Chemistry Reference
In-Depth Information
For example, in the case of linear cost functions, C
0
1
D
C
0
2
D
0,wehave
f
0
C
f
00
x
k
2f
0
f
00
x
k
2f
0
r
kQ
D
and r
kx
D
and the stability condition becomes
x
1
x
2
f
00
2
4f
0
2
1<
.f
0
C
x
1
f
00
/.f
0
C
x
2
f
00
/
4f
0
2
< min
(
1
C
)
:
x
1
x
2
f
00
2
4f
0
2
x
1
x
2
f
00
2
4f
0
2
f
00
.x
1
C
x
2
/
2f
0
f
00
.x
1
C
x
2
/
2f
0
C
; 1
C
It is very easy to check if this condition holds or not.
5.3
Other Learning Processes
In this section we consider other adaptive learning processes. We study situations
where firms adaptively adjust their beliefs about the price function based on the dis-
crepancies between their predicted and actually observed market prices. For the sake
of mathematical simplicity, we will assume in this section that no cost externalities
are present, and that the (inverse) demand and cost functions are linear.
Hence, in this section we consider the demand relationship
A
B
1
B
p
p
D
f.Q/
D
A
BQ or Q.p/
D
f
1
.p/
D
and the cost function
C
k
.x
k
/
D
d
k
C
c
k
x
k
for k
D
1;2;:::;N,wherex
k
is the output of firm k and Q
D
P
lD1
x
l
is the
industry output. We assume that the firms have only limited knowledge of the price
function p
D
f.Q/and over time they repeatedly update their estimates.
We consider three scenarios. First, regarding the demand function of the general
form Q.p/ we assume that the particular values of the reservation price A and the
slope of the price function B are unknown, but firms know the value of the market
saturation point, that is, they know the value of A=B. In the second scenario we
consider the inverse demand function and assume that the firms know the slope B
but do not know the reservation price A. Finally, we study a situation where firms
know the reservation price A, but do not know the slope B. As we will demonstrate,
the possibility of learning as well as the asymptotic behavior of the learning process
strongly depends on the firm's knowledge about the demand parameters and also on
the updating procedure the firms use.
The learning schemes discussed in this chapter have been introduced by
Szidarovszky (2003) and extended by Szidarovszky and Krawczyk (2005) They are
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