15
choose the level of output that maximizes per-period profits in the standard way to find the
Nash equilibrium levels y * (s ,s- ), = 1,2 . We now write the first-stage problem for firm as
max
s ∈S
1-βm (s )m- (s- )
= max
s ∈S
' π ( s, s-i∙) λ
41 - βmi( si )m -i(s _i.)y
which has the first-order condition
(6)
__________1__________(∂∏(si,s-i ) + ∏(sι,s-i)βm-ι(s-i) ∂mi(si)
I-βm1( sι)m- i ( s- i )^ dsi (1-βmi( si)m- i ( s-.)) dsi ^
≤0si ≥0i=1,2
and the corresponding complementary slackness conditions. Note that if it is difficult to detect
quality deviations (i.e., when ω→0), the model predicts that processors will find it optimal
(** 7
s1=0,s2=0)).7 The
same result holds if β = 0 or when the probability that the processor’s rival is caught is close
to one.
The second-order sufficient conditions are
a1,1 a1,2 1
a2,1 a2,2 _ (1 - βm1 m2 )2
∂2E∏1 ∂2E∏1
∂s12 ∂s1∂s 2
∂2 E ∏2 ∂2 E ∏2
∂s2∂s1 ∂s22
ai,i =
∂ 2πi
∂si
∂2m
- βmim-i) + π -y—
∂si2
βm-i ≤0
∂2πi∂πi∂mπiβ∂m∂m
and Ω =ai,ia-i,-i-ai,-ia-i,i>0 , for
ai i =-----(1-βmim i ) +--i- βm i +------—- —i--l
, ''s⅛-i ⅛-i l^s (1-βmim-i) l^s &_i
i=1,2.