Chemistry Reference
In-Depth Information
x
k
D
R
k
.Q
k
/ we have
f
0
.1
C
R
0
k
/
C
R
0
k
f
0
C
x
k
f
00
.1
C
R
0
k
/
h
k
C
f
C
x
k
f
0
h
0
k
R
0
k
R
0
k
f
C
x
k
f
0
.1
C
R
0
k
/
h
0
k
Œx
k
f
d
k
h
0
k
R
0
k
D
0;
implying that
.f
0
C
x
k
f
00
/h
k
C
x
k
f
0
h
0
k
.2f
0
C
x
k
f
00
/h
k
.x
k
f
d
k
/h
0
k
R
0
k
D
:
(4.11)
Notice that the denominator coincides with the derivative of the left hand side of
(4.10). As we have just shown above, this expression is negative. The first term of
the numerator is positive and the second term is negative. Therefore R
0
k
does not
have a definite sign, however it is easy to see that R
0
k
>
1 always holds.
Example 4.1.
Consider linear price and labor functions, f.Q/
D
A
BQ; and
h
k
.x
k
/
D
q
k
C
p
k
x
k
with all coefficients being positive. In this case f
0
D
B;
f
00
D
0;h
0
k
D
p
k
and h
0
k
D
0 so that
B.q
k
C
p
k
x
k
/
C
x
k
.
B/p
k
2B.q
k
C
p
k
x
k
/
q
k
2.q
k
C
p
k
x
k
/
;
R
0
k
D
D
2
and 0. Therefore the r
k
D
R
0
k
. Q
k
/ values satisfy the con-
ditions that hold in the concave oligopoly case, so the asymptotic properties of this
model are the same as those discussed for concave oligopolies in Chap. 2.
1
which lies between
In the general case however the R
0
k
. Q
k
/ values can be positive. Contrary to the
case of isoelastic price functions there is the possibility that more than one firm has
positive r
k
values.
Labor-managed oligopolies were introduced and first discussed by Ward (1958).
Hill and Waterson (1983) investigated profit maximizing and labor-managed mod-
els with identical cost functions. The non-symmetric case was examined by Neary
(1984). The works of Okuguchi (1993) and Okuguchi (1996) contain the most
general existence results.
4.2.1
Discrete Time Models and Local Stability
The dynamic models with discrete time scales have exactly the same general forms
as the best response dynamics with adaptive expectations (1.28)-(1.29) and the par-
tial adjustment towards the best response (1.30) in the case of concave oligopolies.
Therefore the eigenvalue equation is also the same as given in (2.24), which we
repeat here for the sake of convenience:
2
3
Y
s
X
s
1
a
j
.1
C
r
j
/
m
j
j
1
a
j
.1
C
r
j
/
4
1
C
5
D
0:
(4.12)
j
D
1
j
D
1
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