Chemistry Reference
In-Depth Information
periods negative profits may occur. In some of the models we study it will be possi-
ble to ensure non-negative prices simply by selecting suitable parameter values. For
example, for the N-firm oligopoly model with linear inverse demand function a suf-
ficient condition for non-negative prices is
P
kD1
L
k
A=B (see Example 1.1) and
for the model
wit
h quadratic price function and linear costs, we can simply select
P
kD1
L
k
p
A (see Example 1.4).
If the time scales are
continuous
, then output changes are made continuously,
without direct jumps to the best response levels. It is always assumed that in each
time period the output level moves in a direction towards the best response. This
concept is modeled by an N-dimensional system of ordinary differential equations
of the form
0
@
R
k
.
X
l¤k
1
A
x
k
.t/
D
a
k
x
l
.t//
x
k
.t/
.k
D
1;2;:::;N/:
(1.24)
Here a
k
>0is a given constant and also called the
speed of adjustment
of firm k.
This is the continuous time counterpart of the discrete system (1.23), which is also
called the partial adjustment dynamics.
Example 1.9.
Consider again the case of linear oligopolies with linear inverse
demand and linear cost functions, which was discussed earlier in Example 1.1. By
ignoring the non-negativity condition of the outputs and assuming that L
k
D1
for
all k, the best reply of firm k is given as (see (1.6))
1
2
Q
k
C
A
c
k
2B
R
k
.Q
k
/
D
:
Since for all k, R
k
.Q
k
/ is linear with identical derivative, the dynamical systems
(1.22) and (1.23) have the same coefficient matrix, so the asymptotic behavior of
the discrete dynamics with adaptive expectations and with adaptive adjustments are
equivalent. The dynamical system (1.23) for partial adjustment towards the best
response can be written as
0
1
2
X
l
1
A
c
k
2B
@
A
C
.1
a
k
/x
k
.t/;
x
k
.t
C
1/
D
a
k
x
l
.t/
C
(1.25)
¤
k
which is a linear system with coefficient matrix
0
@
1
A
a
2
a
2
1
a
1
:::
a
2
a
2
1
a
2
:::
:
:
:
:
:
:
:
:
:
:
:
:
:
a
2
a
2
:::1
a
N
Search WWH ::
Custom Search