13
2ctψ+4tψ2+8ct2ψ2-t -2ctψpC+8t2ψ2pI+(8t2ψ2-2tψ-1)wC(1)
I(1) 1612ψ2 - 4tψ -1
Regarding the co-op, its problem at this stage is to determine the price that maximizes the welfare
of its pre-innovation membership in both the pre- and post-innovation stages of the game, subject to
raising earnings that can be retained to finance its cost-reducing innovation in stage 2. The capital
x2
x C(1)
---an and the problem of the co-op
2ψ
12
required for the subsequent investment in innovation is IC = — ψtC
at the pre-innovation stage can be expressed as:
(37)
max MW = MW11^ + MW (3/1) = ( wC (1) c ) χC (1) txCc (1) + ( pC c + c ) xC (1) ∣x(' (1)
WC (1) 2 ψ 2
s.t.
ΠC(1)
IC≥0 =>( pC
-wC(1)
- c ) XC (1) - '∖ ≥ 0
2ψ
where MW(1) is the welfare of the pre-innovation membership in stage 1. The optimality conditions for
the co-op’s optimization problem suggest that the co-op will find it optimal to choose its price such that
the investment constraint binds, i.e., the co-op will price its input so that it raises exactly the amount of
capital needed for its innovation activity in stage 2. The best response function of the co-op is then:
4tψpC+wI(1) -t(4cψ+1)
WC (1) = 4 tψ +1
The Nash equilibrium prices and quantities at the pre-innovation stage of the mixed oligopsony
are given by:
(38)
' t-4t2ψ+(4tψ-3)pC+4tψpI
w (1) = 8tψ - 3
-c
(39)
' 8t2ψ2+4tψ-3+8tψ2(pI-pC)
x1 (1) = ( 4 tψ +1)(8 tψ - 3)
4t-12t2ψ+(32t2ψ2-8tψ-3) pC+4tψpI
(40) wCfn =--------------------—--------------------
C(1) (4tψ+1)(8tψ-3)
(41)
` 8tψ[3>tψ -1 + ψ(PC - Pi )]
XC(1) = (4tψ +1)(8tψ - 3)
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