Chemistry Reference
In-Depth Information
where the left hand side is strictly decreasing in
z
k
. The derivative of the best
response function can be determined by implicitly differentiating this equation to
obtain
f
0
.1
C
R
0
k
/
C
R
0
k
f
0
C
R
k
f
00
.1
C
R
0
k
/
C
0
kxx
R
0
k
C
0
kxQ
D
0;
implying that
f
0
C
R
k
f
00
C
0
kxQ
2f
0
C
R
k
f
00
C
0
kxx
R
0
k
D
;
where C
0
kxQ
is the mixed second order partial derivative of C
k
. If in addition to
conditions (A), (B) and (C') we further assume for all k and feasible values of x
k
and Q
k
that
(D) f
0
.x
k
C
Q
k
/
C
x
k
f
00
.x
k
C
Q
k
/
C
0
kxQ
.x
k
;Q
k
/
<C
0
kxx
.x
k
;Q
k
/
f
0
.x
k
C
Q
k
/;
then relation (2.7) remains valid even in this more general case. It can be proved,
similarly to the special case without cost externalities, that under conditions (A),
(B), (C') and (D) there is always a unique Nash equilibrium.
The existence and uniqueness of the Nash equilibrium has been examined by
many authors. Some earlier results used the Brouwer or Kakutani fixed point theo-
rem, which unfortunately is an approach that does not offer computational methods
to find the equilibria, and this would be required in the situation of general price
and cost functions. A comprehensive summary of the most important earlier results
is given in Okuguchi (1976). Okuguchi and Szidarovszky (1999) provide some
extensions of the earlier results that do lead to computational methods to find the
equilibria. The existence and uniqueness proof presented in this section is taken
from Szidarovszky and Yakowitz (1977). Uniqueness and existence results for Nash
equilibria can also be found in Vives (1999), using arguments based on the Tarski
fixed point theorem.
2.2
Discrete Time Models and Local Stability
Consider first the best reply dynamics with adaptive expectations which are gov-
erned by (1.28) and (1.29). A vector ( x
1
;:::; x
N
; Q
1
;:::; Q
N
) is a steady state of
this system if and only if
D
X
l
Q
k
x
l
;
(2.13)
ยค
k
and
D
R
k
. Q
k
/:
x
k
(2.14)
Search WWH ::
Custom Search