# Cournot competition

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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.

## Contents

• 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 ${\displaystyle N}$, the total number of firms in the market, and take the output of the others as given. Each firm has a cost function ${\displaystyle c_{i}(q_{i})}$. 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.

## History

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]

## Graphically finding the Cournot duopoly equilibrium

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.

${\displaystyle p_{1}}$ = firm 1 price, ${\displaystyle p_{2}}$ = firm 2 price
${\displaystyle q_{1}}$ = firm 1 quantity, ${\displaystyle q_{2}}$ = firm 2 quantity
${\displaystyle c}$ = marginal cost, identical for both firms

Equilibrium prices will be:

${\displaystyle p_{1}=p_{2}=P(q_{1}+q_{2})}$

This implies that firm 1's profit is given by ${\displaystyle \Pi _{1}=q_{1}(P(q_{1}+q_{2})-c)}$

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

• 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 - ${\displaystyle r_{1}(q_{2})}$ - with twice the slope of ${\displaystyle d_{1}(q_{2})}$ and with the same vertical intercept. The point at which the two curves (${\displaystyle c}$ and ${\displaystyle r_{1}(q_{2})}$) intersect corresponds to quantity ${\displaystyle q_{1}''(q_{2})}$. Firm 1's optimum ${\displaystyle q_{1}''(q_{2})}$, depends on what it believes firm 2 is doing. To find an equilibrium, we derive firm 1's optimum for other possible values of ${\displaystyle q_{2}}$. Diagram 2 considers two possible values of ${\displaystyle q_{2}}$. If ${\displaystyle q_{2}=0}$, then the first firm's residual demand is effectively the market demand, ${\displaystyle d_{1}(0)=D}$. The optimal solution is for firm 1 to choose the monopoly quantity; ${\displaystyle q_{1}''(0)=q^{m}}$ (${\displaystyle q^{m}}$ is monopoly quantity). If firm 2 were to choose the quantity corresponding to perfect competition, ${\displaystyle q_{2}=q^{c}}$ such that ${\displaystyle P(q^{c})=c}$, then firm 1's optimum would be to produce nil: ${\displaystyle q_{1}''(q^{c})=0}$. This is the point at which marginal cost intercepts the marginal revenue corresponding to ${\displaystyle d_{1}(q^{c})}$.

• It can be shown that, given the linear demand and constant marginal cost, the function ${\displaystyle q_{1}''(q_{2})}$ is also linear. Because we have two points, we can draw the entire function ${\displaystyle q_{1}''(q_{2})}$, see diagram 3. Note the axis of the graphs has changed, The function ${\displaystyle q_{1}''(q_{2})}$ 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.

## Calculating the equilibrium

In very general terms, let the price function for the (duopoly) industry be ${\displaystyle P(q_{1}+q_{2})}$ and firm ${\displaystyle i}$ have the cost structure ${\displaystyle C_{i}(q_{i})}$. 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): ${\displaystyle \Pi _{i}=P(q_{1}+q_{2})\cdot q_{i}-C_{i}(q_{i})}$. The best response is to find the value of ${\displaystyle q_{i}}$ that maximises ${\displaystyle \Pi _{i}}$ given ${\displaystyle q_{j}}$, with ${\displaystyle i\neq j}$, i.e. given some output of the opponent firm, the output that maximises profit is found. Hence, the maximum of ${\displaystyle \Pi _{i}}$ with respect to ${\displaystyle q_{i}}$ is to be found. First take the derivative of ${\displaystyle \Pi _{i}}$ with respect to ${\displaystyle q_{i}}$:

${\displaystyle {\frac {\partial \Pi _{i}}{\partial q_{i}}}={\frac {\partial P(q_{1}+q_{2})}{\partial q_{i}}}\cdot q_{i}+P(q_{1}+q_{2})-{\frac {\partial C_{i}(q_{i})}{\partial q_{i}}}}$

Setting this to zero for maximization:

${\displaystyle {\frac {\partial \Pi _{i}}{\partial q_{i}}}={\frac {\partial P(q_{1}+q_{2})}{\partial q_{i}}}\cdot q_{i}+P(q_{1}+q_{2})-{\frac {\partial C_{i}(q_{i})}{\partial q_{i}}}=0}$

The values of ${\displaystyle q_{i}}$ that satisfy this equation are the best responses. The Nash equilibria are where both ${\displaystyle q_{1}}$ and ${\displaystyle q_{2}}$ are best responses given those values of ${\displaystyle q_{1}}$ and ${\displaystyle q_{2}}$.

### An example

Suppose the industry has the following price structure: ${\displaystyle P(q_{1}+q_{2})=a-(q_{1}+q_{2})}$ The profit of firm ${\displaystyle i}$ (with cost structure ${\displaystyle C_{i}(q_{i})}$ such that ${\displaystyle {\frac {\partial ^{2}C_{i}(q_{i})}{\partial q_{i}^{2}}}=0}$ and ${\displaystyle {\frac {\partial C_{i}(q_{i})}{\partial q_{j}}}=0,j\neq i}$ for ease of computation) is:

${\displaystyle \Pi _{i}={\bigg (}a-(q_{1}+q_{2}){\bigg )}\cdot q_{i}-C_{i}(q_{i})}$

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

${\displaystyle {\frac {\partial {\bigg (}a-(q_{1}+q_{2}){\bigg )}}{\partial q_{i}}}\cdot q_{i}+a-(q_{1}+q_{2})-{\frac {\partial C_{i}(q_{i})}{\partial q_{i}}}=0}$

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

${\displaystyle {\frac {\partial {\bigg (}a-(q_{1}+q_{2}){\bigg )}}{\partial q_{1}}}\cdot q_{1}+a-(q_{1}+q_{2})-{\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}=0}$
${\displaystyle \Rightarrow \ -q_{1}+a-(q_{1}+q_{2})-{\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}=0}$
${\displaystyle \Rightarrow \ q_{1}={\frac {a-q_{2}-{\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}}{2}}}$

By symmetry:

${\displaystyle \Rightarrow \ q_{2}={\frac {a-q_{1}-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}}{2}}}$

These are the firms' best response functions. For any value of ${\displaystyle q_{2}}$, firm 1 responds best with any value of ${\displaystyle q_{1}}$ that satisfies the above. In Nash equilibria, both firms will be playing best responses so solving the above equations simultaneously. Substituting for ${\displaystyle q_{2}}$ in firm 1's best response:

${\displaystyle \ q_{1}={\frac {a-\left({\frac {a-q_{1}-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}}{2}}\right)-{\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}}{2}}}$
${\displaystyle \Rightarrow \ q_{1}^{*}={\frac {a+{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}-2*{\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}}{3}}}$
${\displaystyle \Rightarrow \ q_{2}^{*}={\frac {a+{\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}-2*{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}}{3}}}$

The symmetric Nash equilibrium is at ${\displaystyle (q_{1}^{*},q_{2}^{*})}$. 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 ${\displaystyle P(q_{1}+q_{2})=a-(q_{1}+q_{2})}$ to obtain the equilibrium market price.

## Cournot competition with many firms and the Cournot theorem

For an arbitrary number of firms, ${\displaystyle N>1}$, 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:

Market demand; ${\displaystyle \ p(q)=a-bq=a-bQ=p(Q)}$

Cost function; ${\displaystyle \ c_{i}(q_{i})=cq_{i}}$, for all i

${\displaystyle \ q_{i}=Q/N={\frac {a-c}{b(N+1)}},}$

which is each individual firm's output

${\displaystyle \sum q_{i}=Nq={\frac {N(a-c)}{b(N+1)}},}$

which is total industry output

${\displaystyle \ p=a-b(Nq)={\frac {a+Nc}{N+1}},}$

which is the market clearing price, and

${\displaystyle \Pi _{i}=\left({\frac {a-c}{N+1}}\right)^{2}\left({\frac {1}{b}}\right)}$ , 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.

${\displaystyle \lim _{N\rightarrow \infty }p=c}$

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 ${\displaystyle N}$ firms, when fixed costs for each firm are ${\displaystyle F}$, we have the endogenous number of firms:

${\displaystyle N={\frac {a-c}{\sqrt {Fb}}}-1}$

and a production for each firm equal to:

${\displaystyle q={\frac {\sqrt {Fb}}{b}}}$

This equilibrium is usually known as Cournot equilibrium with endogenous entry, or Marshall equilibrium. [4]

## 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.

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## References

1. Varian, Hal R. (2006). Intermediate microeconomics: a modern approach (7th ed.). W. W. Norton & Company. p. 490. ISBN   0-393-92702-4.
2. 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