The First Order Profit Maximising Condition Reformatted
Profit function of firm i:
 i  R(Qi )  C (Qi )
1.
where, R(Qi) is the revenue function and C(Qi) is the cost function of firm i.
For profit maximisation the derivative of profit is zero, which amounts to saying that
the derivative of revenue is equal to the derivative of cost:
MRi  MC i
2.
where MRi is the marginal revenue and MCi the marginal cost. Revenue is R=pQi,
where p  f (Q ) , where Q is the sum of all individual outputs of the n firms in the
industry ( Q  i 1 Qi ).
n
Marginal revenue, the derivative with respect to the firm’s output level, Qi , is then
n

p
p Q
p  Qi   j i Q j
MRi  p 
Qi  p 
Qi  p 

Qi
Q Qi
Q  Qi
Qi

MRi = p 
p
Qi (1  i )
Q


Qi 

3.
  j i Q j
n
where i 
is the conjectural variation term, i.e. the firm’s conjecture as to
Qi
its rivals’ response to a change in its own output. If we both multiply and divide
p
equation 3 by
then this becomes:
Q
 1  i 
MRi  p1 
S i 



4.
indl1.doc
Q
Q p
, and S i  i is the market share of firm i. Substituting 4 into 2
Q
p Q
gives the profit maximising condition taken orthodox microeconomic theory
reformatted to with referenced to the structure-conduct-performance framework:
where,   
 1  i 
MRi  p1 
S i   MC i



5.
Alternative market structures
 Perfect competition
Q
 0  1  i  0  i  1
Qi
Hence equation 5 becomes
MRi  p  MCi
6.
 Monopoly
Q  Qi 
Q Qi

1
Qi Qi
Hence equation 5 becomes
 1
MR  p1 
 


  MC


7.
 Joint Profit maximisation
Firms seek to maintain market share following the rule:
Q j
Qi
  j i Q j
n
i 
Qi

Qj
Qi


n
j i
Qi
i  1,2,..., n
Qj

Q  Qi Q

1 
Qi
Qi
indl1.doc
 1  i 
Q
1

Qi S i
Hence equation 5 becomes
 1
MRi  p1 
 


  MC i


8.
 Cournot
Each firm is assuming that its rivals output will not change in response to
a change in his own output, hence:
  j i Q j
n
i 
0
Qi
Hence equation 5 becomes
 S
MRi  p1  i




  MC i


9.
The Cournot Model: An Example
Set n=2, MCi=c and p=a-bQ, b> 0 and i=0. Profits are equal to
 i  a  bQ Qi  cQi
and since  
1 a  bQ
, the profit maximising condition 9 becomes:
b Q
a  bQ 1  Qi b 

Q
Q 
c
a  bQ 
a  bQ  bQi  c  a  2bQi  b j i Q j  c 
n
a  c  j i Q j
Qi 

2b
2
n
for i=1,2.
Writing the two equations explicitly
indl1.doc
Q2 a  c

2
2b
Q1
ac
 Q2 
2
2b
Q1 
Solving the system of the two equations we get:
Q1*  Q2* 
In the case of monopoly, since
a  c C 2(a  c) C a  2c
,Q 
,p 
3b
3b
3
1


bQ
, Qi=Q, equation 7 becomes:
a  bQ
a  2bQ  c  Q M 
ac M ac
,p 
2b
2
In the case of perfect competition, since p=c a-bQ=c, and therefore:
a  bQ  c  Q PC 
a  c PC
,p c
b
Stackelberg’s Model
Instead of simultaneity in the firms’ action as in the Cournot model, here the follower
(say, firm 2) moves second playing Cournot:
Q2 
a  c Q1

2b
2
while the leader, (say, firm 1) will maximise its profits by taking into account the
follower’s reaction function as he knows that the latter will act as a Cournot firm,
(correctly) treating the leader’s output as fixed. Hence equation 1 becomes:
Q
ac


 1  pQ1  cQ1  Q1 a  bQ1  Q2   cQ1   a  bQ1   b
 b 1  c Q1 
2b
2


Q 
ac
1  
 b 1 Q1
2 
 2
Consequently,
indl1.doc
d 1 a  c
Q1
a  c F a  c Q1L a  c
L

 2b
 0  Q1 
, Q2 


,
dQ1
2
2
2b
2b
2
4b
Q S  Q1  Q2 
3(a  c) S
3(a  c) a  3c
, p  a b

4b
4b
4
In other words, there is a first mover advantage as the leader produces more relative to
the Cournot model, at the expense of the follower who know produces less, while the
total output (price) is greater (lower) than the total output (price) in the Cournot
model.
indl1.doc