Bertrand–Edgeworth model
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In
microeconomics Microeconomics is a branch of mainstream economics that studies the behavior of individuals and firms in making decisions regarding the allocation of scarce resources and the interactions among these individuals and firms. Microeconomics fo ...
, 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 willing and able to sell at a particular price. This differs from the Bertrand competition model where it is assumed that firms are willing and able to meet all demand. The limit to output can be considered as a physical capacity constraint which is the same at all prices (as in Edgeworth's work), or to vary with price under other assumptions.


History

Joseph Louis François 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 the ...
(1822–1900) developed the model of
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 ...
in oligopoly. This approach was based on the assumption that there are at least two firms producing a homogenous product with constant
marginal cost In economics, the marginal cost is the change in the total cost that arises when the quantity produced is incremented, the cost of producing additional quantity. In some contexts, it refers to an increment of one unit of output, and in others it ...
(this could be constant at some positive value, or with zero marginal cost as in Cournot). Consumers buy from the cheapest seller. The Bertrand–
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 ...
of this model is to have all (or at least two) firms setting the price equal to marginal cost. The argument is simple: if one firm sets a price above marginal cost then another firm can undercut it by a small amount (often called ''epsilon undercutting'', where epsilon represents an arbitrarily small amount) thus the equilibrium is zero (this is sometimes called the Bertrand paradox). The Bertrand approach assumes that firms are willing and able to supply all demand: there is no limit to the amount that they can produce or sell.
Francis Ysidro Edgeworth Francis Ysidro Edgeworth (8 February 1845 – 13 February 1926) was an Anglo-Irish philosopher and political economist who made significant contributions to the methods of statistics during the 1880s. From 1891 onward, he was appointed th ...
considered the case where there is a limit to what firms can sell (a capacity constraint): he showed that if there is a fixed limit to what firms can sell, then there may exist no pure-strategy
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 ...
(this is sometimes called the
Edgeworth paradox To solve the Bertrand paradox, the Irish economist Francis Ysidro Edgeworth put forward the Edgeworth Paradox in his paper "The Pure Theory of Monopoly", published in 1897. In economics, the Edgeworth paradox describes a situation in which two pla ...
).
Martin Shubik Martin Shubik (1926-2018) was an American mathematical economist who specialized in game theory, defense analysis, and the theory of money and financial institutions. The latter was his main research interest and he coined the term "mathematical ...
developed the Bertrand–Edgeworth model to allow for the firm to be willing to supply only up to its profit maximizing output at the price which it set (under
profit maximization In economics, profit maximization is the short run or long run process by which a firm may determine the price, input and output levels that will lead to the highest possible total profit (or just profit in short). In neoclassical economics, ...
this occurs when marginal cost equals price). He considered the case of strictly convex costs, where marginal cost is increasing in output. Shubik showed that if a Nash equilibrium exists, it must be the perfectly competitive price (where demand equals supply, and all firms set price equal to marginal cost). However, this can only happen if market demand is infinitely elastic (horizontal) at the competitive price. In general, as in the Edgeworth paradox, no pure-strategy Nash equilibrium will exist. Huw Dixon showed that in general a
mixed strategy In game theory, a player's strategy is any of the options which they choose in a setting where the outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the action of a player in a game ...
Nash equilibrium will exist when there are
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
costs. Dixon’s proof used the Existence Theorem of
Partha Dasgupta Sir Partha Sarathi Dasgupta (born on 17 November 1942), is an Indian-British economist who is the Frank Ramsey Professor Emeritus of Economics at the University of Cambridge, United Kingdom and Fellow of St John's College, Cambridge. Personal ...
and
Eric Maskin Eric Stark Maskin (born December 12, 1950) is an American economist and mathematician. He was jointly awarded the 2007 Nobel Memorial Prize in Economic Sciences with Leonid Hurwicz and Roger Myerson "for having laid the foundations of mechanism ...
. Under Dixon's assumption of (weakly) convex costs, marginal cost will be non-decreasing. This is consistent with a cost function where marginal cost is flat for a range of outputs, marginal cost is smoothly increasing, or indeed where there is a kink in total cost so that marginal cost makes a discontinuous jump upwards.


Later developments and related models

There have been several responses to the non-existence of pure-strategy equilibrium identified by
Francis Ysidro Edgeworth Francis Ysidro Edgeworth (8 February 1845 – 13 February 1926) was an Anglo-Irish philosopher and political economist who made significant contributions to the methods of statistics during the 1880s. From 1891 onward, he was appointed th ...
and
Martin Shubik Martin Shubik (1926-2018) was an American mathematical economist who specialized in game theory, defense analysis, and the theory of money and financial institutions. The latter was his main research interest and he coined the term "mathematical ...
. Whilst the existence of mixed-strategy equilibrium was demonstrated by Huw Dixon, it has not proven easy to characterize what the equilibrium actually looks like. However, Allen and Hellwig were able to show that in a large market with many firms, the average price set would tend to the competitive price. It has been argued that non-pure strategies are not plausible in the context of the Bertrand–Edgworth model. Alternative approaches have included: *Firms choose the quantity they are willing to sell up to at each price. This is a game in which price and quantity are chosen: as shown by Allen and Hellwig and in a more general case by Huw Dixon that the perfectly competitive price is the unique pure-strategy equilibrium. *Firms have to meet all demand at the price they set as proposed by Krishnendu Ghosh Dastidar or pay some cost for turning away customers. Whilst this can ensure the existence of a pure-strategy Nash equilibrium, it comes at the cost of generating multiple equilibria. However, as shown by Huw Dixon, if the cost of turning customers away is sufficiently small, then any pure-strategy equilibria that exist will be close to the competitive equilibrium. *Introducing
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 co ...
, as proposed by Jean-Pascal Benassy. This is more of a synthesis of
monopolistic competition Monopolistic competition is a type of imperfect competition such that there are many producers competing against each other, but selling products that are differentiated from one another (e.g. by branding or quality) and hence are not perfec ...
with the Bertrand–Edgeworth model, but Benassy showed that if the
elasticity of demand A good's price elasticity of demand (E_d, PED) is a measure of how sensitive the quantity demanded is to its price. When the price rises, quantity demanded falls for almost any good, but it falls more for some than for others. The price elastici ...
for the firms output is sufficiently high, then any
pure strategy In game theory, a player's strategy is any of the options which they choose in a setting where the outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the action of a player in a game ...
equilibrium that existed would be close to the competitive outcome. *"Integer pricing" as explored by Huw Dixon. Rather than treat price as a
continuous variable In mathematics and statistics, a quantitative variable may be continuous or discrete if they are typically obtained by ''measuring'' or '' counting'', respectively. If it can take on two particular real values such that it can also take on all ...
, it is treated as a discrete variable. This means that firms cannot undercut each other by an arbitrarily small amount, one of the necessary ingredients giving rise to the non-existence of a pure strategy equilibrium. This can give rise to multiple pure-strategy equilibria, some of which may be distant from the competitive equilibrium price. More recently, Prabal Roy Chowdhury has combined the notion of discrete pricing with the idea that firms choose prices and the quantities they want to sell at that price as in Allen–Hellwig. * Epsilon equilibrium in the pure-strategy game. In an epsilon equilibrium, each firm is within epsilon of its optimal price. If the epsilon is small, this might be seen as a plausible equilibrium, due perhaps to
menu cost In economics, the menu cost is a cost that a firm incurs due to changing its prices. It is one microeconomic explanation of the price-stickiness of the macroeconomy put by New Keynesian economists. The term originated from the cost when restauran ...
s or
bounded rationality Bounded rationality is the idea that rationality is limited when individuals make decisions, and under these limitations, rational individuals will select a decision that is satisfactory rather than optimal. Limitations include the difficulty o ...
. For a given \varepsilon>0, if there are enough firms, then an epsilon-equilibrium exists (this result depends on how one models the residual demand – the demand faced by higher-priced firms given the sales of the lower-priced firms).


References


Resources


Edgeworth and modern oligopoly, Theory Xavier VivesThe Pure Theory of Monopoly, Francis Edgeworth
* * * * {{DEFAULTSORT:Bertrand-Edgeworth model Economics models Competition (economics) Game theory Oligopoly