Chemistry Reference
In-Depth Information
In other words, ( x
1
;:::;x
N
) is a Nash equilibrium. The same is true obviously
for partial adjustment towards the best response, which are modeled by (1.30). If the
sign-preserving function ˛
k
is a homogeneous linear function of the form ˛
k
.
z
/
D
a
k
z
, then the system (1.28)-(1.29) reduces to (1.20)-(1.21), so we will not discuss
system (1.20)-(1.21) directly, but only as a special case. Similarly, system (1.30)
also reduces to (1.23) in this case.
The local and global stability properties of an equilibrium depend on the partic-
ular adjustment process which is used by the firms to update their quantity choices.
So any stability result to be introduced and proved in this topic is always appli-
cable to the particular dynamical system for which it is proved. In this section
best reply dynamics with adaptive expectations and partial adjustment towards the
best response will be examined. We will return to gradient adjustments later in this
chapter.
The asymptotic stability of the equilibrium will be examined by the technique of
linearization around the equilibrium, which is summarized briefly in Appendix B.
Here we assume that the equilibrium is interior, otherwise the best response func-
tions are not differentiable. In such cases we have to assume that the conditions of
Theorem B.3 are satisfied in a neighborhood of the equilibrium. First we show that
as far as local asymptotic stability is concerned, the conditions for the best reply
dynamics with adaptive expectations and partial adjustments are equivalent since it
turns out that the Jacobians of the two processes have identical nonzero eigenvalues.
The Jacobian of the the best reply dynamics
1
(1.28)-(1.29) has a special structure,
namely
J
11
J
12
(2.15)
J
21
J
22
with
0
@
1
A
0
@
1
A
0
1
a
1
:::r
1
a
1
r
2
a
2
0 :::r
2
a
2
:
:
:
:
:
:
:
:
:
r
N
a
N
r
N
a
N
::: 0
r
1
.1
a
1
/
0
r
2
.1
a
2
/
:
:
:
J
11
;
J
12
D
D
;
0
r
N
.1
a
N
/
0
1
0
1
0a
1
:::a
1
a
2
0 :::a
2
:
:
:
:
:
:
:
:
:
a
N
a
N
::: 0
1
a
1
0
@
A
@
A
1
a
2
:
:
:
J
21
J
22
D
; and
D
;
0
1
a
N
where for all k,
1
See Appendix B for a definition of the Jacobian of a dynamical system. We stress that unless
indicated otherwise the elements of this matrix are evaluated at the steady state of the system,
which is indicated by the overbar.
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