Chemistry Reference
In-Depth Information
5.1
Misspecified Price Functions
As before, let x
k
denote the output of firm k.1
k
N/, C
k
.x
k
/ its cost, so
no externalities are assumed, and p
D
f.Q/ the true price function, where Q
D
P
kD1
x
k
. Assume that the firms only have estimates of the price function f and
let f
k
, denote firm k's belief about f . In order to be realistic, f
k
isassumedtobe
strictly decreasing. In the case of full information each firm k knowsthetrueprice
function. Hence, given the observed market price p, from the equation
p
D
f.Q
k
C
x
k
/
(5.1)
each firm would be able to obtain the output of the rest of the industry, namely
D
f
1
.p/
x
k
:
Q
k
In the case of misspecified price functions, the true price is again given by p
D
f.Q
k
C
x
k
/.Howeverfirmk believes that it is p
D
f
k
.Q
k
C
x
k
/, so its estimate
Q
k
about the output of the rest of the industry satisfies the equation
f
k
.Q
k
C
x
k
/
D
p
D
f.Q
k
C
x
k
/;
implying that
D
. f
k
of /.Q
k
C
x
k
/
x
k
:
Q
k
(5.2)
Firm k also believes that its profit at any time period t
C
1 is
D
x
k
f
k
.x
k
C
Q
k
.t
C
1//
C
k
.x
k
/;
e
'
k
(5.3)
where Q
k
.t
C
1/ is its expectation of the output of the rest of the industry in period
t
C
1. The best response R
k
of this firm given its belief can be obtained in the same
wayasitwasshownbyrelation(1.3)inChap. 1. However in this case, the true price
function f has to be replaced by the believed price function f
k
.
Assuming discrete time scales, using (5.2) and the fact that the firms form adap-
tive expectations on the output of the rest of the industry we obtain the dynamic
model
Q
k
.t/
C
˛
k
. f
k
of /
N
x
l
.t/
!
x
k
.t/
Q
k
.t/
!!
;
X
x
k
.t
C
1/
D
R
k
lD1
(5.4)
. f
k
of /
N
x
l
.t/
!
x
k
.t/
Q
k
.t/
!
; (5.5)
X
Q
k
.t
C
1/
D
Q
k
.t/
C
˛
k
lD1
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