Stackelberg’s model of non-collusive oligopoly –
n-player generalization –
Market Inverse Demand : 𝑝(𝑦) = 𝑎 − 𝑏𝑦 where 𝑎 > 𝑐 > 0
𝑛
∑ 𝑦𝑖 = 𝑦
𝑖=1
𝑐(𝑦) = 𝑐𝑦𝑖 ∀𝑖 = 1 to 𝑛
Profit function for the (𝑛 − 1)𝑡ℎ firm 𝑛
𝑚𝑎𝑥. 𝜋𝑛−1 = {𝑎 − 𝑏 (∑ 𝑦𝑖 )} 𝑦𝑛−1 − 𝑐𝑦𝑛−1 subject to 𝑎 − 𝑏𝑦−𝑛 − 𝑐 = 2𝑏𝑦𝑛
𝑖=1
The Lagrangian can be written as,
𝑛
𝐿 = [{𝑎 − 𝑏 (∑ 𝑦𝑖 )} 𝑦𝑛−1 − 𝑐𝑦𝑛−1 + 𝜆{2𝑏𝑦𝑛 − 𝑎 + 𝑏𝑦−𝑛 + 𝑐}]
𝑖=1
According to FOC, we get the following equations,
𝑑𝐿
= 𝑎 − 2𝑏𝑦𝑛−1 − 𝑏𝑦−(𝑛 −1) − 𝑐 + 𝑏𝜆 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (𝑖)
𝑑𝑦𝑛−1
𝑛
= 𝑎 − 𝑏𝑦𝑛−1 − 𝑐 − 𝑏 (∑ 𝑦𝑖 ) + 𝑏𝜆 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . (𝑖. 𝑖)
𝑖=1
𝑑𝐿
= −𝑏𝑦𝑛−1 + 2𝑏𝜆 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (𝑖𝑖)
𝑑𝑦𝑛
We can re-write (ii) as 𝑦𝑛−1 = 2𝜆. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (𝑖𝑖𝑖)
𝑑𝐿
= 2𝑏𝑦𝑛 − 𝑎 + 𝑏𝑦−𝑛 + 𝑐 = 0
𝑑𝜆
𝑛
or, 𝑏𝑦𝑛 − 𝑎 + 𝑐 + 𝑏 (∑ 𝑦𝑖 ) = 0
𝑖=1
𝑛
or, ∑ 𝑦𝑖 =
𝑖=1
𝑎 − 𝑐 − 𝑏𝑦𝑛
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (𝑖𝑣)
𝑏
Putting (iii) & (iv) in (i.i), we get,
𝑎 − 𝑏𝑦𝑛−1 − 𝑐 − (−𝑐 + 𝑎 − 𝑏𝑦𝑛 ) + 𝑏𝜆 = 0
or, 𝑦𝑛 − 𝑦𝑛−1 = −𝜆
or, 𝑦𝑛−1 = 2𝑦𝑛 [∵ 𝑦𝑛−1 = 2𝜆, from (iii)]
Now, we can say that,
𝑦𝑛−1 = 2𝑦𝑛
𝑦𝑛−2 = 2𝑦𝑛−1 = 22 𝑦𝑛
𝑦𝑛−3 = 23 𝑦𝑛
⋮
⋮
⋮
𝑦3 = 2𝑛−3 𝑦𝑛
𝑦2 = 2𝑛−2 𝑦𝑛
𝑦1 = 2𝑛−1 𝑦𝑛
𝑛
∑ 𝑦𝑖 = [1 + 2 + 22 + 23 +. . . +2𝑛−1 ]𝑦𝑛
𝑖=1
= (2𝑛 − 1)𝑦𝑛 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (𝑣)
Putting (𝑣) in (iv), we get,
(2𝑛 − 1)𝑦𝑛 =
∴ 𝑦𝑛𝑠 =
𝑎 − 𝑐 − 𝑏𝑦𝑛
𝑏
𝑎−𝑐
2𝑛 𝑏
𝑛
∴ 𝑦 𝑠 = ∑ 𝑦𝑖𝑠 = (
𝑖=1
𝑎−𝑐 1 1
1
1
𝑎−𝑐
1
) [ + 2 + 3 . . . + 𝑛] = (
) [1 − 𝑛 ] . . . . (𝑣𝑖)
𝑏
2 2
2
2
𝑏
2
∴ 𝑝(𝑦 𝑠 ) = 𝑎 − 𝑏(𝑦𝑠 ) =
𝑎 + (2𝑛 − 1)𝑐
= 𝑝 𝑠 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (𝑣𝑖𝑖)
2𝑛
(𝑎 − 𝑐)2 2𝑛 − 1
∴ 𝜋𝑠 = 𝑝 𝑦 − 𝑐𝑦 = {
} [ 2𝑛 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (𝑣𝑖𝑖𝑖)
𝑏
2
𝑠 𝑠
𝑠