Reputations, Market Structure, and the Choice of Quality Assurance Systems in the Food Industry



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 ^


0si0i=1,2


and the corresponding complementary slackness conditions. Note that if it is difficult to detect


quality deviations (i.e., when ω0), 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


2E1  2E1

s12       s1s 2

2 E2 2 E2

s2s1       s22


ai,i =


2πi
si


2m

- βmim-i) + π -y—

si2


βm-i0


2πiπimπiβmm

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.



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