# Cournot duopoly model (1838)

Named after French economist Antoine Augustin Cournot (1801-1877), Cournot duopoly model shows two firms that react to one another’s output changes until they eventually reach a position from which neither would wish to depart.

Both firms eventually expand to such a degree that they have equal shares in the market and secure only normal profits.

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, and choose quantities simultaneously;
• 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 , 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.

### Calculating the equilibrium

In very general terms, let the price function for the (duopoly) 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 (as described above): . 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 such that and for ease of computation) is: The maximization problem resolves to (from the general case): 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 (for example, assuming each firm’s cost is a linear function of quantity and thus using the slope of that function in the calculation), 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 , which is each individual firm’s profit.

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 (under appropriate restrictions) 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

• Output is greater with Cournot duopoly than monopoly, but lower than perfect competition.
• Price is lower with Cournot duopoly than monopoly, but not as low as with perfect competition.
• According to this model the firms have an incentive to form a cartel, effectively turning the Cournot model into a Monopoly. Cartels are usually illegal, so firms might instead tacitly collude using self-imposing strategies to reduce output which, ceteris paribus will raise the price and thus increase profits for all firms involved.

### Bertrand versus Cournot

Although both models have similar assumptions, they have very different implications:

• Since the Bertrand model assumes that firms compete on price and not output quantity, it predicts that a duopoly is enough to push prices down to marginal cost level, meaning that a duopoly will result in perfect competition.
• Neither model is necessarily “better.” The accuracy of the predictions of each model will vary from industry to industry, depending on the closeness of each model to the industry situation.
• If capacity and output can be easily changed, Bertrand is a better model of duopoly competition. If output and capacity are difficult to adjust, then Cournot is generally a better model.
• Under some conditions the Cournot model can be recast as a two-stage model, where in the first stage firms choose capacities, and in the second they compete in Bertrand fashion.

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.

Also see: duopoly theory, bertrand duopoly model, bilateral monopoly

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