Chemistry Reference
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and since an increase in the value of "
k
decreases the price estimate the dynamic
processes (5.91) and (5.92) are now modified to become
"
k
.t
C
1/
D
"
k
.t/
a
k
p
k
.k
D
1;2;:::;N/
(5.143)
and
"
k
D
a
k
p
k
.k
D
1;2;:::;N/:
(5.144)
Notice also that p
k
is the same for all firms, so "
1
;:::;"
N
is a steady state of
the dynamical systems (5.143) or (5.144) if and only if
NA
B
A
C
!
N
N
N
N
X
X
X
X
1
"
k
C
B.N
C
1/
c
k
"
k
c
l
c
l
D
0:
l
D
1
k
D
1
k
D
1
l
D
1
Clearly "
1
D D
"
N
D
B satisfies this equation, so the full knowledge of the
price function is a steady state. However, this is single linear equality in the variables
1="
1
;:::;1="
N
; therefore there are infinitely many positive steady states. That is, at
any other steady state there is no discrepancy between expected and actual prices, so
all firms believe that their price functions are correct, but they are not. Therefore no
learning is possible in this case (see Fudenberg and Levine (1998), Marimon (1997),
or Kirman and Salmon (1995)).
5.4
Uncertain Price Functions
In this section we assume that the firms face an uncertain price function, so firm k
believes that the price function is f
k
.Q/
C
k
,where f
k
.Q/ is the estimate of f.Q/
and
k
is a random error. It is also assumed that E.
k
/
D
0 and Var.
k
/
D
k
for
all k. The believed profit of firm k,
D
x
k
. f
k
.Q/
C
k
/
C
k
.x
k
/;
e
'
k
(5.145)
is therefore also random with expectation
'
k
/
D
x
k
f
k
.Q/
C
k
.x
k
/
E.
e
(5.146)
and variance
'
k
/
D
x
k
k
: (5.147)
For the sake of simplicity no externalities are considered. The firms want to max-
imize their expected profits and at the same time to ensure as low variance of the
profit as possible. Therefore at each time period each firm faces a multi-objective
optimization problem where E.
e
Var.
'
k
/ is minimized.
Assume that
k
measures the relative importance of lowering Var.
e
'
k
/ is maximized and Var.
e
e
'
k
/ compared
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