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**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.^{ [1] } It has the following features:

**Antoine Augustin Cournot** was a French philosopher and mathematician who also contributed to the development of economics.

A **duopoly** is a type of oligopoly where two firms have dominant or exclusive control over a market. It is the most commonly studied form of oligopoly due to its simplicity. Duopolies sell to consumers in a competitive market where the choice of an individual consumer can not affect the firm. The defining characteristic of both duopolies and oligopolies is that decisions made by sellers are dependent on each other.

- History
- Graphically finding the Cournot duopoly equilibrium
- Calculating the equilibrium
- An example
- Cournot competition with many firms and the Cournot theorem
- Implications
- Bertrand versus Cournot
- See also
- References

- 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.

In marketing, a **product** is an object or system made available for consumer use; it is anything that can be offered to a market to satisfy the desire or need of a customer. In retailing, products are often referred to as *merchandise*, and in manufacturing, products are bought as raw materials and then sold as finished goods. A service is also regarded to as a type of product.

In economics and marketing, **product differentiation** is the process of distinguishing a product or service from others, to make it more attractive to a particular target market. This involves differentiating it from competitors' products as well as a firm's own products. The concept was proposed by Edward Chamberlin in his 1933 *The Theory of Monopolistic Competition*.

**Collusion** is a secret cooperation or deceitful agreement in order to deceive others, although not necessarily illegal, as a conspiracy. A secret agreement between two or more parties to limit open competition by deceiving, misleading, or defrauding others of their legal rights, or to obtain an objective forbidden by law typically by defrauding or gaining an unfair market advantage is an example of collusion. It is an agreement among firms or individuals to divide a market, set prices, limit production or limit opportunities. It can involve "unions, wage fixing, kickbacks, or misrepresenting the independence of the relationship between the colluding parties". In legal terms, all acts effected by collusion are considered void.

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.

In economics, a **cost curve** is a graph of the costs of production as a function of total quantity produced. In a free market economy, productively efficient firms optimize their production process by minimizing cost consistent with each possible level of production, and the result is a cost curve. profit maximizing firms use cost curves to decide output quantities. There are various types of cost curves, all related to each other, including total and average cost curves; marginal cost curves, which are equal to the differential of the total cost curves; and variable cost curves. Some are applicable to the short run, others to the long run.

In microeconomics, **supply and demand** is an economic model of price determination in a market. It postulates that, holding all else equal, in a competitive market, the unit price for a particular good, or other traded item such as labor or liquid financial assets, will vary until it settles at a point where the quantity demanded will equal the quantity supplied, resulting in an economic equilibrium for price and quantity transacted.

A **monopoly** exists when a specific person or enterprise is the only supplier of a particular commodity. This contrasts with a monopsony which relates to a single entity's control of a market to purchase a good or service, and with oligopoly which consists of a few sellers dominating a market. Monopolies are thus characterized by a lack of economic competition to produce the good or service, a lack of viable substitute goods, and the possibility of a high monopoly price well above the seller's marginal cost that leads to a high monopoly profit. The verb *monopolise* or *monopolize* refers to the *process* by which a company gains the ability to raise prices or exclude competitors. In economics, a monopoly is a single seller. In law, a monopoly is a business entity that has significant market power, that is, the power to charge overly high prices. Although monopolies may be big businesses, size is not a characteristic of a monopoly. A small business may still have the power to raise prices in a small industry.

“ | The state of equilibrium... is therefore stable; i.e. if either of the producers, misled as to his true interest, leaves it temporarily, he will be brought back to it. | ” |

— Antoine Augustin Cournot,Recherches sur les Principes Mathematiques de la Theorie des Richesses (1838), translated by Bacon (1897). |

Antoine Augustin Cournot (1801-1877) 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 (a duopoly).^{ [2] } The model was one of a number that Cournot set out "explicitly and with mathematical precision" in the volume.^{ [3] } 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.^{ [3] } He then showed that a stable equilibrium occurs where these functions intersect (i.e. the simultaneous solution of the best response functions of each firm).^{ [3] }

In game theory, the **best response** is the strategy which produces the most favorable outcome for a player, taking other players' strategies as given. The concept of a best response is central to John Nash's best-known contribution, the Nash equilibrium, the point at which each player in a game has selected the best response to the other players' strategies.

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.^{ [1] } This idea of stability was later taken up and built upon as a description of Nash equilibria, of which Cournot equilibria are a subset.^{ [3] }

This section presents an analysis of the model with 2 firms and constant marginal cost.

In economics, **marginal cost** is the change in the total cost that arises when the quantity produced is incremented by one unit; that is, it is the cost of producing one more unit of a good. Intuitively, marginal cost at each level of production includes the cost of any additional inputs required to produce the next unit. At each level of production and time period being considered, marginal costs include all costs that vary with the level of production, whereas other costs that do not vary with production are fixed and thus have no marginal cost. For example, the marginal cost of producing an automobile will generally include the costs of labor and parts needed for the additional automobile but not the fixed costs of the factory that have already been incurred. In practice, marginal analysis is segregated into short and long-run cases, so that, over the long run, all costs become marginal. Where there are economies of scale, prices set at marginal cost will fail to cover total costs, thus requiring a subsidy. Marginal cost pricing is not a matter of merely lowering the general level of prices with the aid of a subsidy; with or without subsidy it calls for a drastic restructuring of pricing practices, with opportunities for very substantial improvements in efficiency at critical points.

- = firm 1 price, = firm 2 price

- = firm 1 quantity, = firm 2 quantity

- = marginal cost, identical for both firms

Equilibrium prices will be:

This implies that firm 1's profit is given by

- Calculate firm 1's residual demand: Suppose firm 1 believes firm 2 is producing quantity . What is firm 1's optimal quantity? Consider the diagram 1. If firm 1 decides not to produce anything, then price is given by . If firm 1 produces then price is given by . More generally, for each quantity that firm 1 might decide to set, price is given by the curve . The curve is called firm 1's residual demand; it gives all possible combinations of firm 1's quantity and price for a given value of .

- 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 - - with twice the slope of and with the same vertical intercept. The point at which the two curves ( and ) intersect corresponds to quantity . Firm 1's optimum , depends on what it believes firm 2 is doing. To find an equilibrium, we derive firm 1's optimum for other possible values of . Diagram 2 considers two possible values of . If , then the first firm's residual demand is effectively the market demand, . The optimal solution is for firm 1 to choose the monopoly quantity; ( is monopoly quantity). If firm 2 were to choose the quantity corresponding to perfect competition, such that , then firm 1's optimum would be to produce nil: . This is the point at which marginal cost intercepts the marginal revenue corresponding to .

- It can be shown that, given the linear demand and constant marginal cost, the function is also linear. Because we have two points, we can draw the entire function , see diagram 3. Note the axis of the graphs has changed, The function 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 intersection point of the reaction curves. See diagram 4.

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

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 .

Suppose the industry has the following price structure: The profit of firm (with cost structure 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.

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.^{ [4] }

- 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.

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.

An **oligopoly** is a market form wherein a market or industry is dominated by a small number of large sellers (oligopolists). Oligopolies can result from various forms of collusion which reduce competition and lead to higher prices for consumers. Oligopolies have their own market structure.

The **Herfindahl index** is a measure of the size of firms in relation to the industry and an indicator of the amount of competition among them. Named after economists Orris C. Herfindahl and Albert O. Hirschman, it is an economic concept widely applied in competition law, antitrust and also technology management. It is defined as the sum of the squares of the market shares of the firms within the industry, where the market shares are expressed as fractions. The result is proportional to the average market share, weighted by market share. As such, it can range from 0 to 1.0, moving from a huge number of very small firms to a single monopolistic producer. Increases in the Herfindahl index generally indicate a decrease in competition and an increase of market power, whereas decreases indicate the opposite. Alternatively, if whole percentages are used, the index ranges from 0 to 10,000 "points". For example, an index of .25 is the same as 2,500 points.

In economics, **elasticity** is the measurement of the proportional change of an economic variable in response to a change in another. It shows how easy it is for the supplier and consumer to change their behavior and substitute another good, the strength of an incentive over choices per the relative opportunity cost.

In economics, **economic equilibrium** is a situation in which economic forces such as supply and demand are balanced and in the absence of external influences the (equilibrium) values of economic variables will not change. For example, in the standard textbook model of perfect competition, equilibrium occurs at the point at which quantity demanded and quantity supplied are equal. **Market equilibrium** in this case is a condition where a market price is established through competition such that the amount of goods or services sought by buyers is equal to the amount of goods or services produced by sellers. This price is often called the **competitive price** or market clearing price and will tend not to change unless demand or supply changes, and the quantity is called the "competitive quantity" or market clearing quantity. However, the concept of *equilibrium* in economics also applies to imperfectly competitive markets, where it takes the form of a Nash equilibrium.

The **classical general equilibrium model** aims to describe the economy by aggregating the behavior of individuals and firms. Note that the classical general equilibrium model is unrelated to classical economics, and was instead developed within neoclassical economics beginning in the late 19th century.

In economics and in particular neoclassical economics, the **marginal product** or **marginal physical productivity** of an input is the change in output resulting from employing one more unit of a particular input, assuming that the quantities of other inputs are kept constant.

In microeconomics, **marginal revenue** (MR) is the additional revenue that will be generated by increasing product sales by one unit.

**Bertrand competition** is a model of competition used in economics, named after Joseph Louis François Bertrand (1822–1900). It describes interactions among firms (sellers) that set prices and their customers (buyers) that choose quantities at the prices set. The model was formulated in 1883 by Bertrand in a review of Antoine Augustin Cournot's book *Recherches sur les Principes Mathématiques de la Théorie des Richesses* (1838) in which Cournot had put forward the Cournot model. Cournot argued that when firms choose quantities, the equilibrium outcome involves firms pricing above marginal cost and hence the competitive price. In his review, Bertrand argued that if firms chose prices rather than quantities, then the competitive outcome would occur with price equal to marginal cost. The model was not formalized by Bertrand: however, the idea was developed into a mathematical model by Francis Ysidro Edgeworth in 1889.

The **Ramsey problem**, or **Ramsey–Boiteux pricing**, is a Second best policy problem concerning what price a public monopolist or a firm faced with an irremovable revenue constraint should set, in order to maximize social welfare. A closely related problem arises in relation to optimal taxation of commodities.

The **Stackelberg leadership model** is a strategic game in economics in which the leader firm moves first and then the follower firms move sequentially. It is named after the German economist Heinrich Freiherr von Stackelberg who published *Market Structure and Equilibrium * in 1934 which described the model.

**Hotelling's lemma** is a result in microeconomics that relates the supply of a good to the profit of the good's producer. It was first shown by Harold Hotelling, and is widely used in the theory of the firm.

In economics and game theory, the decisions of two or more players are called **strategic complements** if they mutually reinforce one another, and they are called **strategic substitutes** if they mutually offset one another. These terms were originally coined by Bulow, Geanakoplos, and Klemperer (1985).

In economics, the **marginal product of labor** (**MP _{L}**) is the change in output that results from employing an added unit of labor. It is a feature of the production function, and depends on the amounts of physical capital and labor already in use.

The **Brander–Spencer model** is an economic model in international trade originally developed by James Brander and Barbara Spencer in the early 1980s. The model illustrates a situation where, under certain assumptions, a government can subsidize domestic firms to help them in their competition against foreign producers and in doing so enhances national welfare. This conclusion stands in contrast to results from most international trade models, in which government non-interference is socially optimal.

A **markup rule** is the pricing practice of a producer with market power, where a firm charges a fixed mark-up over its marginal cost.

In oligopoly theory, **conjectural variation** is the belief that one firm has an idea about the way its competitors may react if it varies its output or price. The firm forms a conjecture about the variation in the other firm's output that will accompany any change in its own output. For example, in the classic Cournot model of oligopoly, it is assumed that each firm treats the output of the other firms as given when it chooses its output. This is sometimes called the "Nash conjecture" as it underlies the standard Nash equilibrium concept. However, alternative assumptions can be made. Suppose you have two firms producing the same good, so that the industry price is determined by the combined output of the two firms. Now suppose that each firm has what is called the "Bertrand Conjecture" of −1. This means that if firm A increases its output, it conjectures that firm B will reduce its output to exactly offset firm A's increase, so that total output and hence price remains unchanged. With the Bertrand Conjecture, the firms act as if they believe that the market price is unaffected by their own output, because each firm believes that the other firm will adjust its output so that total output will be constant. At the other extreme is the Joint-Profit maximizing conjecture of +1. In this case each firm believes that the other will imitate exactly any change in output it makes, which leads to the firms behaving like a single monopoly supplier.

A **Monopoly price** is set by a Monopoly. A monopoly occurs when a firm lacks any viable competition, and is the sole producer of the industry's product. Because a monopoly faces no competition, it has absolute market power, and thereby has the ability to set a monopoly price that will be above the firm's marginal (economic) cost. Since marginal cost is the increment in total required to produce an additional unit of the product, the firm would be able to make a positive economic profit if it produced a greater quantity of the product and sold it at a lower price.

In labour economics, **Shapiro–Stiglitz theory** of efficiency wages is an economic theory of wages and unemployment in labour market equilibrium. It provides a technical description of why wages are unlikely to fall and how involuntary unemployment appears. This theory was first developed by Carl Shapiro and Joseph Stiglitz.

- 1 2 Varian, Hal R. (2006).
*Intermediate microeconomics: a modern approach*(7th ed.). W. W. Norton & Company. p. 490. ISBN 0-393-92702-4. - ↑ Van den Berg et al. 2011 , p. 1
- 1 2 3 4 Morrison 1998
- ↑ Etro, Federico.
*Simple models of competition Archived 2011-10-05 at the Wayback Machine*, page 6, Dept. Political Economics -- Università di Milano-Bicocca, November 2006

- Holt, Charles.
*Games and Strategic Behavior (PDF version)*, PDF - Tirole, Jean.
*The Theory of Industrial Organization*, MIT Press, 1988. - Oligoply Theory made Simple, Chapter 6 of Surfing Economics by Huw Dixon.

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