Chemistry Reference
In-Depth Information
8
<
:
if f.Q
k
/
C
k
.0/
0;
0
R
k
.Q
k
/
D
if f.L
k
C
Q
k
/
C
L
k
f
0
.L
k
C
Q
k
/
C
k
.L
k
/
0;
(2.3)
L
k
z
k
otherwise;
where
z
k
is the unique solution of the strictly monotonic equation
f.
z
k
C
Q
k
/
C
z
k
f
0
.
z
k
C
Q
k
/
C
k
.
z
k
/
D
0
(2.4)
in the interval .0;L
k
/. Observe that due to our assumptions the left hand side of
(2.4) strictly decreases and is continuous in
z
k
, positive at
z
k
D
0 and negative at
z
k
D
L
k
, therefore there is a unique solution.
In order to analyze the asymptotic behavior of any one of the discrete time
and continuous time dynamical systems (1.19), (1.28)-(1.31) emerging from par-
tial adjustment and best reply behavior, we will need to examine the Jacobian of
the systems, and to do so, we have to determine the derivatives of the best response
functions. These derivatives can be obtained by implicitly differentiating equation
(2.4). Assuming that
z
k
is interior (that is, 0<
z
k
<L
k
), then we have
f
0
.1
C
R
0
k
/
C
R
0
k
f
0
C
R
k
f
00
.1
C
R
0
k
/
C
0
k
R
0
k
D
0;
implying that
f
0
C
R
k
f
00
2f
0
C
R
k
f
00
C
0
k
R
0
k
D
:
(2.5)
Note that the same result could be obtained directly by using
@
2
'
k
=@x
k
@Q
k
@
2
'
k
=@x
k
R
0
k
D
:
(2.6)
Conditions (B) and (C) imply that
1<R
0
k
.Q
k
/
0
(2.7)
for all k and Q
k
. In the first two cases of (2.3), the derivative of R
k
is zero, except
at two possible break-points, so (2.7) is always satisfied. This property will play a
crucial role later in the stability analysis.
Notice that relation (2.4) shows that R
k
.Q
k
/ decreases in Q
k
, that is, larger total
output of the rest of the industry requires smaller output responses from the firms.
Since R
0
k
.Q
k
/ is larger than
1, best responses cannot decrease very rapidly.
We notice that if f.0/>C
k
.0/, that is the reservation price is higher than the
marginal costs at x
k
D
0, then the monopoly quantity (ignoring capacity limits) of
firm k is the solution of equation (2.4) with Q
k
D
0.IfitisbelowL
k
, then it is the
D
0.
best response of firm k at Q
k
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