Solving the best response functions simultaneously, we get the Nash equilibrium levels of innovation for
each firm as:
* 9tψ-2+3ψ(pi-pj)
(13) ri =--------7----1—ς-----
3ψ(9tψ- 2)
Substituting the equilibrium levels of innovation in the expressions for innovation costs and post-
innovation profits, we get the net profits of each firm in stages 2 and 3 as:
(14)
Π*
Πi(2,3)
(9tψ-1)[2-9tψ + 3ψ(Pj -Pi)]
18ψ(2-9tψ)2
Price Competition at the Pre-Innovation Stage (1st Stage of the Game)
In this stage, the two firms seek to determine the input prices that maximize their profits. Since the firms’
payoffs in stages 2 and 3 are not dependent on pre-innovation prices or quantities, the objective of the two
IOFs in stage 1 is to maximize their pre-innovation profits only, i.e.,
(15)
maxΠi(1)
wi(1)
=(pi-c-wi(1))xi(1)
The Nash equilibrium prices, quantities and profits at the pre-innovation stage are then:
(16)
w = 3 ( 2 Pi + Pj)
-c-t
(17)
*
xi(1) =
3t + Pi - Pj
61
(18)
Π*i(1) =
(3t + Pi - Pj )2
181
Table 1 summarizes the subgame perfect equilibrium in the pure oligopsony. It can be seen that
the equilibrium is asymmetric with the differences in input prices, quantities, profits and innovation effort
being determined by the relative quality of the final products produced by the two IOFs (reflected in the
relative prices of these products). In particular, the firm with the higher quality product will offer a higher
price to farmers, will enjoy higher market shares and profits in the 1st and 3rd stages of the game, and will
exert higher innovation effort than its low quality rival.
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