Chemistry Reference
In-Depth Information
P
k
D
1
"
k
P
l
D
1
c
l
.N
C
1/B
X
N
Q
D
x
k
D
;
k
D
1
with the corresponding equilibrium price
N
!
:
X
X
N
1
N
C
1
p
D
A
BQ
D
A
"
l
c
l
(5.118)
lD1
lD1
As in the previous section the firms adjust their beliefs about the price function
based on the discrepancies
.N
C
1/A
!
:
X
N
1
N
C
1
p
k
D
p
e
p
k
D
"
l
"
k
(5.119)
l
D
1
They want to increase "
k
if p
k
>0,andifp
k
<0then they decrease the
value of "
k
,andifp
k
D
0, then they have no reason to change it. This adjustment
concept can be again modeled by the discrete system (5.91) and its continuous coun-
terpart (5.92). Notice that p
k
is a linear function of the state variables "
1
;:::;"
N
,
therefore the corresponding dynamical systems are linear, and in this case local and
global asymptotic stability are equivalent. Similarly to the previous case it is easy
to show that both systems have a unique steady state, "
k
D
A for all k which corre-
sponds to full knowledge of the price function. Notice first that if p
k
D
0 for all
k, then the "
k
values are identical. If
" denotes their common value, then
1
N
C
1
..N
C
1/A
N "
"/
0
D
implying that "
D
A.
Consider first the discrete case. The coefficient matrix has now the special
structure
I C J
where
0
1
2a
1
a
1
:::
a
1
@
A
a
2
2a
2
:::
a
2
1
N
C
1
T
J D
D D C ab
(5.120)
:
:
:
:
:
:
:
:
:
:
:
:
a
N
a
N
:::
2a
N
with
D D
diag
;
a D
T
a
1
N
C
1
;:::;
a
N
N
C
1
a
1
N
C
1
;:::;
a
N
N
C
1
;
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