Chemistry Reference
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full information case. The local asymptotic stability of the equilibrium then can be
examined by following the argument of Sect. 3.1.1, with identical conclusions.
5.1.2
Discrete Time Models and Global Dynamics
In order to illustrate the asymptotic behavior of oligopolies with misspecified price
functions and to show the dependence of the believed equilibrium on the way in
which firms misspecify the price function we study a particular example.
Example 5.3.
Assume again that the true price function is f.Q/
D
A=Q,butfirm
k believes that the price function is f
k
.Q/
D
A=.Q
C
"
k
/ with some "
k
0.
Assume that the firms have linear cost functions denoted by C
k
.x
k
/
D
d
k
C
c
k
x
k
.
Let L
k
denote the capacity limit of firm k.Firmk's expected profit is (assuming
naive expectations)
Ax
k
x
k
C
Q
k
C
"
k
.d
k
C
c
k
x
k
/.
e
'
k
D
Therefore, the best reply is given by
8
<
z
k
0 if
0;
R
k
.Q
k
/
D
z
k
L
k
if
0;
:
z
k
otherwise;
where
s
A.Q
k
C
"
k
/
c
k
.Q
k
C
"
k
/.
z
k
D
(5.14)
Firm k is able to observe the market price p, so it can determine the believed output
of the rest of the industry based on its price function estimate. So from the equation
A
x
k
C
Q
k
C
"
k
D
p;
the firm believes that
A
p
x
k
"
k
:
Then the best response of this firm the
z
k
is given by (5.14). However the price is
p
D
A=.x
k
C
Q
k
/,whereQ
k
is the true output of the rest of the industry, so
Q
k
D
Q
k
D
.x
k
C
Q
k
/
x
k
"
k
D
Q
k
"
k
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