3. Benchmark Case: Innovation and Pricing Decisions in a Pure Oligopsony
Pr ce Compet t on at the Post-Innovat on Stage (3rd Stage of the Game)
In the post-innovation stage of the pure duopsony, the two IOFs seek to determine the input prices that
maximize their profits holding Nash conjectures (i.e., assuming that their decisions will not affect the
behavior of their rival).1 Specifically, the problem of the two IOFs at the 3rd stage of the game can be
expressed as:
(5)
maxΠ (3)
w (3)
=(p -c -w (3))x (3)
where ∈{C, I} , p is the price of Firm ’s final product, and c is the post-innovation marginal
processing cost of Firm . All other variables are as previously defined.
Solving the problem of the two IOFs gives their best response functions as:
wo) = 2 ( w1 (3)+ Pi-t - cû
where j∈{C, I} and ≠j. Solving these best response functions simultaneously and substituting
wC* (3) and wI*(3) into equations (3) and (4) gives the Nash equilibrium prices and quantities as:
(6)
(7)
w⅛) = 3 (2 Pi + Pj
-2ci-cj-3t)
* 31 - ci+ cj + P1 - Pj
x3 _ 61
The equilibrium profits of each IOF are then given by:
(3t-c+c+P-P)2
(8) ∏*(3) =■(-------i j )
i(3) 18t
and are a function of the degree of producer heterogeneity, the prices of the products produced by the two
IOFs, and their post-innovation processing costs. Ceteris Paribus, the lower the post-innovation
processing cost of a firm, the greater its profits at the 3rd stage of the game.
1 The assumption of Nash conjectures is maintained throughout the analysis.