NationMaster - Encyclopedia: Cournot competition

Cournot competition is an economic 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 Î¿Î¯ÎºÏ‰ [okos], house, and Î½ÎÎ¼Ï‰ [nemo], rules hence household management) is the social science that studies the allocation of scarce resources to satisfy unlimited wants. ... Antoine Augustin Cournot Antoine Augustin Cournot (28 August 1801â€‘ 31 March 1877) was a French philosopher and mathematician. ... A true duopoly is a form of oligopoly where only two producers exist in a market. ...

There is more than one firm and all firms produce a homogeneous product;
Firms do not cooperate.
Firms have market power;
The number of firms is fixed;
Firms compete in quantities, and choose quantities simultaneously;
There is strategic behaviour by the firms;

An essential assumption of this model is 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. Each firm has a cost function ci(qi). 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 both firms. Each firm takes the quantity set by its competitors as a given, evaluates its residual demand, and then behaves as a monopoly. In economics, market power is the ability of a firm to alter the market price of a good or service. ... Game theory is often described as a branch of applied mathematics and economics that studies situations where multiple players make decisions in an attempt to maximize their returns. ... 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). ...

1 Graphically finding the Cournot duopoly equilibrium
2 Calculating the equilibrium
- 2.1 An example
3 Cournot competition with many firms and the Cournot Theorem
4 Implications
5 Bertrand versus Cournot
6 Stackelberg versus Cournot
7 See also
8 References

Graphically finding the Cournot duopoly equilibrium

This section presents an analysis of the model with 2 firms and constant marginal cost. In economics and finance, marginal cost is the change in total cost that arises when the quantity produced changes by one unit. ...

p1 = firm 1 price, p2 = firm 2 price

q1 = firm 1 quantity, q2 = firm 2 quantity

c = marginal cost, identical for both firms

Equilibrium prices will be: In economics and finance, marginal cost is the change in total cost that arises when the quantity produced changes by one unit. ... Price of market balance In economics, economic equilibrium is simply a state of the world where economic forces are balanced and in the abscence of external shocks the (equilibrium) values of economic variables will not change. ...

p 1 = p 2 = P (q 1 + q 2)

This implies that firm 1’s profit is given by $Pi 1 = q1(P(q1+q2)-c)$

Calculate firm 1’s residual demand: Suppose firm 1 believes firm 2 is producing quantity q2. What is firm 1's 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.

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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 (c and r1(q2)) 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 the first firm's 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).

A monopoly (from the Greek language monos, one + polein, to sell) is defined as a persistent market situation where there is only one provider of a product or service, in other words a firm that has no competitors in its industry. ... Perfect competition is an economic model that describes a hypothetical market form in which no producer or consumer has the market power to influence prices. ... Image File history File links No higher resolution available. ...

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.

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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.

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

Calculating the equilibrium

In very general terms, let the price function for the (duopoly) industry be $P (q 1 + q 2)$ 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 (or strategies) which produces the most favorable immediate outcome for the current player, taking other players strategies as given. ...

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): $Pi i = P(q_1+q_2).q_i - C_i(q_i)$ . The best response is to find the value of $q i$ that maximises $Pi i$ given $q j$ , with $i ne j$ , i.e. given some output of the opponent firm, the output that maximises profit is found. Hence, the maximum of $Pi i$ with respect to $q i$ is to be found. First derive $Pi i$ with respect to $q i$ :

$frac{partial Pi i }{partial q_i} = frac{partial P(q_1+q_2) }{partial q_i}.qi + P(q1+q2) - frac{partial C_i (q_i)}{partial q_i}$

Setting this to zero for maximisation:

$frac{partial Pi i }{partial q_i} = frac{partial P(q_1+q_2) }{partial q_i}.qi + P(q1+q2) - frac{partial C_i (q_i)}{partial q_i}=0$

The values of $q i$ that satisfy this equation are the best responses. The Nash equilibria are where both $q 1$ and $q 2$ are best responses given those values of $q 1$ and $q 2$ .

An example

Suppose the industry has the following price structure: $P (q 1 + q 2) = a - (q 1 + q 2)$ The profit of firm i (with cost structure $C i (q i)$ such that $frac{partial ^2C_i (q_i)}{partial q_i^2}=0$ and $frac{partial C_i (q_i)}{partial q_j}=0, j ne i$ for ease of computation) is:

$Pi i = bigg(a - (q_1+q_2)bigg).q_i - C_i(q_i)$

The maximisation problem resolves to (from the general case):

$frac{partial bigg(a - (q_1+q_2)bigg) }{partial q_i}.qi + a - (q_1+q_2) - frac{partial C_i (q_i)}{partial q_i}=0$

Without loss of generality, consider firm 1's problem:

$frac{partial bigg(a - (q_1+q_2)bigg) }{partial q_1}.q1 + a - (q_1+q_2) - frac{partial C_1 (q_1)}{partial q_1}=0$

$Rightarrow - q_1 + a - (q_1+q_2) - frac{partial C_1 (q_1)}{partial q_1}=0$

$Rightarrow q_1 = frac{a - q_2 - frac{partial C_1 (q_1)}{partial q_1}}{2}$

By symmetry:

$Rightarrow q_2 = frac{a - q_1 - frac{partial C_2 (q_2)}{partial q_2}}{2}$

These are the firms' best response functions. For any value of $q 2$ , firm 1 responds best with any value of $q 1$ that satisfies the above. In Nash equilibria, both firms will be playing best responses so solving the above equations simultaneously. Substituting for $q 2$ in firm 1's best response: In mathematics, simultaneous equations are a set of equations where variables are shared. ...

$q_1 = frac{a - (frac{a - q_1 - frac{partial C_2 (q_2)}{partial q_2}}{2}) - frac{partial C_1 (q_1)}{partial q_1}}{2}$

$Rightarrow q_1* = frac{a + frac{partial C_2 (q_2)}{partial q_2} - 2*frac{partial C_1 (q_1)}{partial q_1}}{3}$

$Rightarrow q_2* = frac{a + frac{partial C_1 (q_1)}{partial q_1} - 2*frac{partial C_2 (q_2)}{partial q_2}}{3}$

The Nash equilibrium is at $(q 1 * , q 2 * )$ . 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 $P (q 1 + q 2) = a - (q 1 + q 2)$ to obtain the equilibrium market price.

Cournot competition with many firms and the Cournot Theorem

For an arbitrary number of firms, N>1, 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:

$q_i = q = frac{a-c} {(N+1)}$ which is each individual firm's output

$sum q_i = Nq = frac{N(a-c)} {(N+1)}$ which is total industry output

and

$p = frac{a} {N+1} + frac{Nc} {N+1}$ is the market clearing price.

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.

$lim_{Nrightarrow infty} p = c$

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 N firms, when fixed costs for each firm are F, we have the endogenous number of firms:

N=(a-c)/√F-1

and a production for each firm equal to:

q=√F

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. However cartels are usually illegal, so firms have some motive to tacitly collude using self-imposing strategies to reduce output, which (ceteras paribus) raises price and thus increases profits.

Bertrand versus Cournot

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

Bertrand predicts 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.

Bertrand competition is a model of competition used in economics, named after Joseph Louis FranÃ§ois Bertrand (1822-1900). ... A true duopoly is a form of oligopoly where only two producers exist in a market. ... Perfect competition is an economic model that describes a hypothetical market form in which no producer or consumer has the market power to influence prices. ...

Stackelberg versus Cournot

The Stackelberg and Cournot models are similar because in both, competition is on quantity.
The first move gives the leader in Stackelberg a crucial advantage.
There is also the important assumption of perfect information in the Stackelberg game: the follower must observe the quantity chosen by the leader, otherwise the game reduces to Cournot.
This is an example of too much information hurting a player.
In Cournot competition, it is the simultaneity of the game (the imperfection of knowledge) that results in neither player (ceteris paribus) being at a disadvantage.

References

view	Topics in game theory
Definitions Game theory is often described as a branch of applied mathematics and economics that studies situations where multiple players make decisions in an attempt to maximize their returns. ...	Normal form game · Extensive form game · Cooperative game · Information set · Preference In game theory, normal form is a way of describing a game. ... It has been suggested that Game tree be merged into this article or section. ... A cooperative game is a game where groups of players (coalitions) may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players. ... In game theory, an information set is a set that, for a particular player, establishes all the possible moves that could have taken place in the game so far, given what that player has observed so far. ... Preference (or taste) is a concept, used in the social sciences, particularly economics. ...
Equilibrium concepts Price of market balance In economics, economic equilibrium is simply a state of the world where economic forces are balanced and in the abscence of external shocks the (equilibrium) values of economic variables will not change. ... In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ...	Nash equilibrium · Subgame perfection · Bayes-Nash · Trembling hand · Proper equilibrium · Epsilon-equilibrium · Correlated equilibrium · Sequential equilibrium · Quasi-perfect equilibrium · Evolutionarily stable strategy · Risk dominance In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. ... Subgame perfect equilibrium is an economics term used in game theory to describe an equilibrium such that players strategies constitute a Nash equilibrium in every subgame of the original game. ... In game theory, a Bayesian game is one in which information about characteristics of the other players (i. ... The trembling hand perfection is a notion that eliminates actions of players that are unsafe because they were chosen through a slip of the hand. ... Proper equilibrium is a refinement of Nash Equilibrium due to Roger B. Myerson. ... In game theory, an Epsilon-equilibrium is a strategy profile that approximately satisfies the condition of Nash Equilibrium. ... In game theory, a correlated equilibrium is a solution concept that is more general than the well known Nash equilibrium. ... Sequential equilibrium is a refinement of Nash Equilibrium for extensive form games due to David M. Kreps and Robert Wilson. ... Quasi-perfect equilibrium is a refinement of Nash Equilibrium for extensive form games due to Eric van Damme. ... In game theory, an evolutionarily stable strategy (or ESS; also evolutionary stable strategy) is a strategy which if adopted by a population cannot be invaded by any competing alternative strategy. ... Risk dominance and payoff dominance are two related refinements of the Nash equilibrium (NE) solution concept in game theory, defined by John Harsanyi and Reinhard Selten. ...
Strategies In game theory, a players strategy, in a game or a business situation, is a complete plan of action for whatever situation might arise; this fully determines the players behaviour. ...	Dominant strategies · Mixed strategy · Tit for tat · Grim trigger · Collusion In game theory, dominance occurs when one strategy is better or worse than another regardless of the strategies of a players opponents. ... In game theory a mixed strategy is a strategy which chooses randomly between possible moves. ... Tit for Tat is a highly-effective strategy in game theory for the iterated prisoners dilemma. ... Grim Trigger is a trigger strategy in game theory for a repeated game, such as an iterated prisoners dilemma. ... Look up collusion in Wiktionary, the free dictionary. ...
Classes of games	Symmetric game · Perfect information · Dynamic game · Repeated game · Signaling game · Cheap talk · Zero-sum game · Mechanism design · Stochastic game · Nontransitive game In game theory, a symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. ... Perfect information is a term used in economics and game theory to describe a state of complete knowledge about the actions of other players that is instantaneously updated as new information arises. ... In game theory, a sequential game is a game where one player chooses his action before the others chooses theirs. ... In game theory, a repeated game (or iterated game) is an extensive form game which consists in some number of repetitions of some base game (called a stage game). ... Signaling games are dynamic games with two players, the sender (S) and the receiver (R). ... Cheap Talk is a term used in Game Theory for pre-play communication which carries no cost. ... Zero-sum describes a situation in which a participants gain (or loss) is exactly balanced by the losses (or gains) of the other participant(s). ... Mechanism design is a sub-field of game theory. ... In game theory, a stochastic game is a competitive game with probabilistic transitions played by two players. ... A non-transitive game is a game for which the various strategies produce one or more loops of preferences. ...
Games Game theory studies strategic interaction between individuals in situations called games. ...	Prisoner's dilemma · Traveler's dilemma · Coordination game · Chicken · Volunteer's dilemma · Dollar Auction · Battle of the sexes · Stag hunt · Matching pennies · Ultimatum game · Minority game · Rock, Paper, Scissors · Pirate game · Dictator game · Public goods game · Nash bargaining game · Blotto games Will the two prisoners cooperate to minimize total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? In game theory, the prisoners dilemma (sometimes abbreviated PD) is a type of non-zero-sum game in which two players... In game theory, the travelers dilemma (sometimes abbreviated TD) is a type of non-zero-sum game in which two players attempt to maximise their own payoff, without any concern for the other players payoff. ... In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, whereby the players reach an outcome to mutual advantage. ... It has been suggested that Peace war game be merged into this article or section. ... The Volunteers dilemma game models a situation in which each of N players faces the decision of either making a small sacrifice from which all will benefit or freeriding. ... On eBay, where an auction has a starting price of $1 ... The Battle of the Sexes is a two player game used in game theory. ... In game theory, the Stag Hunt is a game first discussed by Jean-Jacques Rousseau. ... Matching Pennies is the name for a simple example game used in game theory. ... The Ultimatum game is an experimental economics game in which two parties interact anonymously and only once, so reciprocation is not an issue. ... Minority Game is a game proposed by Yi-Cheng Zhang and Damien Challet from the University of Fribourg. ... Rock, Paper, Scissors chart Listen to this article (info) play in browser This audio file was created from an article revision dated 2006-07-13, and may not reflect subsequent edits to the article. ... The Pirate Game is a simple mathematical game. ... The dictator game is a very simple game in experimental economics, similar to the ultimatum game. ... The Public goods game is a standard of experimental economics; in the basic game subjects secretly choose how many of their private tokens to put into the public pot. ... The Nash Bargaining Game is a simple two player game used to model bargaining interactions. ... Blotto games (or Colonel Blotto games) constitute a class of two-person zero-sum games in which the players are tasked to simultaneously distribute limited resources over several objects, with the gain (or payoff) being equal to the sum of the gains on the individual objects. ...
Theorems	Minimax theorem · Purification theorems · Folk theorem · Revelation principle · Arrow's Theorem Minimax is a method in decision theory for minimizing the expected maximum loss. ... In game theory, the purification theorem was contributed by Nobel laurate John Harsanyi in 1973[1]. The theorem aims to justify a puzzling aspect of mixed strategy Nash equilibria: that each player is wholly indifferent amongst each of the actions he puts non-zero weight on, yet he mixes them... In game theory, folk theorems are a class of theorems which imply that in repeated games, any outcome is a feasible solution concept, if under that outcome the players minimax conditions are satisfied. ... The revelation principle of economics can be stated as, To any equilibrium of a game of incomplete information, there corresponds an associated revelation mechanism that has an equilibrium where the players truthfully report their types. ... In voting systems, Arrow’s impossibility theorem, or Arrow’s paradox demonstrates the impossibility of designing a set of rules for social decision making that would meet all of a certain set of criteria. ...
Related topics	Mathematics · Economics · Behavioral economics · Evolutionary game theory · Population genetics · Behavioral ecology · Adaptive dynamics · List of game theorists · Social trap · Tragedy of the commons Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Face-to-face trading interactions on the New York Stock Exchange trading floor. ... Nobel Prize in Economics winner Daniel Kahneman, was an important figure in the development of behavioral finance and economics and continues to write extensively in the field. ... Evolutionary game theory (EGT) is the application of game theory in evolutionary biology. ... Population genetics is the study of the distribution of and change in allele frequencies under the influence of the four evolutionary forces: natural selection, genetic drift, mutation, and migration. ... Behavioral ecology is the study of the ecological and evolutionary basis for animal behavior, and the roles of behavior in enabling an animal to adapt to its environment (both intrinsic and extrinsic). ... Adaptive Dynamics is a set of techniques for studying long-term phenotypical evolution developed during the 1990s. ... This is a list of notable economists, mathematicians, political scientists, and computer scientists whose work has added substantially to the field of game theory. ... Social trap is a term used by psychologists to describe a situation in which a group of people act to obtain short-term individual gains, which in the long run leads to a loss for the group as a whole. ... It has been suggested that Tyranny of the Commons be merged into this article or section. ...

Categories: Economics models | Game theory | Competition | Market structure and pricing

Results from FactBites:

Cournot competition - Wikipedia, the free encyclopedia (1323 words)
	Cournot competition is an economic model used to describe industry structure.
	The Stackelberg and Cournot models are similar because in both, competition is on quantity.
	In Cournot competition, it is the simultaneity of the game (the imperfection of knowledge) that results in neither player (ceteris paribus) being at a disadvantage.

Antoine Augustin Cournot - Wikipedia, the free encyclopedia (512 words)
	Antoine Augustin Cournot was born at Gray, Haute-Saone.
	Cournot is credited with the "one monopoly profit" theorem, which says that a monopolist can extract only one premium for being a monopolist, and getting into complementary markets does not pay.
	Cournot is also credited to be one of the sources of inspiration for Leon Walras and his equilibrium theory.

More results at FactBites »

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Contents

Graphically finding the Cournot duopoly equilibrium GA_googleFillSlot("encyclopedia_square");

Calculating the equilibrium

An example

Cournot competition with many firms and the Cournot Theorem

Implications

Bertrand versus Cournot

Stackelberg versus Cournot

See also

References

Graphically finding the Cournot duopoly equilibrium