Peking University
Guanghua School of Management
Industrial Organization, Spring 2013
L. Shi
Suggested Answer to Problem Set 3 (6 Questions)
Question 1: Cournot Competition
(a) Write the two profit functions.
 1  pq1  C1 (q1 )  (100  2q1  2q2 )q1  4q1
 2  pq2  C2 (q2 )  (100  2q1  2q2 )q2  2q2
(b) Find the best response functions for the two firms.
Max (100  2q1  2q2e ) q1  4q1
q1
F .O.C on q1 :
100  4q1  2q2e  4  0
96  2q2e
 q1 
4
This is the response function for firm 1. Similarly,
Max (100  2q e1 2q2 ) q2  2q2
q2
F .O.C on q2 :
100  2q e1 4q2  2  0
 q2 
98  2q e1
4
This is the response function for firm 2.
(c)
q2
48
Firm 1’s
response
function
Firm 2’s
response
function
24.5
49
24
q1
1
q2e , firm 1’s belief of the firm 2’s quantity, has to equal to q2 , what firm 2 actually chooses to produce. This
means that only the point on firm 1’s response function that intersects with the firm 2’s response function is
the equilibrium.
d. Compute the equilibrium quantities and profits for the two firms.
Solve simultaneously the four equations:
96  2q2e
q1 
,
4
98  2q e1
q2 
,
4
q2e  q2 ,
q e1  q1 ,
we get q1 
96  2q
96  2 q2


4
4
e
2
96  2(
98  2 q e1
98  2 q 1
) 96  2(
)
4
4

4
4
Then, we have, q1c , the Cournot quantity firm 1 produces, is 15.67. q2c  16.67 .
1  pq1  C1 (q1 )  (100  2q1  2q2 )q1  4q1  490.78
 2  pq2  C2 (q2 )  (100  2q1  2q2 )q2  2q2  555.44
e. What changes if the marginal cost of firm 2 increased to $4 per unit?
Then
96  2q2e
q1 
,
4
96  2 q e1
q2 
,
4
q2e  q2 ,
q e1  q1 ,
we get q1 
96  2q
96  2 q2


4
4
e
2
96  2(
96  2 q e1
96  2 q 1
) 96  2(
)
4
4

4
4
That is, q1 = q2 =16, which implied a profit level of 512 for each firm.
Price=100-2*32=36.
f. Compare the perfectly competitive, monopoly and the Cournot outputs, prices, and profits when each
firm has a marginal cost of $4 per unit.
Perfect competition:
p=MC=AC=4.
Then using the demand function,
Q=(100-p)/2=48.
Under monopoly:
2
Max(100  2Q )Q  4Q
Q
F .O.C onQ :
100  4Q  4  0
 Q M  24
And price=100-2*24=52.
Profit= (100  2Q)Q  4Q =(100-48)*24-4*24=1152, which is greater than the total profit in a cournot
equilibrium—2*512=1024.
As the market moves from perfect competition to Cournot to monopoly, total quantity decreases, price
increases, profit increases.
Question 2: Bertrand Competition
If firm 1 has constant marginal cost c1 and firm 2 has constant marginal cost c2 , with c1 < c2 , then
(a) show that firm 1 will be the only one selling in the market.
Firm 2 is driven out of the market since no one is buying from it.
(b) what will be the market price? (hint: it depends on whether p1m is greater or lower than c2 where p1m is
the price firm 1 sets assuming firm 2 is not present)
 If p1m < c2 , the firm 1 will charge p1m and firm 2 can not compete; it is as if firm 2 was never
there.
 If p1m > c2 , the firm 1 will charge p1  c2   where  is a very small number. Then firm 2 still
can not compete and firm 1’s profit is maximized.
(c) What are the features that we do not find very realistic in the Bertrand model?
That all consumers flock to the firm which has a lower price.
Question 3: Cournot equlibrium, Stackelberg equilibrium, and Cartel
Market demand: P=100-Q, Q= q1  q2
Cost for firm1, C1 ( q1 )  q12 ,
Cost for firm 2, C2 ( q2 )  2q22
(a) if these two firms were Cournot competitors, what would be the equilibrium output levels, the market
price and the profits of the two firms?
Firm 1 maximizes its profit expecting firm 2 chooses q2e :
Max(100  q1  q2e )q1  q12
q1
F .O.C on q1 :
100  2q1  q2e  2q1  0
100  q2e
 q1 
4
Firm 2 maximizes its profit expecting firm 1 chooses q1e :
3
Max (100  q1e  q2 ) q2  2q22
q2
F .O.C on q2 :
100  q1e  2q2  4q2  0
100  q1e
 q2 
6
Together with
q2e  q2 ,
q1e  q1 ,
we have
q1c  21.74, q2c  13.05, p c  65.2,
 1c  945.04, 2c  510.38,  1c   2c  1455.42
In Cartel equilibrium (from ps2)
q1*  20, q2*  10
p  100  Q  70.
 *  (100  q1  q2 )q1  q12  (100  q1  q2 )q2  2q22  1500
Collusion reduces the quantity supplied, raises prices and the total profit.
(b) if firm 1 is the leader and firm 2 the follower, find the Stackelberg equilibrium (outputs), the market
price, and the profits of the two firms. Compare your results in this case with the Cournot and Cartel
outcomes.
100  q1
Firm 1 chooses quantity to maximize profit knowing that q2 
(the response function of firm 2)
6
100  q1
Max (100  q1 
) q1  q12
q1
6
F .O.C on q1 :
100 2 q1

 2 q1  0
6
6
 q1  22.73
100  2 q1 
Then
100  q1
q2 
 12.88
6
Q  35.61
100  q1
 1  (100  q1 
) q1  q12  946.97
6
100  q1
 2  (100  q1 
) q2  2 q22  497.59
6
 1   2  1444.56
Moving from Stakelberg to Cournot to Cartel, quantity supplied decreases, price increases, and total profit
increases.
Question 4
(a)
Step 1: firm i maximize profit (like in Cournot competition)
4
Max pqi  c( qi )
qi
 Max qi [110  ( q1e  q2e  ...  qi  ...  q ne )]  (25  10qi )
qi
where qe i is firmi ' s belief of other firms ' choiceof quantity , i means any firm except firmi
F .O.C on qi :
qi ( 1)  [110  ( q1e  q2e  ...  qi  ...  q ne )]  10  0
(1)
Step 2: market clears, which means
qe i  qi
(2)
q1  q2  ...  qi  ...  qn  q
(3)
Inserting equations (2) and (3) in (1), we get
q( 1)  [110  (nq)]  10  0
Rearrange, we get
q  (nq)  100
Thus,
100
q
(4) .
n 1
The third condition of market clearing that differentiates Cournot and monopolistic competition is that
 i  0, for every i.
That is, pq  c( q)  0.
 [110  (nq)]q  (10q  25)  0
Equation (4) can also be written as
qn  q  100, i.e.
qn  100  q
Inserting this last equation in equation (5), we have
[110  (100  q)]q  (10q  25)  0.
 (10  q) q  10q  25  0
(5)
 q 2  25
 q  5.
From equation (4), we know
n+1=100/q=20,
so n=19.
And Q=19*5=95 and p=110-Q=$15.
b) The first-best outcome, i.e., the outcome that maximize the total surplus is when MU=MC at that quantity,
i.e., p=MC. P=110-Q and MC=10. Thus, we have 110-Q=10. That is Q𝑓𝑏 (the first best Q)=100.
At this quantity level, the consumer surplus is .5*100*100=5000.
The firm total surplus is 100*(10-10)-25=-25.
So total surplus is 5000-25=4975.
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c) In a monopolistic competition equilibrium, the consumer surplus is .5*95*(110-15)=.5*95*95.
The firms’ total surplus is 19*(15q-10q-25)=19*(5q-25)=0, which is unsurprising – we know that firms’
economic profit is zero in the monopolistic competitive eq. total surplus is thus 4512.5, which is smaller than the
first-best total surplus.
d) A franchise tax of $75 raises fixed costs from $25 to $100. Making this substitution in the calculation
above yields that in the new equilibrium there are only n = 9 firms. Hence q = 10, Q = 90, and p = $20.
e) If unit production costs fall from $10 to $5, there are n = 20 firms in equilibrium, hence q = 5, Q = 100,
and p = $10.
Question 5
Use Figure 7.1 of CP (on page 208).
Question 6
a) Let’s assume c1 (𝑞1 ) = c2 (𝑞2 ) = 0
max π1 (𝑞1 ) = 𝑝1 𝑞1 = 𝑝1 (96 − 2𝑝1 + 0.5𝑝2 )
96 + 0.5𝑝2
F. O. C
𝑝1 =
4
max π2 (𝑞2 ) = 𝑝2 𝑞2 = 𝑝2 (96 − 2𝑝1 + 0.5𝑝2 )
96 + 0.5𝑝1
F. O. C
𝑝2 =
4
192
384
𝑝1 = 𝑝2 =
, 𝑞1 = 𝑞2 =
7
7
b)
max π1 (𝑞1 ) = 𝑝1 𝑞1 = 𝑝1 (96 − 2𝑝1 + 𝑝2 )
96 + 𝑝2
F. O. C
𝑝1 =
4
max π2 (𝑞2 ) = 𝑝2 𝑞2 = 𝑝2 (96 − 2𝑝1 + 𝑝2 )
6
96 + 𝑝1
4
𝑝1 = 𝑝2 = 32, 𝑞1 = 𝑞2 = 64
More generally, when 𝑞1 = 𝑎 − 𝑏1 𝑝1 + 𝑏2 𝑝2 𝑎𝑛𝑑 𝑞21 = 𝑎 − 𝑏1 𝑝2 + 𝑏2 𝑝1 𝑎𝑛𝑑 𝑐1 = 𝑐2 = 𝑚𝑞,
𝑎+𝑚𝑏
we have p=2𝑏 −𝑏1 .
F. O. C
1
𝑝2 =
2
So it is generalizable that the higher the b2, the higher the price.
Product differentiation is more significant in b) since an increase of 1unit in the price of good 2 increases the
quantity demanded of good 1 by 1 unit instead of ½ unit. And because product differentiation is more
significant, the equilibrium price would be expected to be higher in b).
[Remark]
The reason is that the higher b2 reduces the impact of p1 change on demand for product 1, which makes the
demand elasticity w.r.t price lower, which increases the monopoly power, which raises the price.
From the equation, higher b2 means that product 2 is a closer substitute for product 1, which, on the face value,
means less product differentiation.
But consider the case of b2=0. When p2 change, demand for product 1 does not change, can we really say that
the the two goods are differentiated? No, they are just not related, i.e., they not even in the same market.
Also consider the case of b2=2. When p2 change, demand for product 1 change significantly. Can we really say
the two goods are homogeneous? No. If they are homogeneous, one unit increase in p1 reduce q1 by 2, we
should not have the results that one unit increase in p2 (which is p1!) increase the q1 by 2.
This analysis shows that in cases of higher b2, the two goods are more closely related, i.e., competing in the
same market, yet they are distinct products in the market. In this sense, we say higher b2 corresponds to greater
differentiation.
Lower b2 means that the two products are more remotely related, i.e., not competing in the same mkt. Together
the demand elasticity w.r.t price is made higher by lower b2, and lower equilibrium price can it achieve.
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