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  Cournot Duopoly, Cournot-Nash Equilibrium and Generalization to Oligopoly

Cournot Duopoly (Augustine A. Cournot, 1838)

1) Assumptions:

* λi = dQ-i/dqi =  0
* Each firm determines products that maximize their profits taking the products of rivals’ constant.
* Market demand is P = a − Q ( a > 0) and Q = q1 + q2 (Q is the total quantity of spring water sold in the market per unit of time).
* For simplicity, MC = AC = c.

2) Model:

* Firm 2 assumes that firm 1 is producing q1 and it produces q2 as the best response to q1.
* So the economic profit for firm 2 is as follows,

π2 = (P − c)q2 = (a − c − q1 − q2)q2 .

(Δπ2/Δq2) = a-c-q1-2q2= 0   (i). From (i), we get the best response of firm 2 to q1 of firm 1,

q2 = ((a- c- q1)/2)       (ii): Reaction Curve of firm 2

* In the same way we get reaction curve of firm 1.

q1= ((a- c- q2)/2)     (iii): Reaction Curve of firm 1      

1850_cournot duopoly.jpg

Cournot-Nash Equilibrium:

1654_cournot nash equilibrium.jpg

* Suppose firm 1 and firm 2 decide to make a Cartel and act as collective monopolist. subsequently what are the profits of two firms?

As well as what is the monopolistic product? (MR = MC)

Given demand curve is P = a − Q . TR = PQ = (a − Q)Q. AndMR = a − 2Q, and MC = c

∴ MR = a − 2Q = c = MC. So Q m (a- c) /2

If these two firms produce the same quantities, then q1=q2= (a-c)/4.

And Pm = (a+c)/2, πm= (a-c)2/8 , which is bigger than profit at point C (πc = (a-c)2/9).

* If the two firms make a Cartel and determine their quantities cooperatively, then their economic profits are higher than those of Cournot-Nash equilibrium. Therefore, there is a good reason and motive to cartelize or to collude.

* However why do we call C as an equilibrium point not M? Or, why is not M is an equilibrium? For the reason that there is another motive to cheat by increasing quantity to earn more profit at point M.

* If firm 1 produces q 1 = (a-c)/ 4 and keeps the Cartel but firm 2 does not, then the profit function of firm 2 is as follows:

167_cartel formula.jpg

This maximizes economic profit of firm 2. Hence, firm 2 breaks the cartel and produces more. Market price of firm 2’s products is P 2 = (3a + 5c)/8 and

2213_market price of firms product].jpg

* Firm 2 has a motive to cheat firm 1. Vice versa. Therefore one-shot cartel is unstable.

* Cartel is simple to collapse. But, if the cartel is made among few firms for relatively long periods, then it can be sustained longer with efficient and severe punishment or penalty on the violators.

Generalization to Oligopoly:

1617_generalization to ologopoly.jpg

If n = 2  we get the similar results in Cournot Duopoly.

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