = 35, the price is P = 120−2(35) = 50 and
each firm sells less than their capacity. Each firm finds it profitable to undercut
the current price of 50 and the price competition drives the price down to 40
where the market quantity is 40 and each firm is producing at capacity. When
each firm is producing at capacity, one firm finds it profitable to restrict quantity
further to increase the price. The argument is circular and the equilibrium does
not exist.
Bertrand Model With Imperfect Substitutes
4. The inverse demand functions for Firms 1 and 2 are given by pi = 120 − 2qi − 2θqj6=i
with θ = 43 so that
3
p1 = 120 − 2q1 − q2
2
3
p2 = 120 − 2q2 − q1
2
which lead to the corresponding demand functions
240 8
− p1 +
7
7
240 8
− p2 +
q2 =
7
7
q1 =
6
p2
7
6
p1 .
7
The two firms compete in prices and have constant marginal costs of 20.
(a) What is the equilibrium price, firm quantity and market quantity?
Solution: This is a symmetric problem, so we just need to setup a single Firm’s
problem. Firm 1’s profits should be a function of prices since firms are choosing
prices. Note that q1 (p1 , p2 ) says that firm 1’s demand is a function of both firms
1 ,p2 )
prices as indicated above. Also note that dq1 (p
= − 78 from Firm 1’s demand
dp1
Page 4 of 6
(25)
ECNS 406
Homework #: 4
Due: 9/29/2015
equation.
π1
dπ1
dp1
dπ1
dp1
dπ1
dp1
Set
dπ1
dp1
= p1 q1 (p1 , p2 ) − 20q1 (p1 , p2 )
dq1 (p1 , p2 )
dq1 (p1 , p2 )
= q1 (p1 , p2 ) + p1
− 20
dp1
dp1
8
160
= q1 (p1 , p2 ) − p1 +
7
7
240 8
6
8
160
=
− p1 + p2 − p1 +
7
7
7
7
7
= 0 and solve for p1 to get Firm 1’s reaction function.
p1
16
400 6
=
+ p2
7
7
7
3p2
p1 = 25 +
8
Firm 2’s reaction function is p2 = 25 + 3p81 .
Imposing symmetry conditions give p1 = p2 = p∗ and the reaction function is
∗
now p∗ = 25 + 3p8 so p∗ = 40. When p∗ = 40, the demand functions above give
− 27 p∗ = 240
− 80
= 160
. The market quantity is Q∗ = 320
.
q ∗ = 240
7
7
7
7
7
(b) Are the two products strategic substitutes or strategic compliments?
Solution: They are strategic compliments because the slope of the reaction
function is positive.
(c) How does the equilibrium of this model compare to the equilibrium if firms compete
in quantities rather than prices and the goods are perfect substitutes (θ = 1)?
Solution: If firms compete in quantities and θ = 1, the the problems is the
Cournot Duopoly problem solved on the last homework. The equilibrium of
this model has q ∗ = 50
, Q∗ = 100
and P ∗ = 160
. When firms compete in prices
3
3
3
3
with θ = 4 , each firm produces more, the market quantity is higher and the
market price is lower than compared with quantity competition and θ = 1.
Page 5 of 6
ECNS 406
Homework #: 4
Due: 9/29/2015
(d) Is price competition relaxed with the addition of imperfect substitutes?
Solution: Yes, because prices are not equal to marginal cost.
(e) How does the equilibrium of this model compare to the equilibrium if firms compete
in quantities rather than prices and the goods are imperfect substitutes with θ = 34 ?
Solution: Firm 1’s profits should be a function of quantities since firms are
choosing quantities. Note that p1 (q1 , q2 ) says that firm 1’s inverse demand is a
(q1 ,q2 )
function of both firms quantities as indicated above. Also note that dp1 dq
=
1
−2 from Firm 1’s inverse demand equation.
π1
dπ1
dq1
dπ1
dq1
dπ1
dq1
Set
dπ1
dq1
= q1 p1 (q1 , q2 ) − 20q1
dp1 (q1 , q2 )
= p1 (q1 , q2 ) + q1
− 20
dq1
= p1 (q1 , q2 ) − 2q1 − 20
3
= 120 − 2q1 − q2 − 2q1 − 20
2
= 0 and solve for q1 to get Firm 1’s reaction function.
3
4q1 = 100 − q2
2
3q2
q1 = 25 −
8
Firm 2’s reaction function is q2 = 25 − 3q81 .
Imposing symmetry conditions give q1 = q2 = q ∗ and the reaction function is
∗
now q ∗ = 25 − 3q8 so q ∗ = 200
. The market quantity is Q∗ = 400
and the price
11
11
1320−800
520
400
∗
is P = 120 − 2 11 =
=
.
In
this
example,
when
firms
set
quantities,
11
11
the equilibrium price is higher and the quantities are lower.
Page 6 of 6