18
|
all |
a1,2 |
1 |
ds1 |
|
_ a2,1 |
a2,2 _ |
(1-βm1(s1)m2(s2))2 |
_ds2_ |
|
~∂2 E ∏1 |
∂2 E ∏1 ^ |
~∂2 E ∏1 ' -----dω ∂s1∂ω | ||
|
ds2 |
∂s1∂s 2 |
ds1 | ||
|
∂2 E ∏2 |
∂2 E ∏2 |
_ds2_ |
_ |
∂2 E ∏2 ——z—dω [_ ∂s2∂ω J |
|
∂s 2∂s1 |
∂s 22 |
where ai,i and ai,-i, for i = 1,2 are as previously defined. The system can in principle be
solved to obtain
|
(8) |
^ ds1* ^ |
1 |
- a 2,2 |
a1,2 |
~∂2 E ∏1 |
|
∂s1∂ω | |||||
|
= |
, |
, |
∂2 E ∏2 | ||
|
ds * |
Ω |
_ a2,1 |
-a1,1 _ | ||
|
_ dω _ |
∂s 2∂ω |
where again the stability conditions are Ω =(a1,1a2,2 - a1,2a2,1) >0 and ai,i < 0 for i = 1,2 .
The decisions of both processors are equal, so it suffices to analyze the responses of
processor 1. From equation (8), the change in the optimal stringency of assurance for firm 1 is
given by
(9)
ds 1 ( ∂2 E ∏1 ∂2 E ∏2
^^= = _ I - a 2 2 Z + + al 2 Z Z
dω Ω4 , ∂s1∂ω , ∂s 2∂ω
The sign of expression (9) is ambiguous in general. However, we can further analyze
some cases. From the stability conditions, we know that Ω is positive and a2,2 is negative.
Nothing more can be ascertained without imposing further structure. If symmetry is assumed
(as it is here) the cross-partial terms are equal and hence we need to sign only one of them.
We study the case in which QASs are strategic complements.9 From the previous
analysis we know that this case arises when a1,2 > 0 . Using the stability conditions, we
• ʃɪ, ɪ, ɪ (ds* I (∂2E∏1
immediately see that sgn ∣ —- I = sgn ∣------
У dω ) ∂s1∂ω
Cross-partial differentiation of the objective