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also briefly discussed in Szidarovszky et al. (2008). The model studied in Sect. 5.3.2
has been analyzed in Szidarovszky (2004) where the effect of information delay has
also been investigated.
5.3.1
Unknown Slope with Known Market Saturation Point
We assume first that the firms know that the demand function is linear and decreasing.
Firms know the value of the market saturations point A=B, that is the total output
level that renders the price zero, but they have only a misspecified estimate of the
slope 1=B of the demand function. Suppose that in period t firm k has an estimate of
this slope, which we write as 1="
k
.t/. Then, in this case this is equivalent to saying
that firm k's estimate of the price function is f
k
.Q/
D
"
k
.A=B
Q/,wherethe
factor "
k
.t/ is adjusted over time on the basis of observed price data.
Let us consider the situation from the point of view of an arbitrary firm, say
firm k.Given"
k
, each firm k solves the static game. It believes that the profit of
each firm l (including itself) is given as
x
l
"
k
A
x
l
B
Q
l
e
e
.c
l
e
x
l
C
d
l
/:
(5.84)
Basedonthisbelieffirmk it is able to calculate the believed equilibrium outputs
and the equilibrium price. Then this believed price will be compared to the actual
market price the firm receives, and based on the discrepancy between the believed
and actual prices firm k can adjust the shape estimate "
k
.
Assuming an interior optimum, firm k believes that the best response of firm l is
2"
k
Q
l
A
2B
c
l
e
D
x
l
2
;
implying that
A
B
c
l
"
k
Q:
e
x
l
D
(5.85)
By summing these equations for all firms we have
N
X
NA
B
1
"
k
Q
D
N Q;
c
l
l
D
1
so firm k believes that the total output of the industry is
NA
B
!
:
X
N
1
N
C
1
1
"
k
Q
k
D
c
l
(5.86)
l
D
1
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