Perhaps the most famous theory developed by Cournot relates to the introduction of an additional seller of mineral water. In a profoundly original theoretical conception, Cournot laid the groundwork for many other ideas of importance to economics, such as imperfect competition) and the theory of games. Although Cournot's theory of duopoly (two sellers) was later altered and refined (notably by the Englishman Francis Y. Edgeworth and the French mathematician Joseph Bertrand), nothing can mask Cournot's brilliant and original insight.
Cournot considered two sellers, A and B, who both know the total (aggregate) demand curve for their perfectly homogeneous product, mineral water. Otherwise, they are completely uninformed about each other's policies, to the extent that A thinks that B will keep his quantity constant no matter what A does, and B thinks the same thing about A's quantity. Further, both sellers continue to make this assumption no matter how much experience they have to the contrary. In the language of duopoly, this assumption is called a zero output conjectural variation i.e., a conjecture that B will have no output reaction to A's actions. Cournot further assumed that either A or B could supply all the output of mineral water and, moreover, that mineral-water production is costless. He analyzed the problem of output and price determination both mathematically and graphically in the Researches, but our discussion will be in graphical terms.
In order to analyze the duopoly problem, Cournot developed a new tool of graphical analysis, the reaction curve, one of which is reproduced in Figure 2.
Figure 2 depicts a concave reaction function AA, which reveals A's choice of outputs with respect to B's choice of outputs. Specifically, it shows the outputs firm A will select in order to maximize profits, given B's selection of outputs. For example, if B selects output Ob00, A—in order to maximize profits—will want to charge a certain price for an output Oa0. If, on the other hand, B produces quantity Ob1 A will be led by the motive of profit maximization to produce a lower quantity Oa1v and so on for all other quantities B might produce. Whatever quantity B chooses, moreover, A thinks that it will be permanent, and so A acts to maximize his or her profits.
What quantity will A and B end up producing? Clearly, the problem cannot be solved without adding B's reaction function indicating the kind of responses B will make to A's output. The two functions are combined in Figure 3, where B's reaction function is defined in the same manner as A's was above.
Suppose B decides to produce some output—say, Ob0—on the assumption that A will keep output at level Oa0. B would then be maximizing his or her profits at output Ob0. On the assumption that B would hold output at level Ob0, A would maximize profits by producing output Oa1. Such a move would cause B to reassess the situation and to increase his output to Ob1, which maximizes his or her profits on the assumption that A will keep output at Oav However, the assumption proves unfounded (though B, or A, is assumed never to catch on), and the process of output variation to maximize profits goes on as traced by the arrows in Figure 3.
Point E (Figure 3) represents an equilibrium solution for firms A and B, i.e., one to which they will always return if moved away. At point E the duopolists both share profits (Cournot expressed this amount mathematically) and charge a common price that is lower than the price that would obtain under simple monopoly (a fact that Cournot himself noted) but higher than the one that would be charged under competition, with many sellers. Cournot was quick to point out that collusion between the two competitors would result in production of monopoly output and a two-way split of monopoly profits. But Cournot expressed duopoly output precisely: it would be two-thirds the output produced if the market were competitive. In fact, his gen eral expression for output was that it would be n/n + 1 times competitive output. Thus if there were five sellers, quantity sold would be five-sixths of competitive output. If there were 2,000 sellers, output would clearly approach the competitive amount. In this manner Cournot related his duopoly theory to the competitive model.