Chemistry Reference
In-Depth Information
Using the right hand equality in the first equation of this example the above quantity
can be rewritten in terms of the unknown value A as
.A
k
A/x
k
C
A
k
Q
k
A
Q
k
D
:
Firm k can now use this value Q
k
to determine its best response. The expected profit
of firm k is
A
k
x
k
x
k
C
Q
k
.d
k
C
c
k
x
k
/;
where we assume a linear cost function, which is known by the firm. The best
response is therefore given by
e
'
k
D
8
<
z
k
0;
0 if
R
k
.Q
k
/
D
z
k
0;
L
k
if
:
z
k
otherwise;
where
s
A
k
Q
k
c
k
z
k
Q
k
.
D
We will next determine the subjective equilibrium in the case when it is interior and
we will realize that it does not coincide with the full information Nash equilibrium.
The quantities in the subjective equilibrium satisfy the equation
s
A
k
c
k
A
..A
k
A/x
k
C
A
k
Q
k
/
1
A
..A
k
A/x
k
C
A
k
Q
k
/;
x
k
D
where we have simply used the expressions provided above. Using the relation
Q
D
x
k
C
Q
k
, this implies that
s
A
k
c
k
A
.A
k
Q
Ax
k
/;
A
k
Q
A
D
so that the market quantity of firm k can be expressed in terms of the realized
industry output Q as
A
k
.AQ
c
k
Q
2
/
A
2
x
k
D
:
By adding up the expressions for the individual quantities offered by the firms
k
D
1;2;:::;N, we obtain a simple equation for the realized industry output Q,
X
N
X
N
Q
2
A
2
Q
A
Q
D
A
k
A
k
c
k
:
kD1
kD1
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