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 who was inspired by observing competition in a spring water duopoly. It has the following features:
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, the total number of firms in the market, and take the output of the others as given. Each firm has a cost function. Normally the cost functions are treated as common knowledge. The cost functions may be the same or different among firms. 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

first outlined his theory of competition in his 1838 volume Recherches sur les Principes Mathematiques de la Theorie des Richesses as a way of describing the competition with a market for spring water dominated by two suppliers. 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 output levels of the other firm in the market. He then showed that a stable equilibrium occurs where these functions intersect.
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.

Graphically finding the Cournot duopoly equilibrium

This section presents an analysis of the model with 2 firms and constant marginal cost.
Equilibrium prices will be:
This implies that firm 1's profit is given by
In very general terms, let the price function for the industry be and firm have the cost structure. To calculate the Nash equilibrium, the best response functions of the firms must first be calculated.
The profit of firm i is revenue minus cost. Revenue is the product of price and quantity and cost is given by the firm's cost function, so profit is :
. The best response is to find the value of that maximises given, with, i.e. given some output of the opponent firm, the output that maximises profit is found. Hence, the maximum of with respect to is to be found. First take the derivative of with respect to :
Setting this to zero for maximization:
The values of that satisfy this equation are the best responses. The Nash equilibria are where both and are best responses given those values of and.

An example

Suppose the industry has the following price structure: The profit of firm is:
The maximization problem resolves to :
Without loss of generality, consider firm 1's problem:
By symmetry:
These are the firms' best response functions. For any value of, firm 1 responds best with any value of that satisfies the above. In Nash equilibria, both firms will be playing best responses so solving the above equations simultaneously. Substituting for in firm 1's best response:
The symmetric Nash equilibrium is at. Making suitable assumptions for the partial derivatives, the equilibrium quantities can be substituted in the assumed industry price structure to obtain the equilibrium market price.

Cournot competition with many firms and the Cournot theorem

For an arbitrary number of firms,, the quantities and price can be derived in a manner analogous to that given above. With linear demand and identical, constant marginal cost the equilibrium values are as follows:
Market demand;
Cost function;, for all i
which is each individual firm's output
which is total industry output
which is the market clearing price, and
The Cournot Theorem then states that, in absence of fixed costs of production, as the number of firms in the market, N, goes to infinity, market output, Nq, goes to the competitive level and the price converges to marginal cost.
Hence with many firms a Cournot market approximates a perfectly competitive market. This result can be generalized to the case of firms with different cost structures and non-linear demand.
When the market is characterized by fixed costs of production, however, we can endogenize the number of competitors imagining that firms enter in the market until their profits are zero. In our linear example with firms, when fixed costs for each firm are, we have the endogenous number of firms:
and a production for each firm equal to:
This equilibrium is usually known as Cournot equilibrium with endogenous entry, or Marshall equilibrium.

Implications

Although both models have similar assumptions, they have very different implications:
However, as the number of firms increases towards infinity, the Cournot model gives the same result as in Bertrand model: The market price is pushed to marginal cost level.