Cournot competition is an economics model used to describe industry structure. It so called after Antoine Augustin Cournot (1801-1877) after he observed competition in a spring water duopoly. It has the following features: Economics (deriving from the Greek words οίκος [oikos], house, and νέμω [nemo], rules hence household management) is the social science that studies the allocation of scarce resources to satisfy unlimited wants. ... Antoine Augustin Cournot (28 August 1801‑ 31 March 1877) was a French philosopher and mathematician. ...

• There are two firms producing homogeneous products;
• Firms do not cooperate.
• Firms have market power;
• There are barriers to entry;
• Firms compete in quantities, and choose quantities simultaneously;
• There is strategic behaviour by the firms;

Price is a commonly known decreasing function of total output. All firms know N and take the output of the others as given. Each firm has a cost function ci(qi) (cost per unit multiply quantity). Normally the cost functions are treated as common knowledge. The cost functions are normally the same for all firms. The market price is set at a level such that demand equals the total quantity produced by both firms. Homogeneous is an adjective that has several meanings. ... In economics, market power (sometimes called monopoly power) is a market failure which occurs when one or more of the participants has the ability to influence the price or other outcomes in some general or specialized market. ... Barriers to entry is a term used in economics and especially the theory of competition to refer to obstacles placed in the path of a firm who wants to enter a given market. ... The supply and demand model describes how prices vary as a result of a balance between product availability at each price (supply) and the desires of those with purchasing power at each price (demand). ...

Graphically finding the Cournot duopoly equilibrium

p1 = firm 1 price, p2 = firm 2 price

q1 = firm 1 quantity, q2 = firm 2 quantity

c = marginal cost (assumed to be constant)

Equilibrium prices will be: In economics and finance, marginal cost is the cost of increasing the quantity produced (or purchased) by one unit. ... For the 2002 science fiction movie see Equilibrium (2002 movie) Equilibrium or balance is any of a number of related phenomena in the natural and social sciences. ...

p1 = p2 = P(q1+q2)

This implies that firm i’s profit is given by

• Calculate firm 1’s residual demand: Suppose firm 1 believes firm 2 is producing quantity q2. What is firm 1s optimal quantity? Consider the diagram 1. If firm 1 decides not to produce anything, then price is given by P(0+q2)=P(q2). If firm 1 sets produces q1’ then price is given by P(q1’+q2). More generally, for each quantity that firm 1 might decide to set, price is given by the curve d1(q2). The curve d1(q2) is called firm 1’s residual demand; it gives all possible combinations of firm 1’s quantity and price for a given value of q2.

• Determine firm 1’s optimum output: To do this we must find where marginal revenue equals marginal cost. Marginal cost (c) is assumed to be constant. Marginal revenue is a curve - r1(q2) - with twice the slope of d1(q2) and with the same vertical intercept. The point at which the two curves intersect corresponds to quantity q1’’(q2). Firm 1’s optimum q1’’(q2), depends on what it believes firm 2 is doing. To find an equilibrium, we derive firm 1’s optimum for other possible values of q2. Diagram 2 considers two possible values of q2. If q2=0, then firm firms residual demand is effectively the market demand, d1(0)=D. The optimal solution is for firm 1 to choose the monopoly quantity; q1’’(0)=q^m (q^m is monopoly quantity). If firm 2 were to choose the quantity corresponding to perfect competition, q2=qc P(q^c)=c, then firm 1’s optimum would be to produce nil: q1’’(q^c)=0. This is the point at which marginal cost intercepts the marginal revenue corresponding to d1(q^c).

In economics, a monopoly (from the Greek monos, one + polein, to sell) is defined as a persistent market situation where there is only one provider of a kind of product or service. ... Perfect competition is a model in economic theory. ...

• It can be shown that, given the linear demand and constant marginal cost, the function q1’’(q2) is also linear. Because we have two points, we can draw the entire function q1’’(q2), see diagram 3. Note the axis of the graphs has changed, The function q1’’(q2) is firm 1’s reaction function, it gives firm 1’s optimal choice for each possible choice by firm 2. In other words, it gives firm 1’s choice given what it believes firm 2 is doing.

• The last stage in finding the Cournot equilibrium is to find firm 2’s reaction function. In this case it is symmetrical to firm 1’s as they have the same cost function. The equilibrium is the interception point of the reaction curves. See diagram 4.

• The prediction of the model is that the firms will choose Nash equilibrium output levels.

In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, where no player has anything to gain by changing only ones own strategy. ...

Calculating the equilibrium

In very general terms, let the price function for the (duopoly) industry be P(q₁ + q₂) and firm i have the cost structure C_i(q_i). To calculate the Nash equilibrium, the best response functions of the firms must first be calculated. In game theory, the best response is the strategy in a single period that creates the most favorable immediate outcome for the current player, taking other players strategies as given. ...

The profit of firm i is revenue less cost. Revenue is the product of price and quantity and cost is given by the firm's cost structure, so profit is (as described above): . The best response is to find the value of q_i that maximises given q_j, with , i.e. given some output of the opponent firm, the output that maximises profit is found. Hence, the maximum of with respect to q_i is to be found. First derive with respect to q_i:

Setting this to zero for maximisation:

The values of q_i that satisfy this equation are the best responses. The Nash equilibria are where both q₁ and q₂ are best responses given those values of q₁ and q₂.

An example

Suppose the industry has the following price structure: P(q₁ + q₂) = a − b(q₁ + q₂) The profit of firm i (with cost structure C_i(q_i) such that and for ease of computation) is:

The maximisation 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 q₂, firm 1 responds best with any value of q₁ that satisfies the above. In Nash equilibria, both firms will be playing best responses so solving the above equations simultaneously. Substituting for q₂ in firm 1's best response: In mathematics, simultaneous equations are a set of equations where variables are shared. ...

The Nash equilibria are all (q₁ * ,q₂ * ). This yields a market price of 5a/3.

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.

Bertrand versus Cournot

• Although both models have similar assumptions, but both have very different implications.
• Bertrand predicts a duopoly is enough to push prices down to marginal cost level, meaning that duopoly will result in perfect competition.
• Neither model is ‘better’, it depends on the industry as to which is more accurate.
• If capacity and output can be easily changed, Bertrand is a better model of duopoly competition. Or, if output and capacity are difficult to adjust, then Cournot is generally a better model.

A duopoly is a form of oligopoly where only two producers are present in a given market. ... Perfect competition is a model in economic theory. ...

See also

• Nash equilibrium
• game theory
• Bertrand competition
• Stackelberg competition