↑ arvau

Wikipedia / Cournot competition

Cournot competition is an economic model describing an oligopolistic market where firms simultaneously compete by choosing the quantity of goods to produce and sell in the market. It is a static game, whereby a Nash equilibrium is found where no firms desire to unilaterally change their output level when the other firms produce the output levels assigned to them in the (purported) equilibrium.

This model is named after Antoine Cournot, who was inspired to develop a theoretical approach to this setting after observing firm behaviour in a spring water duopoly. The model has the following features:

An essential assumption of this model is the "Cournot conjecture" that each firm aims to maximize profits, 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

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. This is a simultaneous solution such that the corresponding choice in the best response function 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.

Cournot's discussion of oligopoly draws on two theoretical advances made earlier in his writing. Both, accepted into microeconomic theory, are particularly important within subfield of industrial organization where Cournot's assumptions can be relaxed to study various market structures and industries. For example, the Stackelberg Competition model is a subset of Cournot's duopoly game. Cournot's discussion of monopoly influenced later writers such as Edward Chamberlin and Joan Robinson during the 1930s revival of interest in imperfect competition.

Conceptual framework

Cournot presents a mathematical analysis of the equilibrium condition in a model of duopolist behaviour. This model is logical. Cournot formalized the notion of aggregated demand, such that the demand for a commodity is a function of its price. In this model, the demand curve is normally a decreasing function; the second derivative can be assumed to be negative. If so, then a sufficient condition for a maximum.

Under the theory, duopolist seeks independently to maximize profits. This restriction is necessary, to ensure firms do not coordinate. Without coordination, each firm attempts to optimize their individual profits by maximizing revenue.

Suppose that there are two owners of mineral water springs and the aggregate supply. He or she sells the water at this price, passing the proceeds back to the proprietors.

.

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

Finding the equilibrium

In a Cournot game, the equilibrium that maximize their own profits. In equilibrium, there is no incentive for either firm to change away from this equilibrium output quantity. Note that there exists an subgame perfect equilibrium is sufficiently large.

Example 1

Cournot's model of competition is typically presented for the case of a duopoly. Assume each firm faces the same marginal cost is the marginal cost. This assumption tells us that both firms face the same cost-per-unit produced. Therefore, as each firm's profit is equal to its revenues minus costs, where revenue equals the number of units produced multiplied by the market price, we can denote the profit functions for firm 1 and firm 2 as follows:

.

we can write each firm's profit function as:

As firms are assumed to be profit-maximizers, the first-order conditions (F.O.C.s) for each firm are:

which is why we set the above equations equal to zero.

for firms 1 and 2 respectively. So, we obtain:

, and output quantity of rival firms. The functions can be thought of as describing a firm's "Best Response" to the other firm's level of output.

We can now find a Cournot-Nash Equilibrium.

.

Example 2

should be 0, and the mirror-image reasoning applies to their rival. We thus get the equations:

  1. and

  2. .

The equlibirum position is found by solving these two equations simultaneously. This is most easily done by adding and subtracting them, turning them into:

  1. and

  2. .

.

.

is 0, the two equations reduce to:

  1. and

  2. .

of the second equation.

Comparison with monopoly

obtained earlier for monopoly.

. Hence, prices are lower under duopoly than under monopoly, and quantities sold are accordingly higher.

Extension to oligopoly

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.

Reception in theoretical literature

One part of Cournot's original duopoly game was that each firm owner can adjust his supply "en modifiant correctement le prix", or that the price can be adjusted in the same time period by both firms. Francis Edgeworth regarded equality of price in Cournot as "a particular condition, not... abstractly necessary in cases of imperfect competition." 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".

Cournot's economic theory was little noticed until Léon Walras credited him as a forerunner. The French mathematician Joseph Bertrand, when reviewing Walras's Théorie Mathématique de la Richesse Sociale, was drawn to Cournot's book given the acceptance of the theory by Walras himself. Bertrand did criticize Cournot's theory, summarized as:[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.In response to the criticism, Vilfredo Pareto, another influential economist, was unimpressed by Bertrand's critique. Later, Irving Fisher outlined a model of duopoly that was in line with Bertrand's critique, and not Cournot's original model, stating: 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 regarded Bertrand as the first to identify this model, and it has since been referred to as Bertrand competition. Today, this duopoly theory is a key model in game theory, and has application in modern economics for understanding oligopolies.

See also

Notes

References

Sources

Further reading