Cournot competition

Cournot competition is an economic model used to describe an industry structure in which companies compete on the amount of output they will produce, which they decide on independently of each other and at the same time. It is named after Antoine Augustin Cournot (1801–1877) who was inspired by observing competition in a spring water duopoly. It has the following features:

• There is more than one firm and all firms produce a homogeneous product, i.e. there is no product differentiation;
• Firms do not cooperate, i.e. there is no collusion;
• Firms have market power, i.e. each firm's output decision affects the good's price;
• The number of firms is fixed;
• Firms compete in quantities rather than prices; and
• The firms are economically rational and act strategically, usually seeking to maximize profit given their competitors' decisions.

An essential assumption of this model is the "not conjecture" that each firm aims to maximize profits, based on the expectation that its own output decision will not have an effect on the decisions of its rivals. Price is a commonly known decreasing function of total output. All firms know $N$, the total number of firms in the market, and take the output of the others as given. The market price is set at a level such that demand equals the total quantity produced by all firms. Each firm takes the quantity set by its competitors as a given, evaluates its residual demand, and then behaves as a monopoly.

History

The state of equilibrium... is therefore stable; i.e. if either of the producers, misled as to his true interest, leaves it temporarily, he will be brought back to it.

— Antoine Augustin Cournot, Recherches sur les Principes Mathématiques de la Théorie des Richesses (1838), translated by Bacon (1897).

Antoine Augustin Cournot (1801-1877) first outlined his theory of competition in his 1838 volume Recherches sur les Principes Mathématiques de la Théorie des Richesses as a way of describing the competition with a market for spring water dominated by two suppliers (a duopoly). The model was one of a number that Cournot set out “explicitly and with mathematical precision” in the volume. Specifically, Cournot constructed profit functions for each firm, and then used partial differentiation to construct a function representing a firm's best response for given (exogenous) output levels of the other firm(s) in the market. He then showed that a stable equilibrium occurs where these functions intersect (i.e. the simultaneous solution of the best response functions of each firm).

The consequence of this is that in equilibrium, each firm's expectations of how other firms will act are shown to be correct; when all is revealed, no firm wants to change its output decision. This idea of stability was later taken up and built upon as a description of Nash equilibria, of which Cournot equilibria are a subset.

The legacy of the Recherches

Cournot’s economic theory was little noticed until Léon Walras credited him as a forerunner. This led to an unsympathetic review of Cournot’s book by Joseph Bertrand which in turn received heavy criticism. Irving Fisher found Cournot’s treatment of oligopoly “brilliant and suggestive, but not free from serious objections”. He arranged for a translation to be made by Nathaniel Bacon in 1897.

Reactions to this aspect of Cournot’s theory have ranged from searing condemnation to half-hearted endorsement. It has received sympathy in recent years as a contribution to game theory rather than economics. James Friedman explains:

In current language and interpretation, Cournot postulated a particular game to represent an oligopolistic market...

The maths in Cournot’s book is elementary and the presentation not difficult to follow. The account below follows Cournot’s words and diagrams closely. (The diagrams were presumably included as an oversized plate in the original edition, and are missing from some modern reprints.)

Cournot's conceptual framework

Cournot’s discussion of oligopoly draws on two theoretical advances made in earlier pages of his book. Both have passed (with some adjustment) into standard economic theory. His demand curves were rediscovered independently in 1870 by Fleeming Jenkin, who paired them with supply curves to give them their modern form. Cournot’s discussion of monopoly was definitive, influencing later writers such as Edward Chamberlin and Joan Robinson in the 1930s revival of interest in imperfect competition.

The 'Law of Demand' or 'of Sales'

Cournot’s curve of ‘demand or sales’

Cournot was wary of psychological notions of demand, defining it simply as the amount sold of a particular good (helped along by the fact that the French word débit meaning ‘sales quantity’ has the same initial letter as demande). He formalised it mathematically as follows:

We will regard the sales quantity or annual demand D, for any commodity, to be a function F (p) of its price.

It follows that his demand curves do some of the work of modern supply curves, since producers who are able to limit the amount sold have an influence on Cournot’s demand curve.

Cournot remarks that the demand curve will usually be a decreasing function of price, and that the total value of the good sold is p F (p) which will generally increase to a maximum and then decline towards 0. The condition for a maximum is that the derivative of p F (p), i.e. F (p) + p F '(p), should be 0 (where F '(p) is the derivative of F (p)).

The theory of monopolistic production

A monopolist will seek to maximise his or her revenue, which is equal to the total value of the good sold, and the monopolist will therefore choose p so that (d/dp)(p F (p)) = 0.

Cournot's duopoly theory

Monopoly and duopoly

Cournot insists that each duopolist seeks independently to maximise profits, and this restriction is essential, since Cournot tells us that if they came to an understanding between each other so as each to obtain the maximum possible revenue, then completely different results would be obtained, indistinguishable from the consumer’s point of view from those entailed by monopoly.

Cournot's price model

Cournot presents a mathematically correct analysis of the equilibrium condition corresponding to a certain logically consistent model of duopolist behaviour. However his model is not stated and is not particularly natural (Shapiro remarked that observed practice constituted a “natural objection to the Cournot quantity model”), and “his words and the mathematics do not quite match”.

His model can be grasped more easily if we slightly embellish it. Suppose that there are two owners of mineral water springs, each able to produce unlimited quantities at zero price. Suppose that instead of selling water to the public they offer it to a middle-man. Each proprietor notifies the middle-man of the quantity he or she intends to produce. The middle-man finds the market clearing price, which is determined by the demand function F and the aggregate supply. He or she sells the water at this price, passing the proceeds back to the proprietors.

The consumer demand D for mineral water at price p is denoted by F (p); the inverse of F  is written f ; and the market-clearing price is given by p = f (D) where D = D₁ + D₂ and D_i is the amount supplied by proprietor i.

Each proprietor is assumed to know the amount being supplied by his or her rival, and to adjust his or her own supply in the light of it to maximise his or her profits. The position of equilibrium is one in which neither proprietor is inclined to adjust the quantity supplied.

It needs mental contortions to imagine the same market behaviour arising without a middle-man.

Interpretative difficulties

A feature of Cournot’s model is that a single price applies to both proprietors. He justified this assumption saying that “... dès lors le prix est nécessairement le même pour l’un et l’autre propriétaire”. Magnan de Bornier expands on this saying that “the obvious conclusion that only a single price can exist at a given moment” follows from “an essential assumption concerning his model, [namely] product homogeneity”. Both authors are wrong; for instance, one proprietor may supply a limited quantity at a low price and the other absorb residual demand at a higher price. The producers are at liberty to stipulate different prices, and it is Cournot’s job to show that in equilibrium they will not do so.

Later on Cournot writes that a proprietor can adjust his supply “en modifiant correctement le prix”. Again this is nonsense: it is impossible for a single price to be simultaneously under the control of two suppliers. If there is a single price, then it must be determined by the market as a consequence of the proprietors’ decisions on matters under their individual control.

Cournot’s account threw his English translator (Nathanial Bacon) so completely off balance that his words were corrected to “properly adjusting his price”. Edgeworth regarded equality of price in Cournot as “a particular condition, not... abstractly necessary in cases of imperfect competition”. Magnan de Bornier says that in Cournot’s theory “each owner will use price as a variable to control quantity“ without saying how one price can govern two quantities. A. J. Nichol claimed that Cournot’s theory makes no sense unless “prices are directly determined by buyers”. Shapiro, perhaps in despair, remarked that “the actual process of price formation in Cournot’s theory is somewhat mysterious”.

Collusion

Cournot’s duopolists are not true profit maximisers. Either supplier could increase his or her profits by cutting out the middle-man and cornering the market by marginally undercutting his or her rival; thus the middle-man can be seen as a mechanism for restricting competition.

Finding the Cournot duopoly equilibrium

Cournot’s reasoning is easy to follow once we have an appropriate model in mind.

The revenues accruing to the two proprietors are p D₁ and p D₂, i.e. f (D₁ + D₂) D₁ and f (D₁ + D₂) D₂. The first proprietor maximises profit by optimising over the parameter D₁ under his control, giving the condition that the partial derivative of his profit wrt. D₁ should be 0; and the mirror-image reasoning applies to his or her rival. We thus get the equations

f (D₁ + D₂) + D₁ f '(D₁ + D₂) = 0   and   f (D₁ + D₂) + D₂ f '(D₁ + D₂) = 0.

The equlibirum position is found by solving these two equations simultaneously.

Each equation defines a relation between D₁ and D₂ which may be drawn on a graph. If the first proprietor was providing quantity x_l then the second proprietor would adopt quantity y_l from the red curve to maximise his or her revenue. But then, by similar reasoning, the first proprietor will adjust his supply to x_ll to give him or her the maximum return as shown by the blue curve when D₂ is equal to y_l. This will lead to the second proprietor adapting to the supply value y_ll and so forth, until equilibrium is reached at the point of intersection i whose coordinates are (x,y ).

Cournot remarks that the equilibrium is stable, but that if the red and blue curves were interchanged then this would cease to be true. He adds that it is easy to see that the corresponding diagram would be inadmissible since, for instance, it is necessarily the case that m₁ > m₂. To verify this, notice that when D₁ is 0 the two equations reduce to

f (D₂) = 0   and   f (D₂) + D₂ f '(D₂) = 0.

The first of these corresponds to a supply D₂ so large that the price falls to zero, while the second states that the derivative of D₂ f (D₂) wrt. D₂ is 0; but D₂ f (D₂) is the monetary value of an aggregate supply D₂, and the turning point of this supply value is a maximum. Evidently this maximum is reached before the price falls to zero, so the root m₁ of the first equation is necessarily greater than the root m₂ of the second.

Comparison with monopoly

If we add and subtract Cournot’s two simultaneous equations, they become

D₁ = D₂   and   2f (D) + Df '(D) = 0

where D = D₁ + D₂ and p  and D are functionally related via f  in one direction and F  in the other. The first equation tells us that each proprietor supplies half the total, and the second tells us that the effect of duopoly is to determine total output in a certain way. If we re-express the second equation in terms of p, it tells us that F (p) + 2p F '(p) = 0, which can be compared with the equation F (p) + p F '(p) = 0 obtained earlier for monopoly.

If we plot another variable u against p, then we may draw a curve of the function u = –F (p) / F '(p). The monopoly price is the p for which this curve intersects the line u = p , while the duopoly price is given by the intersection of the curve with the steeper line u = 2p. Regardless of the shape of the curve, the intersection with u = 2p occurs to the left – i.e. at a lower price – than the intersection with u = p. Hence prices are lower under duopoly than under monopoly, and quantities sold accordingly higher.

Extension to oligopoly

When there are n proprietors the price equation becomes F (p) + n p F '(p) = 0; the price can be read from the diagram from the intersection of u = n p with the curve. Hence the price diminishes indefinitely as the number of proprietors increases. With an infinite number of proprietors the price becomes zero; or more generally, if we allow for costs of production, the price becomes the marginal cost.

Bertrand's critique

The French mathematician Joseph Bertrand, when reviewing Walras’s ‘Théorie Mathématique de la Richesse Sociale’, was drawn to Cournot’s book by Walras’s high praise of it. He was unappreciative. His summary of Cournot’s theory of duopoly has remained influential:

Cournot assumes that one of the proprietors will reduce his price to attract buyers to him, and that the other will in turn reduce his price even more to attract buyers back to him. They will only stop undercutting each other in this way, when either proprietor, even if the other abandoned the struggle, has nothing more to gain from reducing his price. One major objection to this is that there is no solution under this assumption, in that there is no limit to the downward movement... If Cournot’s formulation conceals this obvious result, it is because he most inadvertently introduces as D and D' the two proprietors’ respective outputs, and by considering them as independent variables, he assumes that should either proprietor change his output then the other proprietor’s output could remain constant. It quite obviously could not.

Pareto was similarly unimpressed by Bertrand’s reasoning, concluding from it that Bertrand ‘wrote his article without consulting the books he criticised’.

Irving Fisher outlined a model of duopoly similar to the one Bertrand had accused Cournot of analysing incorrectly:

A more natural hypothesis, and one often tacitly adopted, is that each [producer] assumes his rival’s price will remain fixed, while his own price is adjusted. Under this hypothesis each would undersell the other as long as any profit remained, so that the final result would be identical with the result of unlimited competition.

Fisher seemed to regard Bertrand as having been the first to present this model, and it has since entered the literature as Bertrand competition.

See also

• Aggregative game
• Bertrand competition
• Bertrand–Edgeworth model
• Conjectural variation
• Game theory
• Nash equilibrium
• Stackelberg competition
• Tacit collusion

References

Further reading

• Holt, Charles. Games and Strategic Behavior (PDF version), PDF
• Tirole, Jean. The Theory of Industrial Organization, MIT Press, 1988.
• Oligoply Theory made Simple, Chapter 6 of Surfing Economics by Huw Dixon.