Chemistry Reference
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the number of labor units in firm k as a function of its production level x
k
, then the
payoff of firm k is given as
x
k
f.Q/
Wh
k
.x
k
/
d
k
h
k
.x
k
/
'
k
.x
1
;:::;x
N
/
D
;
(4.9)
with Q
D
P
lD1
x
l
as before. Here the total production costs are given by
C
k
.x
k
/
D
Wh
k
.x
k
/
C
d
k
:
Notice that no externalities are included in this model.
The existence of the static Nash equilibrium has been proved by Okuguchi (1996)
under realistic conditions. This result has been also discussed in detail in Okuguchi
and Szidarovszky (1999).
In this section we assume that the price function f and the functions h
k
,forall
k, are twice continuously differentiable. Furthermore, we assume that
(A) f
0
.Q/<0;
(B) x
k
f
00
.Q/
C
f
0
.Q/<0;
(C) h
0
k
.x
k
/>0; and h
0
k
.x
k
/
0,
for all k and all feasible values of x
k
and Q.
Condition (C) states that the functions h
k
are convex and increasing, which
means that for additional outputs increasingly more labor units are required.
Consider an interior equilibrium. In its neighborhood the best response of firm k
is the solution of the single variable equation
d
k
h
0
k
.x
k
/
Œf.x
k
C
Q
k
/
C
x
k
f
0
.x
k
C
Q
k
/h
k
.x
k
/
Œx
k
f.x
k
C
Q
k
/
@'
k
@x
k
D
D 0;
h
k
.x
k
/
2
which can be written as
Œf.x
k
C
Q
k
/
C
x
k
f
0
.x
k
C
Q
k
/h
k
.x
k
/
Œx
k
f.x
k
C
Q
k
/
d
k
h
0
k
.x
k
/
D
0: (4.10)
Notice that the derivative of the left hand side with respect to x
k
is given by
.2f
0
C
x
k
f
00
/h
k
.x
k
f
d
k
/h
0
k
:
Under assumptions (A) and (B), the first term is negative. If we make the natu-
ral assumption that at the equilibrium the firms have non-negative payoffs, then the
second term is non-positive, so the derivative of the left hand side of (4.10) is neg-
ative. Therefore in the neighborhood of the equilibrium the best response function
is unique. By implicitly differentiating (4.10) with respect to Q
k
and noting that
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