12
(29) rC
xC (1)
ψ
(30)
rI
t - rC + pI - pC
4tψ - 1
Solving these best response functions simultaneously, we get the equilibrium levels of innovation:
(31)
(32)
rC
'
rI
xC (1)
ψ
tψ-xC(1) +ψ(pI-pC)
ψ(4tψ-1)
The total innovation in the mixed duopsony is then:
(33) r = r + r = tψ + xC(1) (4tψ - 2) + ψ(Pi - Pc )
T C I ψ(4tψ -1)
Plugging rC' and rI' in the expressions for innovation costs, post-innovation profits, and member welfare,
we get:
(34)
` _[xC(1) - ψ + ψ(Pc - pI )]2
I (2,3) = 2ψ (4 tψ -1)
x
(35) MW =I pc -c + —
ι ψ
c IxC (1)
12
__∕γ2
2 txC (1)
x2
xC (1)
2ψ
Price ComPetition at the Pre-Innovation Stage in the Mixed OligoPsony
Unlike the pure oligopsony case, in the mixed duopsony the outcome of price competition in the pre-
innovation stage affects firms’ optimal decisions and payoffs in subsequent stages (see equations (31),
(32), (34) and (35)). Thus, in stage 1 the IOF seeks to determine the input price that maximizes its total
profits (i.e., its profits at the pre-innovation stage plus its profits at the post-innovation stage minus its
innovation costs), i.e.,
(36)
maxΠTI=(PI-c-wI(1))xI(1)
wI(1)
[xC(1) - tΨ + ψ(Pc - Pi )]2
2ψ(4tψ -1)
The best response function of the IOF is given by: