Chemistry Reference
In-Depth Information
Example 4.13.
Consider the special case of symmetric firms, when a
k
a, r
kQ
r
Q
and r
kx
r
x
. Then condition (4.85) reduces to
2
1
r
x
C
r
Q
0<a<
and condition (4.86) becomes
2
a.1
r
x
C
r
Q
Nr
Q
/>0:
These relations hold if the value of a is sufficiently small, in particular if
2
1
C
.r
Q
r
x
/
Nr
Q
;
a<
where the right hand side is always positive.
Example 4.14.
Assume there are N firms with identical capacity limit L,and
assume that the price function is f.Q/
D
LN
Q, so the price is always non-
negative. If the firms have linear cost functions C
k
.x
k
/
D
d
k
C
c
k
x
k
and quadratic
output adjustment costs, K
k
.x
k
x
k
.t//
2
x
k
.t//
D
k
.x
k
where x
k
>x
k
.t/
and zero otherwise, then the profit of firm k may be written as
(
0
if x
k
x
k
.t/;
x
k
.LN
x
k
Q
k
/
.d
k
C
c
k
x
k
/
k
.x
k
x
k
.t//
2
if x
k
>x
k
.t/:
Assume an interior optimum for the profit maximization problem. In the first case
(x
k
x
k
.t/) the first order condition can be written as
LN
2x
k
Q
k
c
k
D
0;
implying that
LN
Q
k
c
k
2
x
k
D
:
This point is below x
k
.t/ if and only if
LN
Q
k
c
k
2x
k
.t/:
In the second case (x
k
>x
k
.t/) the first order condition is
LN
2x
k
Q
k
c
k
2
k
.x
k
x
k
.t//
D
0;
the solution of which is
LN
Q
k
c
k
C
2
k
x
k
.t/
2
C
2
k
x
k
D
:
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