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x
2
L
2
Boundary equilibria with monopoly outputs
x
2
Interior equilibrium
x
2
R
1
.x
2
/
R
2
.x
1
/
x
1
L
1
x
1
x
1
Fig. 1.3
Example 1.2; the Cournot model with linear price function and quadratic cost function
in the case of duopoly .N
2/. The figure shows case (ii) when B
2
>4.B
D
C
e
1
/.B
C
e
2
/ and
x
k
<x
k
, k
D
1;2. Three equilibria occur in this case
and the equilibrium profits are
D
.B
C
e
k
/.x
k
/
2
;
'
k
k
D
1;2:
If in contrast
B
2
>4.B
C
e
1
/.B
C
e
2
/;
so that marginal costs are decreasing strongly, then the uniqueness of the
equilibrium is no longer guaranteed. For example, Fig. 1.3 shows a case where
A
c
k
B
A
c
k
x
k
2.B
C
e
k
/
D
x
k
;
D
<
so that there is an interior equilibrium and there are also two boundary equilib-
ria given by
A
c
1
2.B
C
e
1
/
;0
0;
;
A
c
2
2.B
C
e
2
/
E
1
D
and E
2
D
where we assume again that A>c
k
for both firms. Observe in addition, that
E
k
includes the monopoly output for firm k (k
D
1;2). At the boundary
equilibrium E
k
, the profit of firm k is
.A
c
k
/
2
=.4.B
C
e
k
// > 0:
In the borderline case, when
B
2
D
4.B
C
e
1
/.B
C
e
2
/;
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