Chemistry Reference
In-Depth Information
5.3.2
Unknown Reservation Price with Known Slope
In this section we assume that all firms know the slope B of the price function but
the value of the reservation price f.0/
D
A is unknown. In this case, the firms try to
estimate and learn about the value of A.Let f
k
.Q/
D
"
k
BQ denote the believed
price function of firm k, where the value of "
k
will be repeatedly updated. The
learning process will be similar to that introduced in the previous section.
Consider first the situation from the point of view of firm k. The profit of any
firm l (including itself) is given as
x
l
."
k
B Q
l
B
e
e
x
l
/
.c
l
e
x
l
C
d
l
/;
(5.114)
where the 'tilde' again indicates that we are dealing with quantities based on firm
k's estimate of the price function. So the best response of firm l is
2B
Q
l
"
k
c
l
e
x
l
D
2
;
implying that
"
k
c
l
B
Q:
e
D
x
l
(5.115)
By summing these equations for all l,wefind
N"
k
P
l
D
1
c
l
B
Q
D
N Q;
so firm k believes that the total output of the industry is
N"
k
P
l
D
1
c
l
.N
C
1/B
Q
k
D
:
Therefore firm k will produce the output
"
k
.N
C
1/c
k
C
P
l
D
1
c
l
.N
C
1/B
"
k
c
k
B
Q
k
x
k
D
D
(5.116)
and expects the market price to be
"
k
C
P
l
D
1
c
l
N
C
1
D
f
k
.Q
k
/
D
e
p
k
:
(5.117)
In reality however each firm reasons independently in the same way, the expected
price and produced amount depend on its estimated price function, so the actual
total output of the industry becomes
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