Chemistry Reference
In-Depth Information
Assume that all equilibrium outputs are positive, the other case can be examined
similarly. The first order conditions imply that
@'
k
@x
k
D
@
@x
k
Œx
k
.A
Bx
k
BQ
k
/
.d
k
C
c
k
x
k
/
D
A
2Bx
k
BQ
k
c
k
D
A
Bx
k
BQ
c
k
D
0;
(1.7)
where Q is the total output of the industry. So
A
BQ
c
k
B
A
c
k
B
Q:
x
k
D
D
(1.8)
By summing this last equation over all firms we obtain for Q the single equation
NA
P
i
D
1
c
i
B
Q
D
NQ;
(1.9)
implying that at the equilibrium
NA
P
i
D
1
c
i
.N
C
1/B
Q
D
:
(1.10)
Notice that
N
Q<A=B, so the price is always positive. From (1.8) and (1.10) we can
compute the equilibrium output levels of the firms as
NA
P
i
D
1
c
i
.N
C
1/B
D
A
.N
C
1/c
k
C
P
i
D
1
c
i
.N
C
1/B
A
c
k
B
x
k
D
:
(1.11)
The output levels in (1.11) can be an equilibrium only if they are all non-negative
and below the corresponding capacity limits. The equilibrium price is then
A
C
P
i
D
1
c
i
N
C
1
p
D
A
B Q
D
:
At the equilibrium, the profit of firm k is given by
A
C
P
i
D
1
c
i
N
C
1
!
'
k
D
x
k
p
.d
k
C
c
k
x
k
/
D
x
k
c
k
d
k
A
.N
C
1/c
k
C
!
2
N
X
1
.N
C
1/
2
B
D
c
i
d
k
:
i
D
1
Search WWH ::
Custom Search