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1−
a
1
(1+
r
1
)
1−
a
j
0
(1+
r
j
0
)
1−
a
S
(1+
r
S
)
λ
Fig. 3.3
The oligopoly with isoelastic price function and convex cost functions with partial
adjustment towards the best response with naive expectations. The graphical determination of the
eigenvalues in the case 1<j
0
<s
Notice that conditions g.
1/ > 0 and g.1/ > 0 can be written as (2.22) and
X
N
r
k
1
C
r
k
<1;
k
D
1
respectively.
In the case of complex roots, no similar stability condition can be given. The
possibility of complex roots will be shown later in Example 3.2.
The assumption that C
k
is a convex function in its entire domain guarantees the
existence of a Nash equilibrium. However if this condition is not satisfied every-
where and there is an interior equilibrium, then we have to assume that C
k
>0and
C
0
k
0 in its neighborhood in order to assure local asymptotic stability of that
equilibrium. As an illustration consider a duopoly with linear cost functions and
isoelastic price function.
Example 3.2.
In this example we consider the duopoly case .N
D
2/.Byusingthe
notation of Example 3.1 we assume that the cost function of firm k is C
k
.x
k
/
D
d
k
C
c
k
x
k
.k
D
1;2/, the price function is f.Q/
D
A=Q with some positive con-
stant A and the capacity limits are sufficiently large. The equilibrium is positive,
since condition (3.5), that is c
k
c
1
C
c
2
, is satisfied for both firms. Furthermore
at the equilibrium
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