Chemistry Reference
In-Depth Information
Chapter 3
General Oligopolies
In the previous chapter we analyzed concave oligopolies where the best response
functions were monotonic and therefore the local and global analysis of the corre-
sponding dynamic processes were relatively simple. The examples discussed there
have allowed the reader to become familiar with the major concepts and methods
that we shall use in the rest of the topic. If we drop the simplifying assumptions of
the previous chapter then more complex dynamics may arise. In this chapter we will
present a collection of such models.
We initiate our discussion in Sect. 3.1 where we consider oligopolies with isoe-
lastic price functions and dynamics in discrete time. We give a detailed analysis
of local and global stability of some particular examples. In Sect. 3.2 we return to
the issue of oligopolies with cost externalities, which may display multiple interior
Nash equilibria. The global analysis of some specific examples indicates how the
oligopoly may converge to particular equilibria.
3.1
Isoelastic Price Functions
In this section we assume that the price function is isoelastic, as in Example 1.5. As
in the previous chapters let N denote the number of firms, let x
k
be the output of
firm k.k
D
1;2;:::;N/and Q
D
P
kD1
x
k
the total output of the industry. Then
the price function is f.Q/
D
A=Q with some positive constant A. If no externalities
are assumed and C
k
.x
k
/ denotes the cost of firm k, then its profit is given as
8
<
:
C
k
.0/;
D
0;
if x
k
'
k
.x
1
;:::;x
N
/
D
Ax
k
Q
k
C
x
k
C
k
.x
k
/; if x
k
>0;
where we use again the simplifying notation Q
k
D
P
l¤k
x
l
so that Q
D
Q
k
C
x
k
.
In the following discussion we will assume that for all k, C
k
is twice continuously
differentiable, increasing and convex, so that for all feasible values of x
k
,
(D) C
k
.x
k
/>0and C
0
k
.x
k
/
0.
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