Cournot competition is an
economic
An economy is an area of the Production (economics), production, Distribution (economics), distribution and trade, as well as Consumption (economics), consumption of Goods (economics), goods and Service (economics), services. In general, it is ...
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
Antoine Augustin Cournot (; 28 August 180131 March 1877) was a French philosopher and mathematician who also contributed to the development of economics.
Biography
Antoine Augustin Cournot was born at Gray, Haute-Saône. In 1821 he entered o ...
(1801–1877) who was inspired by observing competition in a spring water
duopoly
A duopoly (from Greek δύο, ''duo'' "two" and πωλεῖν, ''polein'' "to sell") 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 simplicit ...
.
It has the following features:
* There is more than one firm and all firms produce a
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
, i.e., there is no
product differentiation
In economics and marketing, product differentiation (or simply 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 com ...
;
* Firms do not cooperate, i.e., there is no
collusion
Collusion is a deceitful agreement or secret cooperation between two or more parties to limit open competition by deceiving, misleading or defrauding others of their legal right. Collusion is not always considered illegal. It can be used to att ...
;
* Firms have
market power
In economics, market power refers to the ability of a firm to influence the price at which it sells a product or service by manipulating either the supply or demand of the product or service to increase economic profit. In other words, market powe ...
, i.e., each firm's output decision affects the good's price;
* The number of firms is fixed;
* Firms compete in quantities rather than prices; and
* 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. The market price is set at a level such that
demand
In economics, demand is the quantity of a good that consumers are willing and able to purchase at various prices during a given time. The relationship between price and quantity demand is also called the demand curve. Demand for a specific item ...
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
A monopoly (from Greek language, Greek el, μόνος, mónos, single, alone, label=none and el, πωλεῖν, pōleîn, to sell, label=none), as described by Irving Fisher, is a market with the "absence of competition", creating a situati ...
.
History
Antoine Augustin Cournot
Antoine Augustin Cournot (; 28 August 180131 March 1877) was a French philosopher and mathematician who also contributed to the development of economics.
Biography
Antoine Augustin Cournot was born at Gray, Haute-Saône. In 1821 he entered o ...
(1801–1877) first outlined his theory of competition in his 1838 volume ''Recherches sur les Principes Mathématiques de la Théorie des Richesses'' as a way of describing the competition with a market for spring water dominated by two suppliers (a
duopoly
A duopoly (from Greek δύο, ''duo'' "two" and πωλεῖν, ''polein'' "to sell") 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 simplicit ...
). The model was one of a number that Cournot set out "explicitly and with mathematical precision" in the volume.
Specifically, Cournot constructed profit functions for each firm, and then used
partial differentiation
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Par ...
to construct a function representing a firm's
best response
In game theory, the best response is the strategy (or strategies) 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 contribu ...
for given (exogenous) output levels of the other firm(s) in the market.
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).
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.
This idea of stability was later taken up and built upon as a description of
Nash equilibria
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...
, of which Cournot equilibria are a subset.
The legacy of the ''Recherches''
Cournot's economic theory was little noticed until
Léon Walras
Marie-Esprit-Léon Walras (; 16 December 1834 – 5 January 1910) was a French mathematical economist and Georgist. He formulated the marginal theory of value (independently of William Stanley Jevons and Carl Menger) and pioneered the developmen ...
credited him as a forerunner. This led to an unsympathetic review of Cournot's book by
Joseph Bertrand
Joseph Louis François Bertrand (; 11 March 1822 – 5 April 1900) was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, economics and thermodynamics.
Biography
Joseph Bertrand was ...
which in turn received heavy criticism.
Irving Fisher
Irving Fisher (February 27, 1867 – April 29, 1947) was an American economist, statistician, inventor, eugenicist and progressive social campaigner. He was one of the earliest American neoclassical economists, though his later work on debt def ...
found Cournot's treatment of oligopoly "brilliant and suggestive, but not free from serious objections". He arranged for a translation to be made by Nathaniel Bacon in 1897.
Reactions to this aspect of Cournot's theory have ranged from searing condemnation to half-hearted endorsement. It has received sympathy in recent years as a contribution to
game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
rather than economics.
James W. Friedman explains:
In current language and interpretation, Cournot postulated a particular game to represent an oligopolistic market...
The maths in Cournot's book is elementary and the presentation not difficult to follow. The account below follows Cournot's words and diagrams closely. The diagrams were presumably included as an oversized plate in the original edition, and are missing from some modern reprints.
Cournot's conceptual framework
Cournot's discussion of oligopoly draws on two theoretical advances made in earlier pages of his book. Both have passed (with some adjustment) into microeconomic theory, particularly within subfield of
Industrial Organization
In economics, industrial organization is a field that builds on the theory of the firm by examining the structure of (and, therefore, the boundaries between) firms and markets. Industrial organization adds real-world complications to the perf ...
where Cournot's assumptions can be relaxed to study various Market Structures and Industries, for example, the
Stackelberg Competition model. Cournot's discussion of monopoly influenced later writers such as
Edward Chamberlin
Edward Hastings Chamberlin (May 18, 1899 – July 16, 1967) was an American economist. He was born in La Conner, Washington, and died in Cambridge, Massachusetts.
Chamberlin studied first at the University of Iowa (where he was influenced by Fr ...
and
Joan Robinson
Joan Violet Robinson (''née'' Maurice; 31 October 1903 – 5 August 1983) was a British economist well known for her wide-ranging contributions to economic theory. She was a central figure in what became known as post-Keynesian economics.
B ...
during the 1930s revival of interest in
imperfect competition In economics, imperfect competition refers to a situation where the characteristics of an economic market do not fulfil all the necessary conditions of a perfectly competitive market. Imperfect competition will cause market inefficiency when it hap ...
.
The 'Law of Demand' or 'of Sales'
![Cournotdemand](https://upload.wikimedia.org/wikipedia/commons/0/0f/Cournotdemand.svg)
Cournot was wary of psychological notions of demand, defining it simply as the amount sold of a particular good (helped along by the fact that the French word ''débit'', meaning 'sales quantity', has the same initial letter as ''demande'', meaning 'demand' ). He formalised it mathematically as follows:
We will regard the sales quantity or annual demand , for any commodity, to be a function of its price.
It follows that his demand curves do some of the work of modern supply curves, since producers who are able to limit the amount sold have an influence on Cournot's demand curve.
Cournot remarks that the demand curve will usually be a decreasing function of price, and that the total value of the good sold is
, which will generally increase to a maximum and then decline towards 0. The condition for a maximum is that the derivative of
, i.e.,
, should be 0 (where
is the derivative of
).
Cournot's duopoly theory
Monopoly and duopoly
Cournot insists that each duopolist seeks ''independently'' to maximize profits, and this restriction is essential, since Cournot tells us that if they came to an understanding between each other so as each to obtain the maximum possible revenue, then completely different results would be obtained, indistinguishable from the consumer's point of view from those entailed by monopoly.
Cournot's price model
Cournot presents a mathematically correct analysis of the equilibrium condition corresponding to a certain logically consistent model of duopolist behaviour. However his model is not stated and is not particularly natural (
Shapiro
Shapiro, and its variations such as Shapira, Schapiro, Schapira, Sapir, Sapira, Spira, Sapiro, Spiro (name)/Spyro (in Greek), Szapiro/ Szpiro (in Polish) and Chapiro (in French), is a Jewish Ashkenazi surname.
Etymology
The surname is derive ...
remarked that observed practice constituted a "natural objection to the Cournot quantity model"
[, quoted by .]), and "his words and the mathematics do not quite match".
His model can be grasped more easily if we slightly embellish it. Suppose that there are two owners of mineral water springs, each able to produce unlimited quantities at zero price. Suppose that instead of selling water to the public they offer it to a middle man. Each proprietor notifies the middle man of the quantity he or she intends to produce. The middle man finds the market-clearing price, which is determined by the demand function
and the aggregate supply. He or she sells the water at this price, passing the proceeds back to the proprietors.
The consumer demand
for mineral water at price
is denoted by
; the inverse of
is written
and the market-clearing price is given by
, where
and
is the amount supplied by proprietor
.
Each proprietor is assumed to know the amount being supplied by his or her rival, and to adjust his or her own supply in the light of it to maximize his or her profits. The position of equilibrium is one in which neither proprietor is inclined to adjust the quantity supplied.
It needs mental contortions to imagine the same market behaviour arising without a middle man.
Interpretative difficulties
A feature of Cournot's model is that a single price applies to both proprietors. He justified this assumption by saying that "dès lors le prix est nécessairement le même pour l'un et l'autre propriétaire". de Bornier expands on this by saying that "the obvious conclusion that only a single price can exist at a given moment" follows from "an essential assumption concerning his model,
amely
Amely was an American rock band from Orlando, Florida, United States, formed in 2008. The band comprised four members; Petie Pizarro (Vocals/Guitar), Brandon Walden (Guitar), Patrick Ridgen (Bass) and Nate Parsell (Drums). The sound of the band ...
product homogeneity".
Later on Cournot writes that a proprietor can adjust his supply "en modifiant correctement le prix". Again, this is nonsense: it is impossible for a single price to be simultaneously under the control of two suppliers. If there is a single price, then it must be determined by the market as a consequence of the proprietors' decisions on matters under their individual control.
Cournot's account threw his English translator (Nathaniel Bacon) so completely off-balance that his words were corrected to "properly adjusting ''his'' price".
Edgeworth
Edgeworth may refer to:
People
* Edgeworth (surname)
Places
* Edgeworth, Gloucestershire, England
* Edgeworth, New South Wales, Australia
* Edgeworth, Pennsylvania, USA
* Edgeworth Island, Nunavut, Canada
* Edgeworthstown, County Longford, Repu ...
regarded equality of price in Cournot as "a particular condition, not... abstractly necessary in cases of imperfect competition". Jean Magnan de Bornier says that in Cournot's theory "each owner will use price as a variable to control quantity" without saying how one price can govern two quantities. A. J. Nichol claimed that Cournot's theory makes no sense unless "prices are directly determined by buyers".
Shapiro
Shapiro, and its variations such as Shapira, Schapiro, Schapira, Sapir, Sapira, Spira, Sapiro, Spiro (name)/Spyro (in Greek), Szapiro/ Szpiro (in Polish) and Chapiro (in French), is a Jewish Ashkenazi surname.
Etymology
The surname is derive ...
, perhaps in despair, remarked that "the actual process of price formation in Cournot's theory is somewhat mysterious".
Collusion
Cournot's duopolists are not true profit-maximizers. Either supplier could increase his or her profits by cutting out the middle man and cornering the market by marginally undercutting his or her rival; thus the middle man can be seen as a mechanism for restricting competition.
Finding the Cournot duopoly equilibrium
Example 1
Cournot's model of competition is typically presented for the case of a
duopoly
A duopoly (from Greek δύο, ''duo'' "two" and πωλεῖν, ''polein'' "to sell") 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 simplicit ...
market structure; the following example provides a straightforward analysis of the Cournot model for the case of Duopoly. Therefore, suppose we have a market consisting of only two firms which we will call firm 1 and firm 2. For simplicity, we assume each firm faces the same marginal cost. That is, for a given firm
's output quantity, denoted
where
, firm
's cost of producing
units of output is given by
, where
is the marginal cost.
This assumption tells us that both firms face the same cost-per-unit produced. Therefore, as each firm's profit is equal to its revenues minus costs, where revenue equals the number of units produced multiplied by the market price, we can denote the profit functions for firm 1 and firm 2 as follows:
:
:
Note that in the above profit functions we have price as a function of total output which we denote as
and for two firms we must have
. For example's sake, let us assume that price (inverse demand function) is linear and of the form
. So, the inverse demand function can then be rewritten as
.
Now, substituting our equation for price in place of
we can write each firm's profit function as:
:
:
As firms are assumed to be profit-maximizers, the first-order conditions (F.O.C.s) for each firm are:
:
:
The F.O.C.s state that firm
is producing at the profit-maximizing level of output when the marginal cost (
) is equal to the marginal revenue (
). Intuitively, this suggests that firms will produce up to the point where it remains profitable to do so, as any further production past this point will mean that
, and therefore production beyond this point results in the firm losing money for each additional unit produced. Notice that at the profit-maximizing quantity where
, we must have
which is why we set the above equations equal to zero.
Now that we have two equations describing the states at which each firm is producing at the profit-maximizing quantity, we can simply solve this system of equations to obtain each firm's optimal level of output,
for firms 1 and 2 respectively. So, we obtain:
:
:
These functions describe each firm's optimal (profit-maximizing) quantity of output given the price firms face in the market,
, the marginal cost,
, and output quantity of rival firms. The functions can be thought of as describing a firm's "Best Response" to the other firm's level of output.
We can now find a Cournot-
Nash Equilibrium
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...
using our "Best Response" functions above for the output quantity of firms 1 and 2. Recall that both firms face the same cost-per-unit (
) and price (
). Therefore, using this symmetrical relationship between firms we find the equilibrium quantity by fixing
. We can be sure this setup gives us the equilibrium levels as neither firm has an incentive to change their level of output as doing so will harm the firm at the benefit of their rival. Now substituting in
for
and solving we obtain the symmetric (same for each firm) output quantity in Equilibrium as
.
This equilibrium value describes the optimal level of output for firms 1 and 2, where each firm is producing an output quantity of
. So, at equilibrium, the total market output
will be
.
Example 2
The revenues accruing to the two proprietors are
and
, i.e.,
and
. The first proprietor maximizes profit by optimizing over the parameter
under his control, giving the condition that the partial derivative of his profit with respect to
should be 0, and the mirror-image reasoning applies to his or her rival. We thus get the equations:
:
and
:
.
The equlibirum position is found by solving these two equations simultaneously. This is most easily done by adding and subtracting them, turning them into:
:
and
:
, where
.
Thus, we see that the two proprietors supply equal quantities, and that the total quantity sold is the root of a single nonlinear equation in
.
![Cournotduopoly](https://upload.wikimedia.org/wikipedia/commons/c/cb/Cournotduopoly.svg)
Cournot goes further than this simple solution, investigating the stability of the equilibrium. Each of his original equations defines a relation between
and
which may be drawn on a graph. If the first proprietor was providing quantity
, then the second proprietor would adopt quantity
from the red curve to maximize his or her revenue. But then, by similar reasoning, the first proprietor will adjust his supply to
to give him or her the maximum return as shown by the blue curve when
is equal to
. This will lead to the second proprietor adapting to the supply value
, and so forth until equilibrium is reached at the point of intersection
, whose coordinates are
.
Since proprietors move towards the equilibrium position it follows that the equilibrium is stable, but Cournot remarks that if the red and blue curves were interchanged then this would cease to be true. He adds that it is easy to see that the corresponding diagram would be inadmissible since, for instance, it is necessarily the case that
. To verify this, notice that when
is 0, the two equations reduce to:
:
and
:
.
The first of these corresponds to the quantity
sold when the price is zero (which is the maximum quantity the public is willing to consume), while the second states that the derivative of
with respect to
is 0, but
is the monetary value of an aggregate sales quantity
, and the turning point of this value is a maximum. Evidently, the sales quantity which maximizes monetary value is reached before the maximum possible sales quantity (which corresponds to a value of 0). So, the root
of the first equation is necessarily greater than the root
of the second equation.
Comparison with monopoly
We have seen that Cournot's system reduces to the equation
.
is functionally related to
via
in one direction and
in the other. If we re-express this equation in terms of
, it tells us that
, which can be compared with the equation
obtained earlier for monopoly.
![Cournotolig](https://upload.wikimedia.org/wikipedia/commons/5/5f/Cournotolig.svg)
If we plot another variable
against
, then we may draw a curve of the function
. The monopoly price is the
for which this curve intersects the line
, while the duopoly price is given by the intersection of the curve with the steeper line
. Regardless of the shape of the curve, the intersection with
occurs to the left – i.e., at a lower price – than the intersection with
. Hence, prices are lower under duopoly than under monopoly, and quantities sold are accordingly higher.
Extension to oligopoly
When there are
proprietors, the price equation becomes
. The price can be read from the diagram from the intersection of
with the curve. Hence, the price diminishes indefinitely as the number of proprietors increases. With an infinite number of proprietors, the price becomes zero; or more generally, if we allow for costs of production, the price becomes the marginal cost.
Bertrand's critique
The French mathematician
Joseph Bertrand
Joseph Louis François Bertrand (; 11 March 1822 – 5 April 1900) was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, economics and thermodynamics.
Biography
Joseph Bertrand was ...
, when reviewing
Walras's ''Théorie Mathématique de la Richesse Sociale'', was drawn to Cournot's book by Walras's high praise of it. Bertrand was critical of Cournot's reasoning and assumptions, Bertrand claimed that "removing the symbols would reduce the book to just a few pages". His summary of Cournot's theory of duopoly has remained influential:
Cournot assumes that one of the proprietors will reduce his price to attract buyers to him, and that the other will in turn reduce his price even more to attract buyers back to him. They will only stop undercutting each other in this way, when either proprietor, even if the other abandoned the struggle, has nothing more to gain from reducing his price. One major objection to this is that there is no solution under this assumption, in that there is no limit to the downward movement... If Cournot's formulation conceals this obvious result, it is because he most inadvertently introduces as D and D' the two proprietors' respective outputs, and by considering them as independent variables, he assumes that should either proprietor change his output then the other proprietor's output could remain constant. It quite obviously could not.
Pareto was unimpressed by Bertrand's critique, concluding from it that Bertrand 'wrote his article without consulting the books he criticised'.
Irving Fisher
Irving Fisher (February 27, 1867 – April 29, 1947) was an American economist, statistician, inventor, eugenicist and progressive social campaigner. He was one of the earliest American neoclassical economists, though his later work on debt def ...
outlined a model of duopoly similar to the one Bertrand had accused Cournot of analysing incorrectly:
A more natural hypothesis, and one often tacitly adopted, is that each roducerassumes his rival's ''price'' will remain fixed, while his own price is adjusted. Under this hypothesis each would undersell the other as long as any profit remained, so that the final result would be identical with the result of unlimited competition.[, cited by .]
Fisher seemed to regard Bertrand as having been the first to present this model, and it has since entered the literature as
Bertrand competition 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 p ...
.
See also
*
Aggregative game In game theory, an aggregative game is a game in which every player’s payoff is a function of the player’s own strategy and the aggregate of all players’ strategies. The concept was first proposed by Nobel laureate Reinhard Selten in 1970 who ...
*
Bertrand competition 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 p ...
*
Bertrand–Edgeworth model
In microeconomics, the Bertrand–Edgeworth model of price-setting oligopoly looks at what happens when there is a homogeneous product (i.e. consumers want to buy from the cheapest seller) where there is a limit to the output of firms which are wil ...
*
Conjectural variation 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 accomp ...
*
Game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
*
Hotelling's linear city model
*
Nash equilibrium
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...
*
Stackelberg competition
*
Tacit collusion
Tacit collusion is a collusion between competitors, which do not explicitly exchange information and achieving an agreement about coordination of conduct. There are two types of tacit collusion - concerted action and conscious parallelism. In a ...
Notes
References
Further reading
*
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 o
by
Huw Dixon
Huw David Dixon (/hju: devəd dɪksən/), born 1958, is a British economist. He has been a professor at Cardiff Business School since 2006, having previously been Head of Economics at the University of York (2003–2006) after being a professor ...
.
*
Shumpeter, Joseph, ''History of Economic Analysis'' (1954). Discusses Cournot at length.
{{game theory
Economics models
Non-cooperative games
Competition (economics)
Oligopoly