Cournot with N …rms (revisited)
Cournot model with N symmetric …rms, constant unit variable cost
c, and inverse demand function
P (Q ) = a
where Q = ∑N
i =1 qi
The results:
q =
a c
b (1 + N )
p =
π =
()
bQ
a + Nc
1+N
(a c )2
b (1 + N )2
December 1, 2012
1/4
Cournot with N …rms and entry cost
Suppose now that there is also an entry cost: F > 0.
TCi = F + cqi
()
December 1, 2012
2/4
Cournot with N …rms and entry cost
Suppose now that there is also an entry cost: F > 0.
TCi = F + cqi
Once the …rm decided to enter the market, this is a sunk cost
()
December 1, 2012
2/4
Cournot with N …rms and entry cost
Suppose now that there is also an entry cost: F > 0.
TCi = F + cqi
Once the …rm decided to enter the market, this is a sunk cost
So, pro…ts are simply:
π =
()
(a c )2
b (1 + N )2
F
December 1, 2012
2/4
Cournot with N …rms and entry cost
Suppose now that there is also an entry cost: F > 0.
TCi = F + cqi
Once the …rm decided to enter the market, this is a sunk cost
So, pro…ts are simply:
π =
(a c )2
b (1 + N )2
F
So, how many Cournot-…rms will this market have in equilibrium?
()
December 1, 2012
2/4
Cournot with N …rms and entry cost
Suppose now that there is also an entry cost: F > 0.
TCi = F + cqi
Once the …rm decided to enter the market, this is a sunk cost
So, pro…ts are simply:
π =
(a c )2
b (1 + N )2
F
So, how many Cournot-…rms will this market have in equilibrium?
Zero-pro…t condition regulates entry
(a c )2
b (1 +N )2
()
F =0
December 1, 2012
2/4
Cournot with N …rms and entry cost
Suppose now that there is also an entry cost: F > 0.
TCi = F + cqi
Once the …rm decided to enter the market, this is a sunk cost
So, pro…ts are simply:
π =
(a c )2
b (1 + N )2
F
So, how many Cournot-…rms will this market have in equilibrium?
Zero-pro…t condition regulates entry
(a c )2
b (1 +N )2
F =0
Re-arranging
(a c )2
bF
()
= (1 + N )2
December 1, 2012
2/4
Cournot with N …rms and entry cost
Suppose now that there is also an entry cost: F > 0.
TCi = F + cqi
Once the …rm decided to enter the market, this is a sunk cost
So, pro…ts are simply:
π =
(a c )2
b (1 + N )2
F
So, how many Cournot-…rms will this market have in equilibrium?
Zero-pro…t condition regulates entry
(a c )2
b (1 +N )2
F =0
Re-arranging
(a c )2
bF
Yields:
()
(a c )
N = p
bF
= (1 + N )2
1
December 1, 2012
2/4
Cournot with N …rms and entry cost – extension
Suppose now that aggregate market demand is given by:
P (Q ) = (a
S
bQ ) S
1 is some measure of the ‘size of the market’.
When S grows, demand expands for a given price.
()
December 1, 2012
3/4
Cournot with N …rms and entry cost – extension
Suppose now that aggregate market demand is given by:
P (Q ) = (a
S
bQ ) S
1 is some measure of the ‘size of the market’.
When S grows, demand expands for a given price.
Pro…ts in the standard N-…rm Cournot (i.e., without entry cost)
model will now be –homework for yourselves to work it out–
"
#
(a c )2
π =S
b (1 + N )2
()
December 1, 2012
3/4
Cournot with N …rms and entry cost – extension
Suppose now that aggregate market demand is given by:
P (Q ) = (a
S
bQ ) S
1 is some measure of the ‘size of the market’.
When S grows, demand expands for a given price.
Pro…ts in the standard N-…rm Cournot (i.e., without entry cost)
model will now be –homework for yourselves to work it out–
"
#
(a c )2
π =S
b (1 + N )2
h
i
(a c )2
When including entry cost F > 0, we have: π = S
F
2
b (1 +N )
()
December 1, 2012
3/4
Cournot with N …rms and entry cost – extension
Suppose now that aggregate market demand is given by:
P (Q ) = (a
S
bQ ) S
1 is some measure of the ‘size of the market’.
When S grows, demand expands for a given price.
Pro…ts in the standard N-…rm Cournot (i.e., without entry cost)
model will now be –homework for yourselves to work it out–
"
#
(a c )2
π =S
b (1 + N )2
h
i
(a c )2
When including entry cost F > 0, we have: π = S
F
2
b (1 +N )
Solution:
N = (a
()
c)
r
S
bF
1
December 1, 2012
3/4
Cournot with N …rms and entry cost – extension
Suppose now that aggregate market demand is given by:
P (Q ) = (a
S
bQ ) S
1 is some measure of the ‘size of the market’.
When S grows, demand expands for a given price.
Pro…ts in the standard N-…rm Cournot (i.e., without entry cost)
model will now be –homework for yourselves to work it out–
"
#
(a c )2
π =S
b (1 + N )2
h
i
(a c )2
When including entry cost F > 0, we have: π = S
F
2
b (1 +N )
Solution:
r
S
1
bF
So, number of …rms N grows with the size of the market, S
N = (a
()
c)
December 1, 2012
3/4
Strategic Behaviour – Entry Deterrance
Back to Stackelberg Model
Two …rms in a sequential Cournot model: Leader and Follower
Timing:
1
2
Firm 1 chooses q1 .
Firm 2 observes q1 and then chooses q2 .
How do we …nd the solution to model?
By backwards induction: start by stage 2, and then move backwards to
stage 1
We start by solving the problem of Firm 2, given the value of q1
Next, we solve for q1 , given the response of Firm 2 to each possible q1
In the end, everything is solved as a function of q1
()
December 1, 2012
4/4