Chemistry Reference
In-Depth Information
written as (1.1) that we repeat here for the sake of convenience,
'
k
.x
1
;:::;x
N
/
D
x
k
f.Q/
C
k
.x
k
/:
Assume that the price function f and all cost functions are twice continuously
differentiable and satisfy the conditions
(A) f
0
.Q/ < 0;
(B) x
k
f
00
.Q/
C
f
0
.Q/
0;
(C) f
0
.Q/
C
0
k
.x
k
/<0;
for all k and feasible values of x
k
and Q.
Condition (A) means that f.Q/is strictly decreasing in Q, that is, a larger total
output can only be sold for a lower price. Condition (B) is called the decreasing
marginal revenue condition, it states that marginal revenue for firm k decreases for
higher levels of output of the rest of the industry (see for example, Vives (1999)).
Condition (C) relates the lower bound on the convexity/concavity of the cost func-
tion to the degree of negativity of the slope of the price function. It is assumed by
many authors that f.Q/ is concave and C
k
.x
k
/ is convex for all k. In this case
f
0
<0, f
00
0, C
0
k
0, and naturally C
k
>0, since a larger output level requires
higher cost. Conditions (B) and (C) are then clearly satisfied. In fact these condi-
tions are slightly more general, since they can be also satisfied if f is slightly convex
and/or C
k
is slightly concave provided that
f
0
is large enough. We have to men-
tion as well, that conditions (A)-(C) are more restrictive than the simple condition
that the profit functions be concave.
Notice that
@
@x
k
'
k
.x
1
;:::;x
N
/
D
f.x
k
C
Q
k
/
C
x
k
f
0
.x
k
C
Q
k
/
C
k
.x
k
/;
(2.1)
and under these conditions
@
2
@x
k
'
k
.x
1
;:::;x
N
/
D
2f
0
.x
k
C
Q
k
/
C
x
k
f
00
.x
k
C
Q
k
/
C
0
k
.x
k
/<0; (2.2)
D
l¤k
where Q
k
x
l
as in the previous chapter. Hence '
k
is strictly concave in x
k
.
In order to prove the existence of a unique equilibrium under conditions (A)-(C)
and develop dynamic models we have to determine first the best response functions
of the firms.
The concavity of the profit functions implies that the best response functions can
be obtained in the form
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