Cournot's model of non-collusive duopoly

Cournot's Model


Cournot's model is a model of non-collusive duopoly which can be easily extended to cover cases of oligopoly with more than two firms. August Cournot, an early nineteenth century, French economist, was probably the first economist who analysed the problem of equilibrium under duopoly. This model is based on the following explicit or implicit assumptions :


1. There are only two producers producing a homogeneous commodity. Incidently, the commodity assumed is mineral water whose production cost per unit is constant, though zero, it being a free gift of nature.


2. The two producers are assumed to have identical constant costs which, in his example if mineral water, are zero.


3. Both the Producers are assumed to be know with certainly the total demand curve for the commodity.


4. The demand curve is assumed to be negatively sloping straight line.


5. Absence of collusion and therefore presence of free competition between the two producers is assumed.


6. It is assumed that the objective of each producer is to maximize his individual profit. And, the most crucial assumption relates to the expected behaviour reaction of each producer.


7. Each producer assumes that whatever he may do to his own output, the rival producer will stick to the output that he is currently producing.


On the basis of the above assumption Cournot's analysis leads to the conclusion that under non-collusive duopoly the equilibrium output of the duopolistic “group” would be two thirds of the competitive equilibrium output which would be equally shared by the two producers that is, the equilibrium

output of each one of them would be one-third of the competitive output. How this result follows logically from the above assumptions, is explained below : Let us name of two producers as A and B, and let the total demand curve for their homogeneous commodity (mineral water) be represented by the straight line QR in Fig. 2 below. Since the cost per unit (average cost) is assumed to be

constant and zero (mineral water being a free gift of nature), the marginal cost will equal the average cost and will also be constant at zero. This means that the average cost-cum-marginal cost curve will coincide with the horizontal axis OX. Since both producers are assumed to have identical costs, the horizontal axis OX represents the average cost-cum-marginal cost curve of each one of them. Let us assume that producer A enters the market first. He is then, the lone producer and seller of the commodity, the total demand for which is represented by the straight the QR in our Fig. 2 above. A will behave like a monopolist and produce OM quantity at which his marginal cost equals his marginal revenue. When he is the sole producer and seller, the demand curve facing him is QR which is also his average revenue curve. QM is its companion marginal revenue curve which meets the marginal cost curve (i.e. the horizontal axis) at point M, i.e. condition of profit maximization. Hence he will produce OM output which is one-half of OR, and charge OP price.

Now let B enter the market. He observes that one-half of market is already occupied by A. He assumes that whatever be his own output A will stick to the output OM which he is already producing. So he considers that only the remaining one-half of the market represented by the portion MR of the demand curve QR is open to him. He will behave like, a monopolist in this part of the market, producing one-half of MR, that is, MM, output (1 =1/4 of competitive output OR) which maximises his profit. Now the total output is OM1..... and consequently the price is brought down to OP1.This affects the profits of A adversely. So he makes a counter-move on the assumption that B will stick to MM1 (= 1/4 OR) output. So he believes that now only 3/4 of the market is open to him where in he can monopolistically, producing 3/8 of the competitive output OR.

In other words he reduces his output by 1/8 of OR (as 3/8 = 1/2-1/3). This will be met with a counter-move by B who, finding that now only 5/8 of the market is open to him, will produce 1/2 ×5/8= 5/16 of the competitive output in order to maximise his profit. In other words he will increase his output by 1/16 (as 5/14-1/4=1/16) of OR. These moves and counter-moves will go on infinitely till the equilibrium is attained.

The equilibrium will come about through an infinite series of moves and counter-moves in which A will be reducing his output and B will be increasing his output in a sequences like the above. When the equilibrium is attained and further moves and counter-moves come to a stop, the total output will be
(1-1/2+1/4 -1/8+1/16-1/32+1/44.......upto infinity) OR. The series within the brackets are
geometrical series with the ratio which will sum up to 2/3.This means that the equilibrium total output will be 1/3. We have been describing OR as the
equilibrium competitive output because under free and perfect competition, long-period equilibrium takes place at that output at which price equals
 marginal cost as well as average cost. In this model as depicted in Fig. 2, above this condition is satisfied at OR output because in this example costs are zero. Thus the equilibrium total output in this model is 2/3 of the equilibrium competitive output.

Thus we find that the Cournot's model of non-collusive duopoly, the equilibrium total output is 2/3 of the equilibrium competitive output which is equally shared by the two producers. This output is greater than the equilibrium monopoly output and hence the price under duopoly will be, according to this model less than the monopoly price. As we remarked earlier, Cournot's model of duopoly can be extended to cover oligopoly cases with any number of firms. The general formula for this is that the equilibrium total output under oligopoly is n/n+1 of the competitive,output which is equally shared by all the firms, n in the above formula represents the number of firms. If there are three producers, for example, all other assumptions remaining the same, the equilibrium total output according to Cournot's model, would be 3/4th of the competitive output which will be equally shared by all the three producers, each producing 1/4th of the competitive output.