Download 3.2.Cournot Model

3.2. Cournot Model
Matilde Machado
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3.2. Cournot Model
Assumptions:
All firms produce an homogenous product
The market price is therefore the result of
the total supply (same price for all firms)
Firms decide simultaneously how much to
produce
Quantity is the strategic variable. If OPEC was not a
cartel, then oil extraction would be a good example of Cournot competition.
?
The equilibrium concept used is Nash
Equilibrium (Cournot-Nash)
Agricultural products? http://www.iser.osaka-u.ac.jp/library/dp/2010/DP0766.pdf
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3.2. Cournot Model
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3.2. Cournot Model
Graphically:
Let’s assume the duopoly case (n=2)
MC=c
Residual demand of firm 1:
RD1(p,q2)=D(p)-q2. The problem of the firm
with residual demand RD is similar to the
monopolist’s.
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3.2. Cournot Model
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3.2. Cournot Model
Graphically (cont.):
P
p*
MC
D(p)
RD1(q2) =
Residual demand
q*1=
q2
R1(q2)
MR
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3.2. Cournot Model
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3.2. Cournot Model
Graphically (cont.):
q*1(q2)=R1(q2) is the optimal quantity as a
function of q2
Let’s take 2 extreme cases q2:
Case I: q2=0 ⇒RD1(p,0)=D(p) whole demand
Firm 1
⇓
should
the
q*1(0)=qM produce
Monopolist’
s quantity
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3.2. Cournot Model
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3.2. Cournot Model
Case 2: q2=qc ⇒RD1(p,qc)=D(p)-qc
D(p)
c
Residual
Demand
qc
c
MR<MC⇒q*1=0
qc
MR
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3.2. Cournot Model
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3.2. Cournot Model
Note: If both demand and cost functions are
linear, reaction function will be linear as
well.
q1
Reaction function of
firm 1
qM
qc
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3.2. Cournot Model
q2
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3.2. Cournot Model
If firms are symmetric
then the equilibrium
is in the 45º line, the
reaction curves are
symmetric and
q*1=q*2
q1
qc
q1=q2
qM
q*1
E
45º
q*2
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qM
qc
3.2. Cournot Model
q2
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3.2. Cournot Model
Comparison between Cournot, Monopoly and
Perfect Competition
qM<qN<qc
q1
qc
q1+q2=qN
qM
q1+q2=qM
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q1+q2=qc
qM
q1+q2=qN qc
3.2. Cournot Model
q2
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3.2. Cournot Model
Derivation of the Cournot Equilibrium for n=2
Takes the strategy of
P=a-bQ=a-b(q1+q2)
firm 2 as given, i.e. takes
q2 as a constant. Note
MC1=MC2=c
the residual demand
For firm 1:
here
Max Π1 ( q1 , q2 ) = ( p − c ) q1 = ( a − b(q1 + q2 ) − c ) q1
q
1
∂Π1
FOC:
= 0 ⇔ a − bq1 − bq2 − c − bq1 = 0
∂q1
⇔ 2bq1 = a − bq2 − c
⇔ q1 =
Industrial Economics- Matilde Machado
a − c q2
−
2b
2
3.2. Cournot Model
Reaction function of firm 1:
optimal quantity firm 1
should produce given q2. If
q2 changes, q1 changes as
well.
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3.2. Cournot Model
We solve a similar problem for firm 2 and obtain a
system of 2 equations and 2 variables.
a − c q2

q1 = 2b − 2

q = a − c − q1
 2
2b
2
If firms are symmetric, then
q1* = q2* = q* i.e. we impose that the eq. quantity is in the 45º line
*
a
−
c
q
a−c
⇒ q* =
− ⇔ q* =
= q1N = q2N
2b
2
3b
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3.2. Cournot Model
Solution of the
Symmetric
equilibrium 11
3.2. Cournot Model
Solution of the Symmetric equilibrium
q1* = q2* = q*
*
a
−
c
q
a−c
*
*
⇒q =
− ⇔q =
= q1N = q2N
2b
2
3b
Total quantity and the market price are:
2 a−c
Q N = q1N + q2N = 

3 b 
2
a + 2c
N
N
p = a − bQ = a − ( a − c ) =
3
3
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3.2. Cournot Model
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3.2. Cournot Model
Comparing with Monopoly and Perfect Competition
c
N
M
p
<
p
<
p
c
a +2c
3
a +c
2
Where we obtain that:
∂p c ∂p N ∂p M
>
<
∂c ∂c ∂c
=1
Industrial Economics- Matilde Machado
=
2
3
=
In perfect competition
prices increase 1-to-1 with
costs.
1
2
3.2. Cournot Model
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3.2. Cournot Model
In the Case of n≥2 firms:
Max Π1 ( q1 ,...qN ) = ( a − b(q1 + q2 + ... + qN ) − c ) q1
q1
FOC: a − b(q1 + q2 + ... + qN ) − c − bq1 = 0
⇔ q1 =
a − b( q2 + ... + qN ) − c
2b
If all firms are symmetric:
q1 = q2 = ... = qN = q
q=
a − b(n − 1)q − c
a−c
a−c
 1

⇔ 1 + (n − 1)  q =
⇔ qN =
2b
2b
(n + 1)b
 2

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3.2. Cournot Model
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3.2. Cournot Model
Total quantity and the equilibrium price are:
n a − c n→∞ a − c
Q = nq =

→
= qc
n +1 b
b
n a−c
a
n
n →∞
N
N
p = a − bQ = a − b
=
+
c 
→c
n +1 b
n +1 n +1
N
N
If the number of firms in the oligopoly converges to ∞,
the Nash-Cournot equilibrium converges to perfect
competition. The model is, therefore, robust since
with n→ ∞ the conditions of the model coincide with
those of the perfect competition.
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3.2. Cournot Model
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3.2. Cournot Model
DWL in the Cournot model
= area where the willingness to
pN
pay is higher than MC
DWL
1
DWL = ( p N − p c )( Q c − Q N )
2
1 1
n
n a−c 
 a − c
= 
a+
c − c
−

2  n +1
n +1
n +1 b 
 b
1  a−c
n →∞
= 

→0

2b  n + 1 
2
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3.2. Cournot Model
c
QN
qc
When the number of firms
converges to infinity, the
DWL converges to zero,
which is the same as in
Perfect Competition. The
DWL decreases faster than
either price or quantity (rate
of n2)
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3.2. Cournot Model
In the Asymmetric duopoly case with constant marginal
costs.
linear demand P(q1 + q2 ) = a − b(q1 + q2 )
c1 = MC of firm 1
c2 = MC of firm 2
The FOC (from where we derive the reaction functions):
q1 P′(q1 + q2 ) + P(q1 + q2 ) − c1 = 0
−bq1 + a − b(q1 + q2 ) − c1 = 0
⇔

′
q2 P (q1 + q2 ) + P(q1 + q2 ) − c2 = 0
−bq2 + a − b(q1 + q2 ) − c2 = 0
a − bq2 − c1

q
=
 1
2b
Replace q2 in the reaction function
⇔
of firm 1 and solve for q1
q = a − bq1 − c2
2

2b
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3.2. Cournot Model
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3.2. Cournot Model
In the Asymmetric duopoly case with constant marginal
costs.
a − c1 1  a − bq1 − c2 
3
a c2 c1
− 
⇔
q
=
+ −
1

2b
2
2b
4
4b 4b 2b

a + c2 − 2c1
⇔ q1* =
3b
3b
q1 =
Which we replace back in q2:
*
a
−
bq
a 1  a + c2 − 2c1  c2 a − 2c2 + c1
1 − c2
=
− 
=
q2* =
−
2b 2 
3b
3b
 2b
2b
a + c2 − 2c1 a − 2c2 + c1 2a − c2 − c1
+
=
3b
3b
3b
2a − c2 − c1 a + c2 + c1
p* = a − b(q1* + q2* ) = a −
=
3
3
Q* = q1* + q2* =
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3.2. Cournot Model
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3.2. Cournot Model
From the equilibrium quantities we may conclude that:
a + c2 − 2c1
q =
3b
*
1
a − 2c2 + c1
; q =
3b
*
2
If c1<c2 (i.e. firm 1 is more efficient):
a c2 2c1 a 2c2 c1 c2 − c1
− +
− =
>0
q −q = + −
3b 3b 3b 3b 3b 3b
b
*
1
*
2
⇔q >q
*
1
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*
2
In Cournot, the firm with the largest market
share is the most efficient
3.2. Cournot Model
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3.2. Cournot Model
From the previous result, the more efficient firm is also the
one with a larger price-Mcost margin:
p − c1 p − c2
L1 =
>
= L2
p
p
=
Industrial Economics- Matilde Machado
s1
ε
=
s2
ε
3.2. Cournot Model
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3.2. Cournot Model
Comparative Statics:
The output of a firm ↓ when:
a + c j − 2ci
*
qi =
3b
↑ own costs
↓ costs of rival
q2
Shifts the reaction curve
of firm 1 to the left
↑c1
E’
E
↑q*2 and ↓q*1
q1
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3.2. Cournot Model
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3.2. Cournot Model
Profits are:
Π1* = ( p* − c1 ) q1* = ( a − b(q1* + q2* ) − c1 ) q1* =

  a + c2 − 2c1  ( a + c2 − 2c1 )
 2a − c2 − c1 
= a −b
− c1  × 
=

3b
3b
9b




 
Increase with rival’s costs
∂Π1
>0
∂c2
Decrease with own costs
∂Π1
<0
∂c1
2
Symmetric to firm 2.
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3.2. Cournot Model
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3.2. Cournot Model
More generally… for any demand and cost function. There is a negative
externality between Cournot firms. Firms do not internalize the effect
that an increase in the quantity they produce has on the other firms.
That is when ↑qi the firm lowers the price to every firm in the market
(note that the good is homogenous). From the point of view of the
industry (i.e. of max the total profit) there will be excessive
production.
Max Π i (qi , q j ) = qi P (Q) − Ci (qi )
Externality: firms only take into
account the effect of the price change
in their own output. Then their output
is higher than what would be optimal
from the industry’s point of view.
qi
CPO:
∂Π i
=0⇔
∂qi
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qi P′(Q)
+ P (Q) − Ci′(qi ) = 0
effect of the increase in quantity
on the inframarginal units
3.2. Cournot Model
profitability of the
marginal unit
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3.2. Cournot Model
If we define the Lerner index of the market as:
L ≡ ∑ si Li we obtain:
i
∑s L = ∑s
i
i
i
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i
i
si
ε
=
1
s
∑
ε
2
i
=
i
3.2. Cournot Model
H
ε
Is the
Herfindhal
Concentration
Index
24
3.2. Cournot Model
The positive relationship between profitability and the
Herfindhal Concentration Index under Cournot:
Remember the FOC for each firm in that industry can be
written as:
p − ci si
=
p
ε
The Industry-wide profits are then:
n
n
Π = ∑ ( p − ci ) qi =∑
i =1
n
=∑
i =1
si2 p
ε
( p − ci ) × pq
p
i =1
×Q =
pQ
ε
n
∑s
i =1
i
2
i
=
pQ
ε
n
=∑
i =1
si pqi
ε
n
si p qi
=∑
× ×Q =
Q
i =1 ε
H =κH
The concentration index is up to a constant an exact
measure of industry profitability.
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3.2. Cournot Model
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3.2. Cournot Model
Note: The Cournot model is often times criticized because
in reality firms tend to choose prices not quantities.
The answer to this criticism is that when the cournot
model is modified to incorporate two periods, the first
where firms choose capacity and the second where
firms compete in prices. This two period model gives
the same outcome as the simple Cournot model.
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3.2. Cournot Model
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