
222 Supply Chain Management and Logistics
which is strictly positive by (cr). Also, by (r3), (c1), and (c2), we know that
h
n
(x) < 0 will occur when x is large enough. Therefore,
n
is in existence and
unique.
B. Proof of Proposition 8.2
By (r02), (c2), and (8), we know that i
n
(z) is increasing in n at any xed z. Thus,
n
is increasing in n, as it is the unique root of the decreasing function i
n
(z).
C. Proof of Proposition 8.3
By (c2) and (10), we know that h
1n
(z) is increasing in n at any xed z. Thus,
1
is increasing in n, as it the unique root of the decreasing function h
1n
(z).
On the other hand,
=
/
is the unique root for the decreasi ...