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of the competitors. The firms have to compete in the secondary market to ensure
capital, manpower, energy, material, etc. for their production processes. The tech-
nological and intellectual spillover between companies is another cost externality
which adds to the interdependence of the firms. In the literature on oligopoly theory
the interdependence of the firms through their cost functions is either ignored by
assuming that the cost of firm k is C
k
.x
k
/; or it is assumed that the cost of firm k
depends on its own production level x
k
and also on the total production level of the
rest of the industry, which we will denote by Q
k
D
P
l¤k
x
l
so that the cost func-
tion of firm k may be written more generally as C
k
.x
k
;Q
k
/. In the rest of the topic
we will consider various cases where cost externalities arise. Note that under this
assumption the profit of any firm k just depends on its own output and the output
of the rest of the industry, it does not depend on the individual output level of any
competitor. For this reason it is convenient to rewrite the profit function of firm k as
'
k
.x
1
;:::;x
N
/
D
x
k
f.x
k
C
Q
k
/
C
k
.x
k
;Q
k
/:
(1.2)
Taken together, the above set-up yields a static N-person game, where the play-
ers are the firms, the strategy set of firm k is the interval Œ0;L
k
,whereL
k
is the
capacity limit of firm k and its payoff function is given by (1.2). If we assume that
all firms are rational in the sense that they want to maximize their own profits, then
we can derive the firms' best responses. That is, if firm k knows the total production
Q
k
of the rest of the industry, then it will select a production level x
k
that maxi-
mizes its profit (1.2). For each value of Q
k
let R
k
.Q
k
/ denote the set of all optimal
solutions, that is
x
k
; (1.3)
R
k
.Q
k
/
D
j
x
k
D
arg
0x
k
L
k
f
x
k
f.x
k
C
Q
k
/
C
k
.x
k
;Q
k
/
g
max
which is called the
best response
or
best reply mapping
of firm k. In the general
case this is a point-to-set mapping, and in this case it is usually called the
best
reply correspondence
. In the case of a unique optimal solution, R
k
.Q
k
/ is called
the
best reply
or
reaction function
of firm k.The
Nash equilibrium
of the game is
a simultaneous production vector ( x
1
;:::; x
N
) which is a best response for each
firm, under the assumption that all others maintain their corresponding equilibrium
production levels. This concept can be mathematically expressed for all k as,
D
X
l
2
R
k
. Q
k
/ with
Q
k
x
k
x
l
:
(1.4)
¤
k
At the equilibrium all firms simultaneously select their best responses to the cor-
responding equilibrium choices of the competitors. In other words, no firm has any
interest to deviate unilaterally from its equilibrium level.
In the following examples we will show that best responses might have a large
variety of forms, and also, that oligopolies may have no equilibrium at all. Further-
more, in the case of existence there may be multiple equilibria, and the number of
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