Cournot Model (Duopoly)

Cournot Model (Duopoly)

The Cournot model (by Antoine Cournot, 1838) is in terms of duopoly (two sellers) but it can be easily extended to an oligopolistic situation. This model analyses the process of equilibrium in a duopoly situation when each duopolist assumes that his rival will not react when he changes his output to maximise profits. The assumptions of this model are:
There are two sellers in the market.
The products sold by these two sellers are homogeneous.
The market, or total demand curve, is known and it is a straight line.
Each duopolist assumes that his rival’s output will remain constant when he changes his output. Thus, each duopolist assumes his rival will not react to his action. This is, for each duopolist the conjectural variation or seller interdependence, as given by dQ1/dQ2 or dQ2/dQ1, is assumed to be zero. (Q1 and Q2 are the outputs of two sellers).
Each duopolist produces output of which the profits are at the maximum.
The cost of production is zero for both the sellers.
For example, two natural springs of mineral water with healing qualities, are each owned by one seller. The average and marginal costs for each seller are zero and these curves coincide with the X-axis.
Hence, demand or output at zero price shows the competitive output.
 OD is competitive output.
Let the two duopolists be denoted by X and Y. Let Qx and Qy be their respective outputs.
Suppose seller X enters the market first, followed by seller Y.
We analyse the behaviour of X and Y in stages.
In stage I, seller X acts as a monopolist. He faces demand curve CD so that CA is his marginal revenue curve which must be situated halfway between the Y-axis and demand curve. CA cuts OD at A, such that OA = AD = 1/2 OD. At output OA, marginal revenue = marginal cost = zero and profits are at their maximum. Seller X charges price P1 and makes profit = OARP1.
Thus at stage I, we find Qx = 1/2 OD.
Now seller Y enters with the assumption that X will keep his output constant at 1/2 OD. In other words, Y considers his demand curve to be RD which shows the leftover demand after X has supplied OA output.
Hence at Stage I, seller Y finds his demand curve to be RD with RB as his marginal revenue curve. RB cuts the X-axis at B. For seller Y, marginal revenue = marginal cost = zero at output AB. Thus profit maximising output of Y at Stage I, is AB = 1/2 AD = 1/2 (1/2 OD) = 1/4 OD.
Thus in Stage I, Qx = OA = 1/2 OD
and Qy = AB = 1/2 AD = 1/2 (1/2 OD) = 1/4 OD
Seller Y charges price BT = OP2 and makes a maximum profit = ABTK.
However, since X and Y are selling homogenous products, the price will decrease from OP to OP2 for both of them. Profits of X will thus decline to OAKP2.
Assuming seller Y will keep his output Qy constant at 1/4 OD, seller X will have to reduce his output so as to raise the price and his profit.
In stage II, Seller X will produce profit maximising output on the basis of the demand leftover after assuming Qy to be = 1/4 OD. Therefore, in Stage II, X will produce output Qx = 1/2 (OD – AB) = (OD – BD) and BD is 1/2 AD = 1/4 OD.
At stage II, Y decides his profit maximising output assuming Qx will remain constant at (1/2 – 1/8) OD. Hence, Qy at Stage II will be = 1/2 [OD – (1/2 – 1/8) OD].
That is, Qy = 1/2 (OD – 1/2 OD + 1/8 OD)
= (1/2 – 1/4 + 1/16) OD = (1/4 + 1/6) OD.
We can carry on this reasoning further, to Stage III, Stage IV, etc., to find Qx and Qy at each stage.
In short, we find that at each stage, seller X will decrease his output in such a way that it will be equal to one-half of OD minus the output of Y in the previous stage (which is initially zero). On the other hand, Y will increase his output Qy at each stage so that it will be equal to one-half of the difference between OD and Qx at the same stage.
The stagewise changes in Qx and Qy are summarised in Table 12.1.
We find that at each stage, Qx declines by smaller and smaller quantities. In contrast, Qy increases by smaller and smaller amounts at each stage. Both Qx and Qy will, therefore, arrive at some finite values which will give the equilibrium values of Qx and Qy.