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 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

There have been several responses to the non-existence of pure-strategy equilibrium identified by Francis Ysidro Edgeworth and Martin Shubik. 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^[5] 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^[6] and in a more general case by Huw Dixon^[7] 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^[8] or pay some cost for turning away customers.^[9] 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
  pure strategy
  equilibrium that existed would be close to the competitive outcome.
• "Integer pricing" as explored by
  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^[12]
  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.^[13] 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 costs or bounded rationality
  . For a given ${\displaystyle \varepsilon >0}$, 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).